Morfismos, Vol 4, No 2, 2000

VOLUMEN 4 NÚMERO 2 JULIO A DICIEMBRE DE 2000 ISSN: 1870-6525

MORFISMOS Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav Editores Responsables • Isidoro Gitler • Jes´ us Gonz´alez • Isa´ıas L´opez

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VOLUMEN 4 NÚMERO 2 JULIO A DICIEMBRE DE 2000 ISSN: 1870-6525

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Morfismos

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Morfismos

Contenido

Algebraic K-theory and the η-invariant Jos´e Luis Cisneros Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Average optimal strategies in Markov games under a geometric drift condition Heinz-Uwe K¨ uenle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Little cubes and homotopy theory Dai Tamaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Hipergrupos y ´ algebras de Bose-Msner Isa´ıas L´ opez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Sincronizaci´ on de parejas de aut´ omatas celulares J. Guillermo S´ anchez Saint-Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Morfismos, Vol. 4, No. 2, 2000, pp. 1–14

Algebraic K-theory and the η-invariant Jos´e Luis Cisneros Molina

∗

1

Abstract The aim of this paper is to present the main results of J. D. S. Jones and B. W. Westbury on algebraic K-Theory, homology spheres and the η-invariant [6], giving the basic definitions and prerequisites to understand them.

1991 Mathematics Subject Classification: 18F25, 19D06, 53C27, 58J28.

Keywords and phrases: Algebraic K-theory, Quillen’s +-construction, spin geometry, Dirac operator, η-invariant.

1

Introduction

In [6] J. D. S. Jones and B. W. Westbury constructed elements in K3 (C), the 3rd algebraic K-theory group of the field of complex numbers, using homology 3-spheres endowed with a representation of their fundamental group. They also computed the image of such elements under the regulator map, using the η-invariant. The aim of this paper is to present the main results of J. D. S. Jones [6], giving the basic definitions and prerequisites to understand them.

The paper is divided in four parts. In section 2 we define the algebraic K-groups of a ring using Quillen’s +-construction. We also explain how homology n-spheres equipped with a representation of its fundamental group in the general linear group over a ring R define elements in the K-group Kn (R). In section 3 we give the definition of the η-invariant of a self-adjoint elliptic operator on a closed manifold and its variations.

In section 4 we describe the Dirac operator which is a very important example of this kind of operators and the one which we are interested ∗ 1

Invited article Supported by a scholarship from DGAPA, UNAM.

1

2

´ LUIS CISNEROS MOLINA JOSE

in. Finally in section 5 the main results by Jones and Westbury are presented.

2

Algebraic K-Theory

In this section we define the algebraic K-groups and we describe how to construct elements in this groups using homology spheres equipped with a representation of its fundamental group.

2.1

Classifying space of a group

Any discrete group G has a classifying space BG which is a pointed space (i.e. it has a base point ∗) unique up to homotopy equivalence such that: π1 (BG) = G and πi (BG) = 0 for i ̸= 1 i.e. BG is an EilenbergMac Lane space K(G, 1).

From its definition, the universal covering of BG, denoted by EG is contractible. The covering EG → BG is called the universal bundle for G and the space BG satisfies the following universal property: If EG → BG is a universal bundle for G and X is of the homotopy type of a CW-complex with base point x0 (e.g. manifold). Then we have the following one-to-one correspondences [X, BG] ←→ Hom(π1 (X, x0 ), G) ←→ FG (X) where [X, BG] denotes the homotopy classes of maps from X to BG, Hom(π1 (X, x0 ), G) denotes the homomorphisms from π1 (X, x0 ) to G and FG (X) the equivalence classes of principal (flat) G-bundles over X. Note that in the case when G = GLN (C), Hom(π1 (X, x0 ), G) is precisely the set of representations of π1 (X, x0 ) on CN .

2.2

Quillen’s +-construction

In order to define the algebraic K-theory groups of a ring R, we need the +-construction due to Daniel Quillen in the early 1970’s, for which, among other reasons, he was awarded the Fields Medal in 1978.

Theorem 2.2.1 (Quillen). Let X be a connected CW-complex with base point x0 . Let A ⊂ π1 (X) be a perfect normal subgroup (i.e. A = [A, A] and A = [π1 (X), A], where [ , ] is the commutator). Then there is a space X + (depending on A) and a map i : X → X + such that:

ALGEBRAIC K-THEORY AND THE η-INVARIANT

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(a) The map i induces an isomorphism i : π1 (X)/A → π1 (X + ).

(b) For any π1 (X + )-module L one has ∼ =

i∗ : H∗ (X, i∗ L) → H∗ (X + , L).

(c) The pair (X + , i) is determined by a) and b) up to homotopy equivalence.

Let R be a ring with 1. Consider the group GLN (R) of invertible N × N matrices over R. The elementary group EN (R) is the subgroup of GLN (R) generated by the elementary matrices (see [12, 11, 9] for definition).

We have inclusions GLN (R) ⊂ GLN +1 (R) which restrict to inclusions EN (R) ⊂ EN +1 (R) and we can define GL(R) =

!

GLN (R)

N

E(R) =

!

EN (R).

N

Let X = BGL(R). Then π1 (X) = GL(R) and A = E(R) is perfect. Then applying the +-construction we get BGL(R)+ . Define the algebraic K-groups of the ring R by Kn (R) = πn (BGL(R)+ )

for n ≥ 1.

This definition may seem artificial, the reason is because originally the first three groups K0 (R), K1 (R) and K2 (R) were given by algebraic definitions2 and for a while seemed to be no good way to define the “higher K-functors” Ki , i ≥ 3, until Quillen’s work appeared, for a nice account of this facts see [12].

2 In the present definition we are not including K0 (R), in this case is called the “reduced” algebraic K-theory of R

´ LUIS CISNEROS MOLINA JOSE

4

2.3

Homology spheres

It is well known that the homology of the n-sphere S n is given by !

Z q = 0, n n Hq (S ) = 0 q ̸= 0, n.

A homology n-sphere as its name indicates it, is a path-connected space (say with the homotopy type of a CW-complex) with the same homology groups as S n (n ≥ 3).

Let Σ be a homology n-sphere, since 0 = H1 (Σ, Z) = π1 (Σ)/[π1 (Σ), π1 (Σ)] π1 (Σ) can have no abelian quotients and so is perfect. Given a representation α : π1 (X) → GLN (R), let f : Σ → BGLN (R) be the map which induces α on π1 (by the universal property of classifying spaces). Composing this map with the inclusion BGLN (R) → BGL(R) and applying Quillen’s +-construction we get S n ! Σ+ → BGL(R)+ , since the +-construction is functorial by its universal properties. Here ! denotes homotopy equivalence. The homotopy class of this map gives us the element in K-theory [Σ, α] ∈ Kn (R) = πn (BGL(R)+ ).

2.4

The regulator

There is a homomorphism e : K2n+1 (C) → C/Z called the regulator map which satisfies the following properties (i) It is an isomorphism on K1 (C) ∼ = C∗ → C/Z. (ii) The homomorphism e gives an isomorphism of the torsion subgroup of K2n+1 (C) with Q/Z. The aim now is to compute the image of the elements [Σ, α] ∈ K3 (C) under the regulator map. One way to do this is using the η-invariant.

ALGEBRAIC K-THEORY AND THE η-INVARIANT

3

5

The η-invariant

Let X be a closed (compact without boundary) Riemannian manifold and let E be a smooth vector bundle over X with an inner product.

We denote by C ∞ (X, E) the space of smooth sections of E and we can endow it with an inner product ⟨ , ⟩ using the inner product on E and integration. Let A : C ∞ (X, E) → C ∞ (X, E) be an elliptic differential operator and assume that A is self-adjoint, that is ⟨s1 , As2 ⟩ = ⟨As1 , s2 ⟩ for every s1 , s2 ∈ C ∞ (X, E). Then A has a discrete spectrum with real eigenvalues {λ} and we define the η-series of A by η(s; A) =

! (sign λ)|λ|−s

λ̸=0

where the sum is taken over the non-zero eigenvalues of A. This series converges for ℜ(s) sufficiently large. By results of Seeley [13] extends by analytic continuation to a meromorphic function on the whole s-plane and is finite at s = 0.

The number η(0; A) is called the η-invariant of A and is a spectral invariant which measures the asymmetry of the spectrum of A. We also define a refinement of the η-series which takes into account the zero eigenvalues of A ξ(s; A) =

h + η(s; A) 2

where h is the dimension of the kernel of A or in other words, the multiplicity of the 0-eigenvalue of A. Now consider a representation α : π1 (X) → GLN (C). Then α defines ˜ be the universal a flat bundle Vα over X in the following way. Let X ˜ ×π (X) CN i.e. Vα is X ˜ × CN modulo the cover of X. Then Vα = X 1 action of π1 (X), where π1 (X) acts on the first factor with the canonical action of π1 (X) on the universal cover and via the representation α on the second factor. The bundle Vα also has a canonical flat connection ∇α given by the exterior derivative as follows.

A connection is a first order linear differential operator ∇α

C ∞ (X, Vα ) −−→ Ω1 (X, Vα )

´ LUIS CISNEROS MOLINA JOSE

6

which satisfies the Leibnitz rule ∇α f s = df ⊗ s + f ⊗ ∇α s for every f ∈ C ∞ (X, R) and every s ∈ C ∞ (X, Vα ).

By the previous construction of the bundle Vα we have that C ∞ (X, Vα ) ∼ = ∞ ˜ CN )α and Ω1 (X, Vα ) ∼ ˜ CN )α , where the spaces C ∞ (X, ˜ CN ) α C (X, = Ω1 (X, ˜ CN )α are, respectively, the sections and 1-forms which are and Ω1 (X, equivariant under the action of π1 (X) via the representation α. On the other hand, the exterior derivative d ˜ CN ) → ˜ CN ) C ∞ (X, Ω1 (X,

sends invariant sections to invariant 1-forms. Hence the connection ∇α is given by α

=d ˜ CN )α −∇ ˜ CN )α ∼ C ∞ (X, Vα ) ∼ −−−→ Ω1 (X, = C ∞ (X, = Ω1 (X, Vα ).

Using this connection we can couple the operator A to Vα to get an operator Aα : C ∞ (X, E ⊗ Vα ) → C ∞ (X, E ⊗ Vα ) and as before we define the functions3 η(s; α, A) = η(s; Aα ),

ξ(s; α, A) = ξ(s; Aα )

and their reduced forms η˜(s; α, A) = η(s; α, A) − N η(s; A),

˜ α, A) = ξ(s; α, A) − N ξ(s; A) ξ(s;

where N is the dimension of the representation α.

Once more, following [2, Section 2] we can see that the functions ˜ α, A) are finite at s = 0 and if we reduce modulo Z η˜(s; α, A) and ξ(s; then η˜(α, A) = η˜(0; α, A) ∈ C/Z, 3

˜ A) = ξ(0; ˜ α, A) ∈ C/Z ξ(α,

The operator Aα is not self-adjoint any more, unless the representation α is unitary. Nonetheless, Aα has self-adjoint symbol and that allows us to define the η and ξ functions, see [2, p. 90].

ALGEBRAIC K-THEORY AND THE η-INVARIANT

7

are homotopy invariants of A. The reason for regarding values in C/Z and not just in C is that if we vary A continuously the dimension of ker A is not a continuous function of A. However the jumps of ξ(s; A) are due to eigenvalues changing sign as they cross zero and therefore the jumps are integer jumps.

Note that if we fix the manifold X and the operator A, the invariant ˜ A) only depends on the representation α of the fundamental group ξ(α, of X or equivalently on the flat bundle Vα aver X.

4

The Dirac operator

In this section we describe a particular example of a self-adjoint elliptic differential operator called the Dirac operator which is the one we shall use to compute C/Z-valued invariants of elements of the K-groups of any subring of C. The Dirac operator is very important by itself and plays a central role in the Atiyah-Singer Index Theorem, in the Seiberg-Witten theory and many other things. The main references for the material in this section are [7, 1].

4.1

Clifford algebras

Let V be a finite dimensional real vector space with a non-degenerate, symmetric bilinear form q : V ⊗ V → R. Let {e1 , . . . , en } be an orthogonal basis for V then the Clifford algebra Cl(V, q) is the algebra over R, with unit, generated by the ei , subject to the relations e2i = −q(ei , ei )

ei ej = −ej ei

i ̸= j.

For the special case when V = Rn and q is the standard inner product we denote the algebra Cl(V, q) by Cln and its complexification

by ClnC = Cln ⊗ C. Example 4.1.1.

Cl0 = R

with basis

1

Cl1 = C

with basis

1, e1

Cl2 = H

with basis

1, e1 , e2 , e1 e2

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´ LUIS CISNEROS MOLINA JOSE

The group Spin(n) is defined as a subgroup of the group of units of Cln and it is the non-trivial double covering of SO(n) and for n > 2 it is its universal covering.

Now lets restrict ourselves to odd dimensional vector spaces, in this case, the complexified Clifford algebra ClnC has two inequivalent irreducible complex representations and when they are restricted to Spin(n) they give isomorphic irreducible complex representations of Spin(n). We denote such a representation space by S.

4.2

Spin structures

Let X be an odd dimensional oriented closed Riemannian manifold. The Riemannian metric and the orientation give a reduction of the structure group of the tangent bundle T X of X to SO(n). A spin structure on X is a lift of the structure group SO(n) of T X to Spin(n).

A spin structure on X provide us with a principal Spin(n)-bundle Q which is a double cover of the principal SO(n)-bundle P associated to the tangent bundle T X. The restriction to the fibre of this double cover ϖ : Q → P is the double covering Spin(n) → SO(n).

Now consider the spin representation S of Spin(n) and let S(X) = Q ×Spin(n) S be the vector bundle over X associated to the principal Spin(n)-bundle Q. The bundle S(X) is called the spinor bundle of X and its sections are called spinor fields. We denote the space of spinor fields by C ∞ (X, S(X)).

Let Cl(T ∗ X) be the bundle over X whose fibre at x is Cl(Tx∗ X), the Clifford algebra of the cotangent space at x with the inner product given by the Riemannian metric.

There is a pairing C : Cl(T ∗ X) ⊗ S(X) → S(X) which is called Clifford multiplication.

T ∗ X → Cl(T ∗ X) then we get a pairing

If we consider the inclusion

T ∗ X ⊗ S(X) → S(X).

4.3

The Dirac operator

The Riemannian structure of X provides us with the Riemannian connection on the tangent bundle. This connection can be seen as a 1-form

ALGEBRAIC K-THEORY AND THE η-INVARIANT

9

β on the principal SO(n)-bundle P with values in the Lie algebra so(n).

Since Spin(n) and SO(n) have the same Lie algebra, the double covering ϖ : Q → P given by the spin structure on X gives us a 1-form ϖ∗ (β) which defines a connection on Q called the spin connection. This connection induces a covariant derivative ∇ : C ∞ (X, S(X)) → C ∞ (X, T ∗ X ⊗ S(X)) on spinor fields.

Composing ∇ with Clifford multiplication C : C ∞ (X, T ∗ X ⊗ S(X)) → C ∞ (X, S(X)) we obtain the Dirac operator D = C ◦ ∇ : C ∞ (X, S(X)) → C ∞ (X, S(X)).

It is a self-adjoint, first order, elliptic partial differential operator.

As in the previous section, a representation α : π1 (X) → GLN (C) defines a bundle Vα with a flat connection ∇α . In this case we can define the twisted Dirac operator Dα by the composition C ∞ (X, S(X) ⊗ Vα )

∇⊗Id+Id⊗∇α

−→

C⊗Id

−→

C ∞ (X, T ∗ X ⊗ S(X) ⊗ Vα ) C ∞ (X, S(X) ⊗ Vα )

where ∇ ⊗ ∇α is the product connection on the bundle S(X) ⊗ Vα and Id is the identity map.

5

The results of Jones and Westbury

The relation between the value of the regulator map on the classes [Σ, α] ∈ K3 (C) and the η-invariant of the Dirac operator of the homology sphere Σ is given by the following theorem: Theorem 5.1.1 (Jones–Westbury). ˜ D) e([Σ, α]) = ξ(α, where D is the Dirac operator on Σ.

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In [6] Jones and Westbury give a formula to compute e[Σ, α] when Σ is a Seifert homology sphere. Let (a1 , . . . , an ) be an n-tuple of pairwise coprime integers. The Seifert homology 3-sphere Σ(a1 , . . . , an ) is a 3manifold which admits an action of the circle S 1 which is free except for n exceptional orbits which have isotropy groups Ca1 , . . . , Can where Cm ⊂ S 1 is the cyclic subgroup of order m embedded in S 1 as the mth roots of unity.

In order to give the aforementioned formula we need to know a bit about the fundamental group of Σ(a1 , . . . , an ). Let T (a1 , . . . , an ) be the generalised triangle group which is defined by the following generators and relations T (a1 , . . . , an ) = ⟨x1 , . . . , xn | xa11 = · · · = xann = x1 . . . xn = 1⟩.

This group is perfect and it has a universal central extension T˜(a1 , . . . , an ) which fits into an exact sequence 1 → C∗ → T˜(a1 , . . . , an ) → T (a1 , . . . , an ) → 1 where C∗ is an infinite cyclic group, except for the case of T (2, 3, 5) where C∗ ∼ = Z2 .

In terms of generators and relations T˜(a1 , . . . , an ) = ⟨h, x1 , . . . , xn | [xi , h] = 1, xa11 = h−b1 , . . . , xann = h−bn , x1 . . . xn = h−b0 ⟩ where h is the generator of the centre of T˜(a1 , . . . , an ).

The bi satisfy the relation ! " b1 bn + ··· + =1 a1 . . . an −b0 + a1 an and we have that π1 (Σ(a1 , . . . , an )) = T˜(a1 , . . . , an ).

Let α : π1 (Σ(a1 , . . . , an )) → GLN (C) be a representation, since the group π1 (Σ(a1 , . . . , an )) is perfect every complex representation α must have image in SLN (C). We shall consider only those representations in which the central element h acts as a scalar multiple of the identity, for instance, that is the case when α is irreducible and in general for any decomposable representation.

ALGEBRAIC K-THEORY AND THE η-INVARIANT

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Suppose α(h) = λh I where λh is a scalar, then, since α(h) ∈ SLN (C) rh λh = ζN

is a N th root of unity. Here ζd = e2πi/d ∈ C is the standard primitive dth root of unity. Now consider the matrices α(xj ),

j = 1, . . . , n.

a

In view of the relations xj j = h−bj the eigenvalues λ1 (j), . . . , λN (j) satisfy the equation −b

λk (j)aj = λh j .

There are aj roots of this equation and we define sk (j) by N s (j)−bj rh

λk (j) = ζN ajk

.

We refer to the numbers sk (j),

1 ≤ j ≤ n, 1 ≤ k ≤ N

as the type of the representation α.

Now we have Theorem 5.1.2 (Jones-Westbury).

Let α : π1 (Σ(a1 , . . . , an )) → SLN (C) be a representation of the fundamental group of the Seifert homology sphere Σ(a1 , . . . , an ) in which the central element h acts as a scalar multiple of the identity. Let sk (j),

1 ≤ j ≤ n, 1 ≤ k ≤ N

be the type of the representation α; then 2N ℜ(e[Σ(a1 , . . . , an ), α]) = −

N ! n !

N ! a(sk (j) − sl (j))2 j=1 k=1 l=1

2a2j

where a = a1 . . . an . ˜ D) This formula was obtained using the fact that the invariants ξ(α, are cobordism invariants, so it is enough to compute them on a simpler manifold which is cobordant to the Seifert homology sphere (see [6]). The cobordism invariance follows from the index theorem for flat bundles in [2].

Using the previous theorem they also prove the following results

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12

Theorem 5.1.3 (Jones-Westbury). Every element in K3 (C) of finite order is of the form [Σ(p, q, r), α] for some representation α : π1 (Σ(p, q, r)) → SL2 (C).

Now let Z[ζd ] be the ring of algebraic numbers in the cyclotomic field Q(ζd ). Then combining the results of Borel [3], Merkurjev and Suslin [10] and Levine [8] we have that K3 (Z[ζd ]) = Z/w2 (d) ⊕ Zr2 where w2 (d) = lcm(24, 2d) and r2 is the number of complex places of Q(ζd ). In particular note that if (6, d) = 1 the torsion subgroup of K3 (Z[ζd ]) is exactly Z/24d.

Theorem 5.1.4 (Jones-Westbury).

If (6, d) = 1 there exists a representation α : π1 (Σ(2, 3, d)) → SL2 (Z[ζd ]) such that the element [Σ(2, 3, d), α] ∈ K3 (Z[ζd ]) is a generator of the torsion subgroup.

Example 5.1.5. The Seifert homology sphere P = Σ(2, 3, 5) is called the Poincar´e 3-sphere. Its fundamental group, known as the binary icosahedral group, is a subgroup of SU(2) and the matrices which occur in this subgroup can all be chosen to have coefficients in the ring Z[ζ5 ]. This gives a representation α of π1 (P ) in SL2 (Z[ζ5 ]), and using theorem 5.1.2 we get e[P, α] =

1 .

120

From this we deduce that the generator of the torsion subgroup of K3 (Z[ζ5 ]) is given by [P, α] where α is the natural representation of π1 (P ).

6

Further research and progress

˜ D) directly from its definition, without One could try to compute ξ(α, using the fact that it is a cobordism invariant and expect an improved formula which works for all the representations of π1 (Σ) and which also gives the imaginary part. I established a first step in this direction in ˜ D) directly from its definition for the Poincar´e [4, 5] computing ξ(α,

ALGEBRAIC K-THEORY AND THE η-INVARIANT

13

sphere, which is the only homology 3-sphere with finite fundamental group. The method not only works for the Poincar´e sphere but for any quotient S 3 /Γ of the 3-sphere by a finite subgroup Γ (3-dimensional spherical space forms), so we get a formula to compute the η-invariant of the Dirac operator of S 3 /Γ twisted by any representation of Γ, where Γ is any finite subgroup of S 3 .

Acknowledgement I would like to thank Professor John Jones for his comments on this work.

Jos´e Luis Cisneros Molina Instituto de Matem´ aticas, UNAM, Unidad Cuernavaca, Apdo. Postal #273-3, Adm. de Correos #3, C.P. 62251, Cuernavaca, Morelos, Mexico. jlcm@matcuer.unam.mx

References [1] M. F. Atiyah, R. Bott, and A. Shapiro. Clifford modules. Topology, 3, Suppl. 1, (1964), 3–38.

[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and riemannian geometry III. Mathematics Proceedings of the Cambridge Philosophycal Society, 79 (1976), 71–99.

[3] A. Borel. Stable real cohomology of arithmetic groups. Annales ´ Scientifiques de l’Ecole Normale Sup´erieure, 7 (1974), 235–272.

[4] J. L. Cisneros Molina. The η-invariant of twisted Dirac operators of S 3 /Γ. To appear in Geometriae Dedicata, (2000).

[5] J. L. Cisneros-Molina. The regulator, the Bloch group, hyperbolic manifolds, and the η-invariant. PhD thesis, University of Warwick, Coventry, England, September (1998).

[6] J. D. S. Jones and B. W. Westbury. Algebraic K-theory, homology spheres, and the η-invariant. Topology, 34(4) (1994), 929–957.

[7] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn. Spin Geometry. Princeton University Press, Princeton, New Jersey, (1989).

14

´ LUIS CISNEROS MOLINA JOSE

[8] M. Levine. The indecomposable K3 of fields. Annales Scien´ tifiques de l’Ecole Normale Sup´erieure, 22 (1989), 255–344.

[9] E. Lluis-Puebla. Algebra homol´ ogica, Cohomolog´ıa de Grupos y K-Teor´ıa Algebraica Cl´ asica. Addison-Wesley Iberoamericana, (1990).

[10] A. S. Merkurjev and A. A. Suslin. On K3 of a field. LOMI preprint M-2-07, (1987).

[11] J. W. Milnor. Introducton to Algebraic K-Theory. Study 72.

Princeton University Press, Princeton, New Jersey, (1971).

[12] J. Rosenberg. Algebraic K-Theory and Its Applications. Graduate Texts in Mathematics 147. Springer-Verlag, (1994).

[13] R. T. Seeley. Complex powers of an elliptic operator. In Singular Integrals, Proc. Symp. Pure Math., volume 10, pages 288– 307. American Mathematical Society, Providence, Rhode Island, U.S.A., (1967).

Morfismos, Vol. 4, No. 2, 2000, pp. 15–31

Average optimal strategies in Markov games under a geometric drift condition ∗ Heinz-Uwe Ku ¨enle

Abstract Zero-sum stochastic games with the expected average cost criterion and unbounded stage cost are studied. The state space is an arbitrary Borel set in a complete separable metric space but the action sets are finite. It is assumed that the transition probabilities of the Markov chains induced by stationary strategies satisfy a certain geometric drift condition. It is shown that the average optimality equation has a solution and that both players have optimal stationary strategies.

1991 Mathematics Subject Classification: 91A15 Keywords and phrases: Markov games, Borel state space, average cost criterion, geometric drift condition, unbounded costs

1

Introduction

In this paper two-person stochastic games with the expected average cost criterion are studied. The state space is a standard Borel space, that is, an arbitrary Borel set in a complete separable metric space.

The action sets of both players are finite. Such a stochastic game can be described in the following way: The state xn of a dynamic system is periodically observed at times n = 1, 2, . . .. After an observation at time n the first player chooses an action an from the action set A(xn ) and afterwards the second player chooses an action bn from the action set B(xn ) dependent on the complete history of the system at this time.

The first player must pay cost k 1 (xn , an , bn ), the second player must pay ∗

Invited article

15

16

¨ HEINZ-UWE KUENLE

k 2 (xn , an , bn ), and the system moves to a new state xn+1 in the state space X according to the transition probability p(· | xn , an , bn ).

Stochastic games with Borel state space and average cost criterion are considered by several authors. Related results are given by Maitra and Sudderth [7], [8], [9], Nowak [13], Rieder [15] and K¨ uenle [6] in the case of bounded costs (payoffs). The case of unbounded payoffs is treated by Nowak [14] and K¨ uenle [4]. The assumptions in this paper concerning the transition probabilities are related to Nowak’s assumptions: Nowak assumes that there is a Borel set C ∈ X and for every stationary strategy pair (π ∞ , ρ∞ ) a measure µ such that C is µ-small with respect to the Markov chain induced by this strategy pair. We assume that C is only a µ-petite set with respect to a resolvent of this Markov chain; as against this, we demand that µ is independent of the corresponding strategy pair. (For the definition of ”small sets” and ”petite sets” see [10].)

The paper is organised as follows: In Section 2 the mathematical model of Markov games is presented. Section 3 contains the assumptions on the transition probabilities and on the stage costs, and also some preliminary results. In Section 4 we study the expected average cost of a fixed stationary strategy pair. We show that the Poisson equation has a solution. In Section 5 we prove that the average cost optimality equation has a solution and both players have optimal stationary strategies.

2

The Mathematical Model

In this section we introduce the mathematical model of the stochastic game considered in this paper.

Definition 2.1 M = ((X, σX ), (A, σA ), A, (B, σB ), B, p, k 1 , k 2 , E, F) is called a Markov game if the elements of this tuple have the following meaning: — (X, σX ) is a standard Borel space, called the state space.

— A is a countable set and σA is the power set of A. A(x) ∈ A denotes a finite set of actions of the first player for every x ∈ X. A is called the action space of the first player and A(x) is called the admissible action set of the first player at state x ∈ X.

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

17

— B is a countable set and σB is the power set of B. B(x) ∈ B denotes a finite set of actions of the second player for every x ∈ X. B is called the action space of the second player and B(x) is called the admissible action set of the second player at state x ∈ X. — p is a transition probability from σX×A×B to σX , the transition law.

— k i , i = 1, 2 , are σX×A×B -measurable functions, called stage cost functions.

— Let Hn = (X × A × B)n × X for n ≥ 1, H0 = X. h ∈ Hn is called the history at time n.

A transition probability πn from σHn to σA with πn (A(xn ) | x0 , a0 , b0 , . . . , xn ) = 1 for all (x0 , a0 , b0 , . . . , xn ) ∈ Hn is called a decision rule of the first player at time n.

A transition probability ρn from σHn ×A to σB with ρn (B(xn ) | x0 , a0 , b0 , . . . , xn ) = 1 for all (x0 , a0 , b0 , . . . , xn ) ∈ Hn is called a decision rule of the second player at time n.

A decision rule of the first [second] player is called Markov iff a transition probability π ˜n from σHn to σA [˜ ρn from σHn to σB ] exists such that πn (· | x0 , a0 , b0 , . . . , xn ) = π ˜n (· | xn ) [ ρn (· | x0 , a0 , b0 , . . . , xn ) = ρ˜n (· | xn )] for all (x0 , a0 , b0 , . . . , xn ) ∈ Hn × A. (Notation: We identify πn as π ˜n and ρn as ρ˜n .)

E and F denote non-empty sets of Markov decision rules.

A decision rule of the first [second] player is called deterministic if a function en : Hn → A [fn : Hn → B] exists such that πn (en (hn ) | hn ) = 1 for all hn ∈ Hn [ρn (fn (hn ) | hn ) = 1 for all (hn ) ∈ Hn ].

A sequence Π = (πn ) or P = (ρn ) of decision rules of the first or second player is called a strategy of that player.

Strategies are called deterministic, or Markov iff all their decision rules have the corresponding property.

A Markov strategy Π = (πn ) or P = (ρn ) is called stationary iff π0 = π1 = π2 = . . . or ρ0 = ρ1 = ρ2 = . . .. (Notation: Π = π ∞ or P = ρ∞ .) We assume in this paper that the sets of all admissible strategies are E∞ and F∞ . Hence, only Markov strategies are allowed.

But by means of dynamic programming methods it is also possible to get corresponding results for Markov games with larger sets of admissible strategies. If E and F are the sets of all Markov decision rules

18

¨ HEINZ-UWE KUENLE

(in the above sense) then we have a Markov game with perfect (or complete) information. In this case the action set of the second player may depend also on the present action of the first player. If E is the set of all Markov decision rules but F is the set of all Markov decision rules which do not depend on the present action of the first player then we have a usual Markov game with independent ! action choice. Let i,N i Ω := X × A × B × X × A × B × . . . and K (ω) := N j=0 k (xj , aj , bj ) for ω = (x0 , a0 , b0 , x1 , . . . ) ∈ Ω, i = 1, 2, N ∈ N. By means of the Ionescu-Tulcea Theorem (see, for instance, [11]), it follows that there exists a suitable σ-algebra F in Ω and for every initial state x ∈ X and strategy pair (Π, P ), Π = (πn ), P = (ρn ), a unique probability measure Px,Π,P on F according to the transition probabilities πn , ρn and p.

Furthermore, K i,N is F-measurable for all i = 1, 2, N ∈ N. We set " i,N K i,N (ω)Px,Π,P (dω) (2.1) VΠP (x) = Ω

and ΦiΠP (x) = lim inf N →∞

1 V i,N (x) N + 1 ΠP

(2.2)

if the corresponding integrals exist.

Definition 2.2 A strategy pair (Π∗ , P ∗ ) is called a Nash equilibrium pair iff Φ1Π∗ P ∗ ≤ Φ1ΠP ∗ Φ2Π∗ P ∗ ≤ Φ2Π∗ P for all strategy pairs (Π, P ).

In this paper we will consider especially zero-sum Markov games, that means k 1 = −k 2 . In this case we call a Nash equilibrium pair also an N := V 1,N , Φ 1 optimal strategy pair. We set k := k 1 , VΠP ΠP := ΦΠP . ΠP

3

Assumptions and Preliminary Results

In this paper we use the same notation for a substochastic kernel and for the ”expectation operator” with respect to this kernel, that means: If (Y, σY ) and (Z, σZ ) are standard Borel spaces, v : Y × Z → R a

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

19

σY×Z -measurable function, and q a substochastic kernel from (Y, σY ) to (Z, σZ ) then we put ! q(dz | y)v(y, z) for all y ∈ Y qv(y) := Z

if this integral is well-defined.

Furthermore, we define the operator T by T u = k + pu for all σX -measurable u : X → R for which pu exists, that means, ! p(dξ | x, a, b)u(ξ) T u(x, a, b) = k(x, a, b) + X

for all x ∈ X, a ∈ A, b ∈ B. N exists, then we get Let Π = (πn ) ∈ E∞ , P = (ρn ) ∈ F∞ . If VΠP N VΠP = π 0 ρ0 k +

N " j=1

π0 ρ0 p · · · pπj ρj k.

For π ∈ E, ρ ∈ F we put (πρp)n := πρp(πρp)n−1 where (πρp)0 denotes the identity. Let ϑ ∈ (0, 1). We set for every π ∈ E, ρ ∈ F, x ∈ X, and Y ∈ σX Qϑ,π,ρ (Y | x) := (1 − ϑ)

∞ "

ϑn (πρp)n IY (x)

n=0

where IY is the characteristic function of the set Y .

We remark that for a stationary strategy pair (π ∞ , ρ∞ ) the transition probability Qϑ,π,ρ is a resolvent of the corresponding Markov chain.

Assumption 3.1 There are: a nontrivial measure µ on σX ; a set C ∈ σX ; a σX -measurable function W ≥ 1; and constants ϑ ∈ (0, 1), α ∈ (0, 1), and β ∈ R, with the following properties: (a) Qϑ,π,ρ ≥ IC · µ for all π ∈ E and ρ ∈ F,

¨ HEINZ-UWE KUENLE

20

(b) pW ≤ αW + IC β, (c) |k(x, a, b)| < ∞.

W (x) A(x),b∈B B (x) x∈X,a∈A sup

Assumption 3.1 (a) means that C is a ”petite set”, (b) is called ”geometric drift towards C ” (see Meyn and Tweedie [10]). We assume in this paper that Assumption 3.1 is satisfied.

Lemma 3.2 There are a σX -measurable function V with 1 ≤ W ≤ V ≤ W + const, and a constant λ ∈ (0, 1) with Qϑ,π,ρ V ≤ λV + IC · µV

(3.1)

ϑpV ≤ λV.

(3.2)

and

Proof: Without loss of generality we assume β > 0. Let β ′ := W ′ := W + β ′ , and α′ :=

β ′ +α

β ′ +1 .

ϑ 1−ϑ β,

Then it holds that α′ ∈ (α, 1) and

pW ′ = pW + β ′ ≤ αW + β ′ + βIC

≤ α′ W − (α′ − α)W + α′ β ′ + (1 − α′ )β ′ + βIC ≤ α′ W ′ − (α′ − α) + (1 − α′ )β ′ + βIC = α′ W ′ + β ′ + α − α′ (β ′ + 1) + βIC = α′ W ′ + βIC .

(3.3)

Now let W ′′ := W ′ − β ′ IC = W + β ′ (1 − IC ). Then we get from (3.3 ) p(W ′′ + β ′ IC ) = pW ′

≤ α′ W ′ + βIC

= α′ W ′′ + α′ β ′ IC + βIC 1−ϑ ′ β IC = α′ W ′′ + α′ β ′ IC + ϑ α′ ϑ + 1 − ϑ ′ β IC = α′ W ′′ + ϑ β′ ≤ α′ W ′′ + IC . ϑ

(3.4)

21

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

We put α′′ := it follows:

1−ϑ 1−α′ ϑ .

pW ′′ ≤

Then it holds that α′ =

α′′ +ϑ−1 α′′ ϑ .

For β ′′ := α′′ β ′

α′′ + ϑ − 1 ′′ β ′′ β ′′ W IC . − pI + C α′′ ϑ α′′ α′′ ϑ

Hence, α′′ ϑpW ′′ ≤ (α′′ + ϑ − 1)W ′′ − ϑβ ′′ pIC + β ′′ IC .

Then (1 − ϑ)W ′′ ≤ α′′ W ′′ + β ′′ IC − ϑp(α′′ W ′′ + β ′′ IC ).

This implies (1 − ϑ)W ′′ ≤ α′′ W ′′ + β ′′ IC − ϑπρp(α′′ W ′′ + β ′′ IC ) for every π ∈ E, ρ ∈ F. Hence, Qϑ,π,ρ W

′′

= ≤

∞ !

n=0 ∞ !

(1 − ϑ)ϑn (πρp)n W ′′ ϑn (πρp)n (α′′ W ′′ + β ′′ IC )

n=0

− ′′

∞ !

ϑn (πρp)n (α′′ W ′′ + β ′′ IC )

n=1 ′′

= α W + β ′′ IC .

(3.5) β ′′

β′

α′′ +γ

We choose ϑ′ ∈ (ϑ, 1) and set γ := max{ µ(X) , ϑ′ −ϑ }, λ′ := 1+γ , λ := max{λ′ , ϑ′ }. It follows that α′′ < λ′ ≤ λ < 1 and λ′ − α′′ = (1 − λ′ )γ.

Hence, (λ − α′′ )W ′′ ≥ λ′ − α′′ ≥ (1 − λ′ )γ ≥ (1 − λ)γ.

(3.6)

We put V := W ′′ + γ. Obviously, V ≥ W ′′ ≥ 1 and V ≥ γ. Then it follows Qϑ,π,ρ V

= Qϑ,π,ρ W ′′ + γ ≤ α′′ W ′′ + IC · β ′′ + γ

≤ α′′ W ′′ + IC · γµ(X) + γ ≤ α′′ W ′′ + IC · µV + γ

≤ α′′ W ′′ + IC · µV + (λ − α′′ )W ′′ + λγ (see (3.6 ))

= λ(W ′′ + γ) + IC · µV

= λV + IC · µV.

¨ HEINZ-UWE KUENLE

22

Hence, (3.1 ) is proved. ′ From γ ≥ ϑ′β−ϑ it follows ϑ′ γ ≥ ϑγ + β ′ .

(3.7)

Then = ϑpW ′′ + ϑγ

ϑpV

≤ α′ ϑW ′′ + β ′ + ϑγ (see (3.4 )) ≤ α′ ϑW ′′ + ϑ′ γ (see (3.7 )) ≤ ϑ′ (W ′′ + γ)

= ϑ′ V

≤ λV. Hence, (3.2 ) is also proved. ✷

4

Properties of Stationary Strategy Pairs

For a function u : X → R we put ∥u∥V := supx∈X |u(x)| V (x) . Furthermore, we denote by V the set of all σX -measurable functions u with ∥u∥V < ∞.

In the following we will assume that on V that metric is given which is induced by the weighted supremum norm ∥ · ∥V . Then V is complete.

Lemma 4.1 ∥

sup n∈N,π∈E,ρ∈F

(πρp)n V ∥V < ∞.

Proof: From Assumption 3.1(b) it follows that (πρp)n W ≤ αn W +

1 β.

1−α

By Lemma 3.2 we get (πρp)n V ≤ (πρp)n W + const ≤ αn W + const′ ≤ αn V + const′ .

The statement is implied by this. ✷ Let Tw be the operator given by Tw u(x, a, b) := (1 − ϑ)(ϑk(x, a, b) + w(x)) + ϑpu(x, a, b) for all u ∈ V, x ∈ X, a ∈ A , b ∈ B. We note that Tw has essentially the same structure as the cost operator T used in stochastic dynamic

23

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

programming and stochastic game theory. This implies that some of our proofs are very similar to known proofs. Therefore we will restrict ourselves to a few remarks in these cases. (A very good exposition of basic ideas and recent developments in stochastic dynamic programming can be found in the books of Hern´andez- Lerma and Lasserre [1], [2].)

Obviously, Tw u = (1 − ϑ)ϑT (

u ) + (1 − ϑ)w.

1−ϑ

(4.8)

Lemma 4.2 Let w ∈ V, π ∈ E, ρ ∈ F . Then the functional equation u = πρTw u

(4.9)

has a unique solution uw = Sπρ w ∈ V and it holds: Sπρ w = lim (πρTw )n u = (1 − ϑ) n→∞

∞ !

ϑn (πρp)n (ϑπρk + w)

(4.10)

n=0

for every u ∈ V. Proof: We note that πρTw V ⊆ V. From (3.2 ) it follows that πρTw is contracting on V with modulus λ. The rest of the proof follows by Banach’s Fixed Point Theorem. ✷ We define a new operator Sγ,π,ρ by Sγ,π,ρ w := −(1 − IC )γ + Sπρ w − IC µw

(4.11)

for π ∈ E, ρ ∈ F, w ∈ V where Sπρ is the operator defined by the functional equation (4.9 ). The following lemma gives some properties of this operator.

Lemma 4.3

(a) Sγ,π,ρ V ⊆ V.

(b) Sγ,π,ρ is isotonic. (c) Sγ,π,ρ is contracting.

Proof: (a) is obvious.

¨ HEINZ-UWE KUENLE

24

(b) Using (4.10 ) we get Sγ,π,ρ w = −(1 − IC )γ + (1 − ϑ) = −(1 − IC )γ + (1 − ϑ) +(Qϑ,π,ρ − IC µ)w.

∞ !

n=0 ∞ !

ϑn (πρp)n (ϑπρk + w) − IC µw ϑn+1 (πρp)n πρk

n=0

(4.12)

The statement follows from Assumption 3.1 (a). (c) By Lemma 3.2 and (4.12 ) we get for u, v ∈ V |Sγ,π,ρ u − Sγ,π,ρ v| = |(Qϑ,π,ρ − IC µ)(u − v)|

≤ (Qϑ,π,ρ − IC µ)V ∥u − v∥V

≤ λV ∥u − v∥V . ✷

(4.13)

Lemma 4.4 The operator Sγ,π,ρ has in V a unique fixed point uγ,π,ρ . µuγ,π,ρ is continuous and non-increasing in γ.

Proof: The existence and uniqueness of the fixed point follows from Lemma 4.3 by Banach’s Fixed Point Theorem. From Sγ,π,ρ v ≥ Sγ ′ ,π,ρ v for γ ≤ γ ′ , and the isotonicity of Sγ,π,ρ it follows that uγ,π,ρ ≥ uγ ′ ,π,ρ .

Hence, µuγ,π,ρ ≥ µuγ ′ ,π,ρ . Furthermore, for arbitrary γ, γ ′

|uγ,π,ρ − uγ ′ ,π,ρ | = |(1 − IC )(γ ′ − γ) + (Qϑ,π,ρ − IC µ)(uγ,π,ρ − uγ ′ ,π,ρ )| ≤ |γ − γ ′ |V + λ∥uγ,π,ρ − uγ ′ ,π,ρ ∥V V

Hence, ∥uγ,π,ρ − uγ ′ ,π,ρ ∥V ≤ |γ − γ ′ | + λ∥uγ,π,ρ − uγ ′ ,π,ρ ∥V and |µuγ,π,ρ − µuγ ′ ,π,ρ | ≤ ∥uγ,π,ρ − uγ ′ ,π,ρ ∥V µV |γ − γ ′ | µV . ✷ ≤ 1−λ

Theorem 4.5 There exists a constant g and v ∈ V such that g + v = πρk + πρpv.

It holds: g = Φ π ∞ ρ∞ .

(4.14)

25

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

Proof: From Lemma 4.4 it follows that there is a γ ∗ with γ ∗ = µuγ ∗ ,π,ρ .

Hence, uγ ∗ ,π,ρ = Sγ ∗ ,π,ρ uγ ∗ ,π,ρ = −(1 − IC )γ ∗ + Sπρ uγ ∗ ,π,ρ − IC µuγ ∗ ,π,ρ = Sπρ uγ ∗ ,π,ρ − γ ∗ .

(4.15)

Let w∗ := uγ ∗ ,π,ρ . If we put w = w∗ in (4.9 ), then we get Sπρ w∗ = (1 − ϑ)(ϑπρk + w∗ ) + ϑπρpSπρ w∗ .

It follows by (4.15 ) that w∗ + γ ∗ = (1 − ϑ)(ϑπρk + w∗ ) + ϑπρp(w∗ + γ ∗ ).

Therefore, ϑw∗ + (1 − ϑ)γ ∗ = (1 − ϑ)ϑπρk + ϑπρpw∗ .

For g =

γ∗ ϑ ,

v=

w∗ 1−ϑ

we get (4.14 ). From (4.14 ) it follows

Ng =

N −1 ! n=0

(πρp)n πρk + (πρp)N v − v.

If we consider Lemma 4.1 we get N −1 1 ! (πρp)n πρk = Φπ∞ ρ∞ .

N →∞ N

g = lim

✷

n=0

5

Existence of optimal stationary strategies

We give first a lemma which concerns a certain auxiliary one-stage game.

The results of this lemma are well-known and can be derived, for instance, from the results in [12].

Lemma 5.1 Let u : X × A × B → R a σX×A×B -measurable function |u(x,a,b)| < ∞. Then it holds: with supx∈X,a∈A A(x),b∈B B (x) V (x) (a) inf π∈E supρ∈F πρu = supρ∈F inf π∈E πρu ∈ V. (b) There are π ∗ ∈ E, ρ∗ ∈ F with π ∗ ρu ≤ π ∗ ρ∗ u ≤ πρ∗ u for all π ∈ E, ρ ∈ F.

¨ HEINZ-UWE KUENLE

26

For a function v : X × A → R (v : X × B → R) we put Lv := inf π∈E πv (U v := supρ∈F ρv). We can now prove the following lemma concerning an auxiliary functional equation.

Lemma 5.2 The functional equation u =

inf sup{(1 − ϑ)(ϑπρk + w) + ϑπρpu}

π∈E ρ∈F

= LU Tw u = (1 − ϑ)ϑLU T (

u ) + (1 − ϑ)w 1−ϑ

(5.16)

has for every w ∈ V a unique solution u∗ =: Sw in V. Proof: Let w ∈ V. Then it follows from Lemma 5.1 that LU Tw V ⊆ V. Because πρTw is contracting on V, it holds for u, v ∈ V: πρTw u ≤ πρTw v + λ∥u − v∥V V. Since L and U are isotonic it follows: LU Tw u ≤ LU Tw v + λ∥u − v∥V V. Because u and v can be interchanged, we get that LU Tw is also contracting. The statement follows by Banach’s Fixed Point Theorem. ✷ In the following lemma Sπρ and S are the operators defined by the functional equations (4.9 ) and (5.16 ).

Lemma 5.3 For every w ∈ V there are π ∗ ∈ E, ρ∗ ∈ F with Sπ∗ ,ρ w ≤ Sw ≤ Sπ,ρ∗ w

(5.17)

for all π ∈ E, ρ ∈ F . Furthermore, Sw := inf sup Sπρ w. π∈E ρ∈F

(5.18)

Proof: It follows from Lemma 5.1 that there are π ∗ ∈ E, ρ∗ ∈ F such that π ∗ ρT (

uw uw ) ≤ LU T ( ) 1−ϑ 1−ϑ uw ≤ πρ∗ T ( ) 1−ϑ

(5.19)

27

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

where uw = Sw. Hence, π ∗ ρTw uw ≤ LU Tw uw = uw ≤ πρ∗ Tw uw

(5.20)

for all π ∈ E, ρ ∈ F. Assume that (π ∗ ρTw )n uw ≤ uw ≤ (πρ∗ Tw )n uw

(5.21)

for n ∈ N. Then it follows from (5.20 ) that uw ≤ πρ∗ Tw ((πρ∗ Tw )n uw = (πρ∗ Tw )n+1 uw .

(5.22)

uw ≥ (π ∗ ρTw )n+1 uw .

(5.23)

Analogously,

From (5.22 ) and (5.23 ) it follows by mathematical induction that (5.21 ) holds for all n ∈ N. For n → ∞ we get (5.17 ).

(5.18 ) follows immediately from (5.17 ). ✷ We define a new operator Sγ by Sγ w := −(1 − IC )γ + Sw − IC µw for π ∈ E, ρ ∈ F, w ∈ V, γ ∈ R. The following lemma gives some properties of this operator.

Lemma 5.4

(a) Sγ V ⊆ V.

(b) Sγ is isotonic. (c) Sγ is contracting with modulus λ. (d) Sγ has in V a unique fixed point vγ . It holds limn→∞ (Sγ )n u = vγ for every u ∈ V. Moreover, vγ is isotonic and continuous in γ.

Proof: (a) is obvious. (b) From (4.11 ) and (5.18 ) it follows that Sγ w = inf sup Sγ,π,ρ w. π∈E ρ∈F

By Lemma 4.3 we get the statement.

¨ HEINZ-UWE KUENLE

28

(c) Let w′ , w′′ ∈ V. By Lemma 5.3 it follows that there are π ′′ ∈ E, ρ′ ∈ F, such that Sw′ ≤ Sπ,ρ′ w′ Sw′′ ≥ Sπ′′ ,ρ w′′ for all π ∈ E, ρ ∈ F. Hence, Sγ w′ − Sγ w′′ = −(1 − IC )γ + Sw′ − IC µw′

−(−(1 − IC )γ + Sw′′ − IC µw′′ )

≤ −(1 − IC )γ + Sπ′′ ,ρ′ w′ − IC µw′

−(−(1 − IC )γ + Sπ′′ ,ρ′ w′′ − IC µw′′ )

= Sγ,π′′ ,ρ′ w′ − Sγ,π′′ ,ρ′ w′′

≤ λV ∥w′ − w′′ ∥V

since Sγ,π′′ ,ρ′ is contracting (see Lemma 4.3). Because w′ and w′′ can be interchanged, we get the statement. (d) The existence of a unique fixed point vγ ∈ V and limn→∞ (Sγ )n u = vγ for every u ∈ V follows from Banach’s Fixed Point Theorem. For γ ′ ≤ γ it holds Sγ w ≤ Sγ ′ w = Sγ w + (1 − IC )(γ − γ ′ ) ≤ Sγ w + (γ − γ ′ )V. Assume that for n > 1 Sγn−1 vγ ′ ≤ vγ ′ ≤ Sγn−1 vγ ′ +

γ − γ′ V. 1−λ

Then it follows Sγn vγ ′ ≤ Sγ ′ Sγn−1 vγ ′ ≤ Sγ ′ vγ ′ = vγ ′ ≤ Sγ ′ (Sγn−1 vγ ′ + ≤ Sγ (Sγn−1 vγ ′ + ≤ Sγn vγ ′ +

γ − γ′ V) 1−λ

γ − γ′ V ) + (γ − γ ′ )V 1−λ

λ(γ − γ ′ ) V + (γ − γ ′ )V (see (c)) 1−λ

29

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

= Sγn vγ ′ +

γ − γ′ V. 1−λ

Hence, by mathematical induction we find that this inequality holds for all n ∈ N. For n → ∞ it follows v γ ≤ v γ ′ ≤ vγ +

γ − γ′ V. 1−λ

The rest of the statement is implied by this.

✷

Theorem 5.5 There are g = const and v ∈ V with g + v = LU T v.

(5.24)

It holds g = inf∞ sup ΦΠP . Π∈E

P ∈F∞

Furthermore, there is an optimal stationary strategy pair.

Proof: From Lemma 5.4 it follows that µvγ is non-increasing in γ.

Therefore, there is a γ ∗ with γ ∗ = µvγ ∗ . vγ ∗

= Sγ ∗ v γ ∗ = −(1 − IC )γ ∗ + Svγ ∗ − IC µvγ ∗ = Svγ ∗ − γ ∗ .

(5.25)

Let w∗ := vγ ∗ . If we put w = w∗ in (5.16 ) then we get Sw∗ = LU ((1 − ϑ)(ϑk + w∗ ) + ϑpSw∗ ).

It follows by (5.25 ) w∗ + γ ∗ = LU ((1 − ϑ)(ϑk + w∗ ) + ϑp(w∗ + γ ∗ )).

Therefore, ϑw∗ + (1 − ϑ)γ ∗ = LU ((1 − ϑ)ϑk + ϑpw∗ ). ∗

∗

w we get (5.24 ).

For g = γϑ , v = 1−ϑ From (5.24 ) and Lemma 5.1 it follows that there are π ∗ ∈ E, ρ∗ ∈ F, with

π ∗ ρn T v γ ∗ − g ≤ v γ ∗ ≤ π n ρ∗ T v γ ∗ + ε − g

¨ HEINZ-UWE KUENLE

30

for all Π = (πn ) ∈ E∞ , P = (ρn ) ∈ F∞ . It follows ≤

vγ ∗

π ∗ ρ0 T π ∗ ρ1 T · · · π ∗ ρN T vγ ∗ − (N + 1)g

≤ π0 ρ∗ T π1 ρ∗ T · · · πN ρ∗ T vγ ∗ − (N + 1)g

For N → ∞ we get ΦΠρ∗∞ ≤ g ≤ Φπ∗∞ P

for all Π ∈ E∞ , P ∈ F∞ . This implies

g = inf∞ sup ΦΠP Π∈E

P ∈F∞

and the optimality of (π ∗∞ , ρ∗∞ ). ✷

Heinz-Uwe K¨ uenle Brandenburgische Technische Universit¨ at Cottbus Institut f¨ ur Mathematik PF 101344 D-03013 Cottbus GERMANY Phone: +49 (0355) 69 3151 Fax: +49 (0355) 69 3164 kuenle@math.tu-cottbus.de

References [1]

Hern´andez-Lerma, O.; Lasserre, J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria. Applications of Mathematics 30. Springer-Verlag, New York, 1996

[2]

Hern´andez-Lerma, O.; Lasserre, J. B.: Further Topics on Discrete-Time Markov Control Processes. Applications of Mathematics 42. Springer-Verlag, New York, 1999

[3]

K¨ uenle, H.-U.: Stochastische Spiele und Entscheidungsmodelle.

Teubner-Texte zur Mathematik 89. Teubner-Verlag, Leipzig, 1986

[4]

K¨ uenle, H.-U.: Stochastic games with complete information and average cost criterion. Advances in Dynamic Games and Applications [edited by J.A. Filar, V. Gaitsgory, K. Mizukami] (Annals of the International Society of Dynamic Games, Vol. 5) Birkh¨auser, Boston, 2000, 325– 338

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES

31

[5]

K¨ uenle, H.-U.: Equilibrium strategies in stochastic games with additive cost and transition structure. International Game Theory Review, 1, 1999, 131–147

[6]

K¨ uenle, H.-U.: On multichain Markov games. Annals of the International Society of Dynamic Games. Birkh¨auser, to appear

[7]

Maitra, A.; Sudderth, W.: Borel stochastic games with limsup payoff. Ann. Probab., 21, 1993, 861–885

[8]

Maitra, A.; Sudderth, W.: Finitely additive and measurable stochastic games. Internat. J. Game Theory, 22, 1993, 201–223

[9]

Maitra, A.; Sudderth, W.: Finitely additive stochastic games with Borel measurable payoffs. Internat. J. Game Theory, 27, 1998, 257–267

[10]

Meyn, S. P.; Tweedie, R. L.: Markov Chains and Stochastic Stability. Communication and Control Engineering Series. SpringerVerlag, London, 1993

[11]

Neveu, J.: Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco 1965

[12]

Nowak, A. S.: Minimax selection theorems. J. Math. Anal. Appl.

103, 1984, 106–116.

[13]

Nowak, A. S.: Zero-sum average payoff stochastic games with general state space. Games and Econ. Behavior 7, 1994, 221–232

[14]

Nowak, A. S.: Optimal strategies in a class of zero-sum ergodic stochastic games.Math. Meth. Oper. Res. 50, 1999, 399–419

[15]

Rieder, U.: Average optimality in Markov games with general state space. Proc. 3rd International Conf. on Approximation and Optimization, Puebla, 1995

Morfismos, Vol. 4, No. 2, 2000, pp. 33–60

Little cubes and homotopy theory

∗

Dai Tamaki .

Abstract This is a survey article on the theory of configuration spaces of little cubes, which is naturally related to the homotopy commutativity of double or more highly iterated loop spaces. The article begins with historical background and then surveys important applications to homotopy theory.

1991 Mathematics Subject Classification: 55-02, 55P35 Keywords and phrases: Configuration space, loop space, quasifibration

1

Introduction

A little n-cube is the image of an affine embedding c : In → In whose image has all edges parallel to the corresponding edges of the standard n-cube I n .

c

∗

Invited article

33

34

DAI TAMAKI

The configuration space of j little n-cubes Cn (j) is the set of j-tuples (c1 , · · · , cj ) of little n-cubes whose images have disjoint interiors from each other. Cn (j) is topologized with the compact-open topology.

Since a little cube is determined by its image, it is safe to discuss on little cubes by drawing a picture like:

c2

c1

The reader might notice that he or she has seen this kind of picture in a textbook of algebraic topology.

It is an elementary theorem in homotopy theory that the n-th homotopy group of a pointed space X, πn (X), is an Abelian group, or equivalently the n-th loop space on X, Ωn X, is a homotopy commutative H-space, if n ≥ 2.

In most textbooks on algebraic topology, this fact is explained as follows: Suppose an element α ∈ πn (X) is represented by a continuous map f : (I n , ∂I n ) −→ (X, ∗), where ∗ denotes the base point of X. Pick up another element β ∈ πn (X) represented by g : (I n , ∂I n ) −→ (X, ∗).

The product of α and β in πn (X), α + β, is represented by the map f + g : (I n , ∂I n ) −→ (X, ∗) defined by the following picture.

35

LITTLE CUBES AND HOMOTOPY THEORY

g

X

.......................................................... ............. ....... ....................................................................................................... ...... ........................ . ..... .................. ........ ........................ . . . . ..... . . . . . . . . . . . . . . . ......... .... ... ........ . . . . . . . . . . . . . . . . .... . . . ........ ................ ........ .... ... .... . . . . . . . . . . . . . . ........ ..... .... .... ... .... . .... . ... ... ... .... ... ... . . ... .... ... . ... . . .... ... .... ... ... ..... . . . .... ........ ... ... ..... . . . . . . . ... . ...... ........ ....................................... . . . ........ . ... ........ . . .... .... ......... . . . ........ . ... . . . . . . . . .......... ......... ...... ... . . . . ............ .. . . . .......... . . . . . ............... ............................. ........... ... . . . . . ............. .. ............................... . . . . . . . . . . . . . . . . . ................ . . . . . . . ............................ ..... ............. . . . . . . ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........................................................................

⑦

f

!∗

✯

On the other hand, β + α is represented by the map g + f : (I n , ∂I n ) −→ (X, ∗) defined by the following picture.

X

g

.................................................................... .................................. .................. ................................................. .................... .............. ........ ............. ......................... ................ ........... ............. ...... ............ ..... . . . . . ....... ................... . . . . . ..... ......... ....... ....... . . . . . . . .... . . . . . . . ......... ..... .... ... . . . . . . . . . . . . . . . ........ ................... .... .... .... .. . . . . . . . .... . . . . . . . .... .... ........ .... ... . .... . ... ... ... .... ... . . . ... ... ... .... . . ... ... ... .... .... ... . .... . . . . . ......... ..... ... ..... ... . . . . . . . ..... ....... ... ....................................... . . . ...... . . . . . . ... ......... .... .... ......... . . . . . . . . . .. . . . . . . .......... .......... ......... ............ ............. ........... . . . . . . . . . . ... . . . ................................. ............. ... . . .. . ................. .. . . .............................................................................................. ......................................... . . . . .................................................

⑦

f

!∗

✯

If n ≥ 2, a homotopy connecting these two maps can be obtained by rotating two (rectangular) cubes in I n .

f

g

g

✲

f

✲

g

✲

f

g

f

36

DAI TAMAKI

Notice that domains of f + g and g + f can be regarded as elements of Cn (2) and the above picture can be interpreted as a path in Cn (2), since we need to keep two cubes separated while rotating.

In fact, Cn (2) is path-connected (more generally Cn (j) is known to be (n − 2)-connected). The path-connectivity of Cn (2) implies the commutativity of πn (X) or the homotopy commutativity of Ωn X. This can be considered as the first application of the topology of little cubes to homotopy theory.

It is natural to extend this idea to study higher homotopy commutativity of iterated loop spaces. Boardman and Vogt began the first systematic study in this direction [5, 6]. A few years later, Peter May established a concrete relationship between iterated loop spaces and little cubes by proving the recognition principle [40]. The study of little cubes also led him to introduce the notion of operad, which turned out to be an important object naturally appearing in many fields of mathematics and mathematical physics. But we are not going to cover operad in this article. Those who are interested in operad are recommended to take a look at [36], for example.

The following is the organization of this article: Section 2. Basic Definitions and Fundamental Facts: recalls basic definitions and classical results used in later sections.

Section 3. Stable Splitting of Iterated Loop Spaces: reviews the development of homotopy theory stimulated by Snaith’s theorem of stable splitting of Ωn Σn X. Section 4. Constructing Maps and Spaces: shows that various important maps and spaces have been constructed by using little cubes.

Section 5. Problems: is a collection of open problems related to little cubes.

2

Basic Definitions and Fundamental Facts

Let us begin with a precise definition of little cube.

LITTLE CUBES AND HOMOTOPY THEORY

37

Definition 2.1.1 For a positive integer n, a little n-cube c is a map I n −→ I n which can be decomposed into the following form: c = ℓ 1 × · · · × ℓn where each ℓi : I −→ I is an orientation preserving Affine embedding. The space of little n-cubes is denoted by Cn (1) and topologized with compact-open topology. Thus Cn (1) is a subspace of Map(I n , I n ).

The configuration space of j little n-cubes Cn (j) is defined as follows: Cn (j) = {(c1 , · · · , cj ) ∈ Cn (1)j |ci (IntI n ) ∩ ck (IntI n ) = ∅ if i ̸= k} Cn (j) admits a natural action of the symmetric group of j-letters, Σj , by permuting the indices of little cubes.

Cn = {Cn (j)}j is called the little n-cube operad.

We also need the case n = ∞ to study infinite loop spaces. Define C∞ (j) = colim Cn (j) −→ n

where colimit is taken over the inclusion maps induced by I n × {0} "→ I n+1 .

Obviously C∞ (j) inherits the action of Σj .

Thanks to the following fact, we do not have to struggle with the compact-open topology on Cn (j).

Lemma 2.1.2 Define a map ξn : Cn (1) −→ I 2n by ξn (c) = (c( 41 , · · · , 14 ), c( 43 , · · · , 34 )), then ξn is an embedding of Cn (1) as an open subset of I 2n .

As is stated in Introduction, the following is one of the most fundamental properties of little cube.

Lemma 2.1.3 Cn (j) is (n − 2)-connected. Hence C∞ (j) is contractible.

The following map plays a central role in studying the relationship between the configuration space of little cubes Cn (j) and an n-fold loop space Ωn X.

38

DAI TAMAKI

Definition 2.1.4 For a pointed space X, define θn,j : Cn (j) × (Ωn X)j −→ Ωn X as follows: For (c1 , · · · , cj ) ∈ Cn (j) and ω1 , · · · , ωj ∈ Ωn X, θn,j (c1 , · · · , cj ; ω1 , · · · , ωj ) : I n −→ X is defined to be ωi on each cube ci (I n ) and maps the outside of little cubes to the basepoint. More precisely θn,j (c1 , · · · , cj ; ω1 , · · · , ωj )(t) = =

!

n ωi (c−1 i (t)) if t ∈ ci (I ) for some 0 ≤ i ≤ j, ∗ otherwise.

θn,j is obviously Σj -equivariant, hence we have an induced map on the quotient, which is denoted by the same notation: θn,j : Cn (j) ×Σj (Ωn X)j −→ Ωn X These maps {θn,j } satisfy the following compatibility conditions.

Lemma 2.1.5 Let ∗ denote the constant loop to the basepoint of Ωn X, then, for (c1 , · · · , cj ) ∈ Cn (j) and ω1 , · · · , ωi−1 , ωi+1 , · · · , ωj ∈ Ωn X, we have θn,j (c1 , · · · , cj ; ω1 , · · · , ωi−1 , ∗, ωi+1 , · · · , ωj )

= θn,j−1 (c1 , · · · , ci−1 , ci+1 , · · · , cj ; ω1 , · · · , ωi−1 , ωi+1 , · · · , ωj ) Definition 2.1.6 For a pointed space Y , an action of Cn is a collection of maps θj : Cn (j) ×Σj Y j −→ Y for j = 0, 1, · · · satisfying the same relations as in Lemma 2.1.5.

In other words, an n-fold loop space has an action of Cn . May proved that, conversely, the existence of an action of Cn on a pointed space Y can be used as a criterion for Y to be equivalent to an n-fold loop space.

Theorem 2.1.7 (Recognition Principle) Let X be a path-connected space with a nondegenerate basepoint. Then X has a weak homotopy type of an n-fold loop space if and only if it has an action of little ncube operad.

LITTLE CUBES AND HOMOTOPY THEORY

39

The relations in Lemma 2.1.5 can be used to pierce the spaces Cn (j) ×Σj (Ωn X)j together to get a single space Cn (Ωn X) as follows.

Definition 2.1.8 Let Y be a pointed space with basepoint ∗. Generate ! an equivalence relation ∼ on j Cn (j) ×Σj Y j by relations (c1 , · · · , cj ; y1 , · · · , yi−1 , ∗, yi+1 , · · · , yj )

∼ (c1 , · · · , ci−1 , ci+1 , · · · , cj ; y1 , · · · , yi−1 , yi+1 , · · · , yj ).

Define Cn (Y ) by ⎛ ⎞ $ j Cn (Y ) = ⎝ Cn (j) ×Σj Y ⎠ / ∼ . j

Corollary 2.1.9 {θn,j } induces a well-defined continuous map θn : Cn (Ωn X) −→ Ωn X. Note that Cn (Y ) can be defined for any pointed space Y and we have a natural map σn : Y −→ Cn (Y ) induced by the inclusion of Cn (1) × Y . The reader can easily check that θn ◦ σn ≃ id if Y = Ωn X. In other words, if Y is equivalent to an n-fold loop space Ωn X, Y is a retract of Cn (Y ). σn

✲ Cn (Y )

Y ❅

❅

id

❅

θn

❅ ❘ ❄ ❅

Y. Notice that the evaluation map on Ωn X eval : Σn (Ωn X) −→ X and the Freudenthal suspension E n : X −→ Ωn Σn X

40

DAI TAMAKI

make the following diagram commutative Ωn X

E n✲

◗

◗

Ωn Σn (Ωn X)

id

◗ ◗

◗ # ◗

Ωn eval

❄

Ωn X. This similarity between the functors Cn and Ωn Σn can be explained by the following important theorem of May.

Theorem 2.1.10 (Approximation Theorem) Let X denote a pathconnected space with a nondegenerate basepoint. Then we have a natural weak homotopy equivalence Cn (X) ≃ Ωn Σn X. Recognition Principle is important because it is the origin of the theory of operad. But for practical applications in homotopy theory, Approximation Theorem is far more useful, because it gives us a combinatorial model for iterated loop spaces and thus making them easier to handle.

Besides May’s model, some other combinatorial models for Ωn Σn X have been discovered. The oldest is the reduced product of James [27] which is a model for ΩΣX. Milgram [41] constructed a model for Ωn Σn X generalizing James’ construction. Segal [43] suggested to use the configuration space of distinct points instead of little cubes. The resulting space is equivalent to the little cube model. May’s approach was improved by J. Caruso and S. Waner [11] to prove an approximation theorem for nonconnected spaces. There are also simplicial models found by Milnor [42] for n = 1, by Barratt and Eccles [2] for n = ∞ and by Jeff Smith [44] for general n.

These are essentially equivalent to each other. Among these models, however, the “little cube model” (and its variants, like the one using configuration space of distinct points) has been most popular. This is partly because Victor Snaith used the little cube model to prove the stable splitting theorem, which is the subject of the next section.

LITTLE CUBES AND HOMOTOPY THEORY

3

41

Stable Splitting of Iterated Loop Spaces

3.1

Snaith’s Theorem

A stable splitting (in fact, a splitting after a single suspension) of ΩΣX was proved by James in 1955 [27] using a combinatorial construction of ΩΣX, called the James construction or reduced product. Important tools in classical homotopy theory, like the (James)-Hopf invariant H : ΩS n+1 −→ ΩS 2n+1 and EHP fibration E

H

S n −→ ΩS n+1 −→ ΩS 2n+1 have been constructed by using this splitting.

The little cube model of Ωn Σn X can be regarded as a generalization of the James construction. In fact, Snaith [45] proved a stable splitting theorem for Ωn Σn X by using the little cube model.

Theorem 3.1.1 For a pointed space X with a nondegenerate basepoint and a positive integer n, we have a weak stable homotopy equivalence (1)

∞ !

Ω n Σn X ≃ S

j=1

Cn (j)+ ∧Σj X ∧j .

This holds also for n = ∞.

Note that Cn (X) is naturally filtered by the number of cubes ⎛

Fm Cn (X) = ⎝

$

j≤m

⎞

Cn (j) ×Σj X j ⎠ / ∼

and the right hand side of the above splitting is the “subquotient” of Cn (X) with respect to this filtration.

Later, Cohen, May and Taylor generalized Snaith’s theorem to more general settings [15]. For those who are curious about how to construct these stable splittings, the appendix of [12] will be of interest.

It is also worthwhile to note that these splittings can be obtained by a very simple argument in the category of spectra. This fact, proved

by R.L. Cohen [24], depends on the existence of the Σj -equivariant half smash product of spectra constructed by Lewis, May and Steinberger [37].

42

DAI TAMAKI

3.2

Applications of Stable Splitting

After the stable splitting theorem was proved by Snaith, the case n = 2 and X = S 2k−1 became a subject of intense study in 70’s. Mahowald [38] found a new infinite family in the 2-primary components of the stable homotopy groups of spheres by using the summands of the stable splitting of Ω2 S 9 . An odd primary analogue was obtained by R.L. Cohen [23].

These facts suggest that the cohomology of the summands of the Snaith splitting of Ω2 S 2k+1 is equipped with an important module structure over the Steenrod algebra. To be more precise, we need the following definition.

Definition 3.2.1 For any prime p and integer k ≥ 0, define a module M (k, p) over the mod p Steenrod algebra as follows: M (k, p) =

!

A2 /(χ(Sq i )|i > k) if p = 2, Ap /(χ(β ϵ P i )|i > k, ϵ = 0, 1) if p is odd,

where χ denotes the canonical anti-automorphism on Ap .

Theorem 3.2.2 For any prime p and integer k ≥ 0, there exists a p-local spectrum B(k, p) satisfying the following properties: 1. H ∗ (B(k, p); Z/pZ) ∼ = M (k, p) as modules over the Steenrod algebra.

2. Let jk : B(k, p) −→ HZ/pZ be a generator of H 0 (B(k, p); Z/pZ).

Then for any CW complex X, the induced map of generalized homology theories (jk )∗ : B(k, p)∗ (X) −→ H∗ (X; Z/pZ) is surjective for q ≤ 2k + 1 if p = 2 and for q ≤ 2p(k + 1) − 1 if p is odd.

Furthermore such a spectrum is unique up to homotopy.

Definition 3.2.3 B(k, p) is called the Brown-Gitler spectrum [7].

Mahowald in [38] conjectured that the stable summands of Ω2 S 2k+1 localized at 2 realize the Brown-Gitler spectra. This fact was proved by Brown and Peterson [8].

LITTLE CUBES AND HOMOTOPY THEORY

43

Theorem 3.2.4 Localized at 2, we have the following homotopy equivalence C2 (j)+ ∧Σj (S 2k−1 )∧j ≃ Σj(2k−1) B([ 2j ]).

Odd primary analogs are proved by R. Cohen [23].

Theorem 3.2.5 Localized at an odd prime p, we have the following homotopy equivalence when j = mp + r for p > r > 0: C2 (pj)+ ∧Σpj (S 2k−1 )∧pj ≃ Σj(2pk−2) B(m).

There is another way of describing the summands of the stable splitting of Ωn S n+k closely related to the above theorems.

Consider the vector bundle pn,j : Cn (j) ×Σj Rj −→ Cn (j)/Σj .

Sj

It is easy to see that the Thom complex of pn,j , T (pn,j ), is Cn (j)+ ∧Σj which is a stable summand of Ωn S n+1 . More generally, we have T (pn,j ⊕ · · · ⊕ pn,j ) = Cn (j)+ ∧Σj S jk . !

"# k

$

It is worthwhile to note that the infinite families in the stable homotopy groups found by Mahowald and R. Cohen comes from the triviality of p2,j ⊕p2,j proved by F. Cohen, Mahowald and Milgram [14]. The order of pn,j in general was studied in [13].

Mahowald’s construction of infinite families in the stable homotopy groups of spheres is one of the most important applications of Ω2 S 2k+1 in stable homotopy theory. Note that the classical applications of the James splitting of ΩS 2k+1 live in the unstable world: For example, the mod p Hopf invariant Hp : ΩS 2n+1 −→ ΩS 2np+1 is defined to be the adjoint to the following composition ΣΩS 2n+1

⎛ ⎞ ' ≃ Σ ⎝ (S 2n )∧j ⎠ −→ Σ(S 2n )∧p = S 2np+1 . j

In order to get an unstable application of the splitting of Ω2 S 2k+1 , for example a “secondary Hopf invariant”, we need to desuspend the

44

DAI TAMAKI

Snaith splitting. However, as stated in [12], Ω2 Σ2 X does not split in finitely many suspensions. This is one of the crucial differences between ΩΣX and Ωn Σn X for n ≥ 2.

This is the difficulty in applying Ωn Σn X in unstable homotopy theory when n ≥ 2. But F. Cohen, May and Taylor [15] proved that each piece can be split off in finitely many suspensions.

Theorem 3.2.6 Fix a positive integer k, then there exists a positive integer L (depending on k) and a map ΣL Cn (X) −→ ΣL Cn (k)+ ∧Σk X ∧k which is equivalent to the composition of the Snaith splitting and the projection on the k-th component after suspending infinitely many times.

Furthermore F. Cohen proved that L can be taken to be 2k when n = 2 [12]. This allowed him to construct a map σn : W (n) −→ Ω2p W (n + 1) which can be considered to be the “secondary double-suspension”, where W (n) is the homotopy theoretic fiber of the double suspension map E 2 : S 2n−1 −→ Ω2 S 2n+1 . σn played an essential role in the work of Mahowald and Thompson [39, 48] where they determined the unstable v1 -periodic homotopy groups of spheres.

B. Gray began a systematic study in this direction in [28]. He introduced the notion of EHP spectra which is an unstable way of studying Toda-Smith spectra V (n) in the fashion of Cohen-Moore-Neisendorfer [18, 19, 20]. In order his program to be accomplished, we need to find various unstable maps between loop spaces. As in the case of the classical EHP sequence, those maps could be constructed by using unstable splitting of loop spaces. In fact, Gray is very close to proving the existence of an EHP spectrum for V (0). His construction would be completed, if we could prove Ω2 Σ2 X localized at an odd prime splits after suspending twice, which is conjectured in [28]. However nothing is known about localization of Ωn Σn X.

45

LITTLE CUBES AND HOMOTOPY THEORY

4

Constructing Maps and Spaces

As we have seen in the previous section, little cube model played an important role in the construction and applications of the stable splittings of iterated loop spaces.

It is often helpful to have extra geometric information even though what we need is abstract homotopy-theoretic results.

As is suggested by Quillen’s work on closed model category, homotopy theory (stable or unstable) is fairly formal. It can be axiomatized nicely. But we need concrete models for practical applications.

Little cubes can be used to construct important spaces and maps.

The followings are some of the examples.

4.1

Fibrations

May proved the Approximation theorem (Theorem 3.1.1) by constructing a little cube model for the path-loop fibration: Ωn Σn X −→ P Ωn−1 Σn X −→ Ωn−1 Σn X. Definition 4.1.1 For a pair of pointed space (X, A), define En (X, A) to be the subspace of Cn (X) consisting of elements (c1 , · · · , cj ; x1 , · · · , xj ) satisfying the following properties: if xi1 , · · · , xiℓ ̸∈ A then (prn−1 (ci1 ), · · · , prn−1 (ciℓ )) ∈ Cn−1 (ℓ), where prn−1 is the projection onto the last (n − 1)-coordinates.

Theorem 4.1.2 (May) 1. If X is a path-connected pointed space with nondegenerate basepoint, we have a quasifibration (2)

Cn (X) −→ En (CX, X) −→ Cn−1 (ΣX)

and natural maps α : Cn (X) −→ Ωn Σn X

α ˜ : En (CX, X) −→ P Ωn−1 Σn X making the following diagram commutative up to homotopy Cn (X) α

❄

Ωn Σn X

✲ En (CX, X) α ˜

❄ ✲ P Ωn−1 Σn X

✲ Cn−1 (ΣX) α

❄ ✲ Ωn−1 Σn X.

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DAI TAMAKI

2. En (CX, X) is contractible. Therefore α is a weak homotopy equivalence.

Since path-loop fibration is a principal fibration with contractible total space, principal fibrations with fiber Ωn Σn X are pull-back of the path-loop fibration. Thus it is natural to expect to obtain a little cube model of such a fibration by pulling back May’s quasifibration (2). Unfortunately, however, taking a pull-back in general does not preserve quasifibration.

The quasifibration (2) is not just a quasifibration. The base space is filtered and the quasifibration satisfies a kind of local-triviality on each successive difference of the filtration.

Thanks to this additional structure, a pull-back of (2) has a chance to be a quasifibration. In fact, S.-C. Wong proved the following [49]: Definition 4.1.3 For a pointed space X, define W (k, n, ΣX) to be the homotopy fiber of the Freudenthal suspension map: Ωk−1 E n : Ωk−1 Σk X −→ Ωn+k−1 Σn+k X. Namely W (k, n, ΣX) is defined by the following pull-back diagram: ✲

W (k, n, ΣX)

❄

Ω

k−1

k

Σ X

n Ωk−1 E✲

P Ωn+k−1 Σn+k X ❄ n+k−1

Ω

Σn+k X.

Definition 4.1.4 For a pair of pointed spaces (X, A), define ξ(k, n; X, A) by the following pull-back diagram: ξ(k, n; X, A)

✲ En+k−1 (X, A) πn+k

π(k,n)

❄

Ck−1 (X/A)

σ n✲

❄

Cn+k−1 (X/A).

Theorem 4.1.5 (Wong) For k ≥ 1, n ≥ 0 and a strong NDR pair (X, A), π(k, n) : ξ(k, n; X, A) −→ Ck−1 (X/A) is a quasifibration with fiber Cn+k (A). Thus we have a weak homotopy equivalence ξ(k, n; X, A) ≃ W (k, n; X/A).

LITTLE CUBES AND HOMOTOPY THEORY

47

Wong used ξ(k, n; X, A) to prove a stable splitting of W (k, n; X/A).

Since the secondary-suspension σn was constructed by splitting off the p-adic piece of Ω2 S 2n+1 after the 2p-fold suspension, it would be interesting if we could desuspend Wong’s stable splitting to construct “tertiary suspension map”.

Wong’s idea can be also applied to the following case.

Since Cn (X) has a structure of a monoid by concatenation and the equivalence Cn (X) ≃ Ωn Σn X is an equivalence of Hopf spaces, the p-th power map p× : Ωn Σn X −→ Ωn Σn X is equivalent to the p-th power map on Cn (X). Thus we have a diagram ?

✲ En (CX, X) πn

❄

Cn (ΣX)

❄ ✲ Cn (ΣX).

p×

in which all maps are defined in terms of little cubes.

The homotopy fiber of p× : Ωn Σn+1 X −→ Ωn Σn+1 X is the mapping space from the mod p Moore space Map∗ (P n+1 (p), Σn+1 X). Following Wong’s idea K. Iwama recently proved the following [29].

Theorem 4.1.6 Let X be a pointed space with (X, ∗) a strong NDR pair. Define En (p)(CX, X) by the following pull-back diagram ✲ En (CX, X)

En (p)(CX, X)

πn

πn (p)

❄

Cn (ΣX)

p×

❄ ✲ Cn (ΣX).

Then πn (p) is a quasifibration with fiber Cn+1 (X). Therefore we have a weak homotopy equivalence En (p)(CX, X) ≃ Map∗ (P n+1 (p), Σn+1 X), where P n+1 (p) = S n ∪p en+1 .

48

DAI TAMAKI

It is not known if En (p)(CX, X) has a stable splitting. Although the proof of the above theorem is parallel to that of Wong’s, the proof of stable splitting by Wong cannot be applied in this case. The difficulty comes from the fact that the p-th power map on Cn (ΣX) does not preserve the filtration.

4.2

Kahn-Priddy Transfer

As the Snaith splitting suggests, little cubes are useful for handling stable maps in concrete ways. Another example of applications of little cube in stable homotopy theory is the construction of transfer by D.S. Kahn and S.B. Priddy.

Transfer is a map going to the wrong direction in the homology or cohomology, classically in group cohomology. It is a well-known fact that they can be realized as stable maps between spaces. One of the examples is constructed by Kahn and Priddy by using the little cube model of Ω∞ Σ∞ X [30, 31].

Suppose we are given an N -fold covering space p : E −→ B. The symmetric group of N -letters acts on each fiber by deck transformation and p can be considered to be a fiber bundle with fiber π1 (B)/π1 (E) and structure group ΣN . Since C∞ (N ) is a contractible space with a free ΣN -action, the projection π : C∞ (N ) −→ C∞ (N )/ΣN is a universal ΣN -bundle. Thus we have the following pull-back diagram E

ϕ

N ✲ C (N ) × ∞ ΣN (π1 (B)/π1 (E))

p

π

❄

B

ϕ ¯

❄ ✲ C∞ (N )/ΣN .

Define Φ : B −→ C∞ (N ) ×ΣN E N by Φ(p(x)) = (ϕ(x), xτ1 , · · · , xτN )

LITTLE CUBES AND HOMOTOPY THEORY

49

where {τ1 , · · · , τN } is a choice of coset representatives of π1 (B)/π1 (E).

Composed with C∞ (N ) ×ΣN E N −→ C∞ (E) we have

Φ

B+ −→ C∞ (E+ ) ≃ Ω∞ Σ∞ (E+ ).

The Kahn-Priddy transfer is the stable map adjoint to this map: p! : Σ∞ B+ −→ Σ∞ E+ .

Let p be a prime number. Kahn and Priddy applied this construction to the covering EΣp −→ BΣp to get the famous Kahn-Priddy theorem which states π∗S (BΣp ) −→ π∗S (S 0 ) is surjective on the p-primary components.

4.3

Strong Convergent Cobar Spectral Sequence

Little cubes can be used to construct a spectral sequence converging to the homology of iterated loop spaces: (3)

n−1 n E2 ∼ = Cotorh∗ (Ω Σ X) (h∗ , h∗ ) =⇒ h∗ (Ωn Σn X).

The following is a quick review of the construction by the author [46].

Throughout this subsection h∗ (−) denotes a multiplicative homology theory.

In order to construct a spectral sequence (3), the most natural idea would be to try to find a filtration on Ωn Σn X with which the E 1 -term of the resulting spectral sequence becomes the algebraic cobar construction on the coalgebra h∗ (Ωn−1 Σn X). Thanks to the stable splitting (1), it enough to define a filtration on each Cn (j), separately.

Suppose we have a filtration {F−q Cn (j)} on each Cn (j). Define F−s Cn (X) =

! j

F−s Cn (j)+ ∧Σj X ∧j .

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DAI TAMAKI

{F−s Cn (X)} is a stable filtration for Ωn Σn X. The E 1 -term of the spectral sequence defined by this filtration is 1 = h−s+t (F−s Cn (X), F−s−1 Cn (X)) E−s,t

= =

! j

h−s+t (F−s Cn (j)+ ∧Σj X ∧j , F−s−1 Cn (j)+ ∧Σj X ∧j )

j

˜ −s+t (F−s Cn (j)/F−s−1 Cn (j) ∧Σ X ∧j ). h j

!

˜ ∗ (Ωn−1 Σn X) has sumOn the other hand, the tensor algebra on h mands of the following form: ˜ ∗ ((Cn−1 (j1 ) × · · · × Cn−1 (js ))+ ∧Σ ×···×Σ (ΣX)∧(j1 +···+js ) ). h j1 js Thus what we need is filtrations on {Cn (j)} so that F−s Cn (j) − F−s−1 Cn (j) becomes s vertically aligned stacks of cubes. In order to define such filtrations, we need auxiliary functions on Cn (j).

Let pr1 : Cn (1) −→ C1 (1) be the map induced by the projection onto the first coordinate. It is not difficult to find a function d : Cn (1) × Cn (1) −→ [0, 1] with the following properties. d(c, c′ ) = 0 ⇐⇒ pr1 (c)( 12 ) ̸∈ pr1 (c′ )([0, 1]) or pr1 (c′ )( 12 ) ̸∈ pr1 (c)([0, 1]) d(c, c′ ) = 1 ⇐⇒ pr1 (c)( 12 ) = pr1 (c′ )( 12 ).

For c = (c1 , · · · , cj ) ∈ Cn (j), a stack in c is a subset of {c1 , · · · , cj }.

We say that a stack {ci |i ∈ S} is stable under gravity if and only if d(ci1 , ci2 ) ̸= 0 for any i1 , i2 ∈ S. With these terminologies we define a filtration on little cubes as follows.

Definition 4.3.1 Let F0 Cn (j) = F−1 Cn (j) = Cn (j). For q > 0, c = (c1 , · · · , cj ) ∈ F−s−1 Cn (j) if and only if, for any partition {1, · · · , j} = " " S1 · · · Sq with S1 ̸= ∅, · · · , Sq ̸= ∅, at least one of the stacks corresponding to S1 , · · · , Sq is not stable under gravity.

This filtration is called the gravity filtration. The spectral sequence induced by this filtration is called the gravity spectral sequence.

LITTLE CUBES AND HOMOTOPY THEORY

51

On the contrary to our intension, it is not known whether the E 1 term of the spectral sequence defined by this filtration is isomorphic to the cobar construction. However the following fact proved in [46] is enough to identify the E 2 -term with Cotor.

Theorem 4.3.2 There exists a stable filtration on En (CX, X) with the following isomorphism of differential graded h∗ -modules, if h∗ (Ωn−1 Σn X) is flat over h∗ : E 1 (En ) ∼ = E 1 (Cn ) ⊗h∗ h∗ (Cn−1 (ΣX)),

where {E r (En )} is the spectral sequence induced by the filtration on En (CX, X) and {E r (Cn )} is the spectral sequence induced by the gravity filtration on Cn (X).

Furthermore (E 1 (En ), d1 ) is acyclic.

Corollary 4.3.3 If h∗ (Ωn−1 Σn X) is flat over h∗ , we have the following isomorphism n−1 n E 2 (Cn ) ∼ = Cotorh∗ (Ω Σ X) (h∗ , h∗ ).

For a pointed space X, if h∗ (X) is a flat h∗ -module, we have a spectral sequence, so-called the classical Eilenberg-Moore spectral sequence with E2 ∼ = Cotorh∗ (X) (h∗ , h∗ ).

The E ∞ -term of this spectral sequence is not in general directly related to h∗ (ΩX). However, the gravity spectral sequence does calculate h∗ (Ωn Σn X), since it is a direct sum of the spectral sequences defined by finite filtrations.

Theorem 4.3.4 The gravity spectral sequence converges to h∗ (Ωn Σn X).

Although the gravity spectral sequence has this important property, it also has a disadvantage: the E 1 -term is mysterious. In order to use the gravity spectral sequence for practical computations, it is important to find good generators in the E 2 -term. In the case of the classical Eilenberg-Moore spectral sequence, we have the Dyer-Lashof operations which are defined in terms of the cobar construction. The following fact helps us to find generators in the E 2 -term of the gravity spectral sequence.

Theorem 4.3.5 ([47]) If h∗ (Ωn−1 Σn X) is flat over h∗ , the gravity spectral sequence is isomorphic to the classical Eilenberg-Moore spectral sequence from the E 2 -term on.

52

4.4

DAI TAMAKI

Little Cubes with Overlappings Allowed

We have been considering little cubes disjoint from each other, so far. If we remove this disjointness condition, we obtain loop spaces of different types.

Definition 4.4.1 Define Dn (j) = Cn (1)j . This is the space of j little n-cubes with overlappings allowed.

c3 c2

c1 c4

The symmetric group of j-letters acts on Dn (j) by renumbering of cubes and the definition of Cn (X) can be applied without modification to get a functor Dn (X).

Since the little cube operad Cn acts on Dn (X) in a natural way, Dn (X) is an n-fold loop space by the recognition principle of May. In fact, we can easily see that this is equivalent to a classical construction due to Dold and Thom, i.e. infinite symmetric product, and thus it has a structure of an infinite loop space.

Definition 4.4.2 For a pointed space X and a nonnegative integer i, define ⎛ ⎞ SPi (X) = ⎝

i #

j=0

X j /Σj ⎠ / ∼

where the equivalence relation “∼” is generated by the relation removing basepoint.

SPi (X) is called the i-th symmetric product of X. When i = ∞, it is called the infinite symmetric product.

LITTLE CUBES AND HOMOTOPY THEORY

53

Since there is no restriction on the movements of cubes, Dn (j) is Σj -equivariantly contractible. Thus we have Proposition 4.4.3 For any pointed space X, Dn (X) is homotopy equivalent to SP∞ (X).

The following is a classical theorem of Dold and Thom [25], Theorem 4.4.4 For a path-connected pointed CW-complex X, we have the following natural isomorphism ! n (X) πn (SP∞ (X)) ∼ =H

As an immediate corollary, we have

Theorem 4.4.5 For a pointed connected CW-complex X of finite type, we have the following homotopy equivalence SP∞ (X) ≃

∞ "

K(πn (X), n).

n=0

Cn (X) and Dn (X) are two extreme cases: cubes are disjoint from each other in Cn (j), while any kinds of overlapping are allowed in Dn (j). π∗ (C∞ (X)) is the stable homotopy groups of X, while π∗ (D∞ (X)) is the (reduced) integral homology groups of X. Let us consider the following intermediate objects.

Definition 4.4.6 Let Dni (j) be the subspace of Dn (j), consisting of j little cube (c1 , · · · , cj ) satisfying the following property: each t ∈ I n is contained in the interior of the image of at most i cubes.

Dni (j) inherits the action of Σj . For a pointed space X, define i Dn (X) in the same way as Cn (X) is defined.

Now we have successive inclusions Cn (X) = Dn0 (X) ⊂ Dn1 (X) ⊂ · · · ⊂ Dn∞ (X) = Dn (X).

The commutativity of the following diagram ⊂

Cn (X) ❅ ■ ❅

❅

✲ Dn (X) "

X

" ✒ "

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DAI TAMAKI

and the fact that the induced map on homotopy groups ! ∗ (X) π∗ (X) −→ π∗ (Dn (X)) ∼ = π∗ (SP∞ (X)) ∼ =H

coincides with the Hurewicz homomorphism implies that the Hurewicz homomorphism factors through π∗ (Dni (X)) ◗ ❦◗

✲ H ! ∗ (X) ✸ ✑✑

⊂

π∗ (Dni (X))

◗

◗

✑

✑

π∗ (X).

F. Kato determined the homotopy type of Dni (X) [33].

Theorem 4.4.7 For a path-connected pointed space X with (X, ∗) a strong NDR pair, we have the following weak homotopy equivalence Dni (X) ≃ Ωn SPi Σn X. The idea of the proof is to extend May’s construction of the quasifibration (2).

Since " ∗ if j ≤ i i D0 (j) = ∅ if j > i we have D0i (X) = SPi (X).

Once we have a quasifibration i (ΣX) Dni (X) −→ E −→ Dn−1

with E contractible, we have the desired weak homotopy equivalence i (ΣX) ≃ · · · ≃ Ωn D0i (Σn X) = Ωn SPi Σn X. Dni (X) ≃ ΩDn−1

Definition 4.4.8 For a pair of pointed spaces (X, A), define Eni (X, A) to be the subspace of Cni (X) consisting of elements (c1 , · · · , cj ; x1 , · · · , xj ) satisfying the following properties: if xi1 , · · · , xiℓ ̸∈ A then i (ℓ). (prn−1 (ci1 ), · · · , prn−1 (ciℓ )) ∈ Cn−1

LITTLE CUBES AND HOMOTOPY THEORY

55

Proposition 4.4.9 For a pointed space X with (X, ∗) a strong NDR pair, the projection i Eni (CX, X) −→ Dn−1 (ΣX)

is a quasifibration with fiber Dni (X).

Furthermore Eni (CX, X) is contractible if X is path-connected.

Sadok Kallel [32] independently studied an analogous construction C i (Rn ; X) by using (labelled) configuration space of points in Rn , instead of little cubes, and proved a homotopy equivalence C i (Rn ; X) ≃ Ωn SPi Σn X for connected CW complexes together with a “delooped version” of this homotopy equivalence.

We should point out that most of the constructions using little ncubes can be also done by using configuration space of points in Rn .

One of the advantages of using configuration space of points rather than little cubes is that Rn can be replaced with any space M . Kallel also studied C d (M ; X) where M is a stably parallelizable smooth manifold with nonempty boundary.

5

Problems

We conclude this article with open problems related to little cubes.

1. Find a little cube model of the fiber of the secondary suspension σn : W (n) −→ Ω2p W (n + 1).

2. Find a stable splitting of Map∗ (P n (p), Σn X).

3. Prove that, localized away from 2, Ω2 Σ2 X splits into a wedge of the (localized) Snaith summands after suspending twice.

4. For each prime p, find a combinatorial model for Ωn Σn X localized at or away from p.

5. Compute the homology of Ωn SPi Σn X.

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DAI TAMAKI

Acknowledgment The author is very grateful to Jesus Gonzalez, CINVESTAV, for inviting him to write this article for Morfismos, which gave him a good chance to review the development of the theory of little cubes. He would also like to thank the referees for informing him of Kallel’s work and for pointing out ambiguous assumptions in some theorems.

Dai Tamaki Department of Mathematical Sciences Faculty of Science, Shinshu University 3-1-1 Asahi, Matsumoto, 390-8621 JAPAN rivulus@math.shinshu-u.ac.jp

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[35] Kuhn Nicholas J., The homology of the James-Hopf maps., Illinois Journal of Mathematics, 27 (1983), 315–333.

[36] Loday J.-L., Stasheff J.D. and Voronov A.A. ed. Operads: Proceedings of Renaissance Conferences., Contemporary Mathematics 202, American Mathematical Society, 1997.

[37] Lewis L.G., May J.P. and Steinberger M. Equivariant Stable Homotopy Theory., Lecture Notes in Mathematcis, vol. 1213, Springer-Verlag, 1986.

[38] Mahowald M.E., A new infinite family in 2 π∗S ., Topology, 16 (1977), 249–256.

[39] Mahowald M.E., The image of J and the EHP sequence., Annals of Mathematics, 116 (1982), 65–112.

[40] May J.P., The Geometry of Iterated Loop Spaces., Lecture Notes in Mathematics, vol. 271, Springer-Verlag, 1972.

[41] Milgram R.J., Iterated loop spaces., Annals of Mathematics, 84 (1966), 386–403.

[42] Milnor J.W., On the construction F K., Reprinted in J.F. Adams, “Algebraic Topology – A Student’s Guide”, London Mahtematical Society Lecture Notes Series 4, Camridge University Press, 1972.

[43] Segal G.B., Configuration spaces and iterated loop spaces., Inventiones Mathematicae, 21 (1973), 213–221.

[44] Smith J.H., Simplicial group models for Ωn S n X., Israel Journal of Mathematics, 66 (1989), 330–350.

[45] Snaith V.P.A., Stable decomposition for Ωn S n X., Journal of the London Mathematical Society (2), 7 (1974), 577–583.

[46] Tamaki D. A dual Rothenberg-Steenrod spectral sequence., Topology, 33 (1994), 631–662.

60

DAI TAMAKI

[47] Tamaki D. Remarks on the cobar-type Eilenberg-Moore spectral sequences., preprint.

[48] Thompson R.D., The v1 -periodic homotopy groups of an unstable sphere at odd primes., Transactions of the American Mathematical Society, 319 (1990), 535â&#x20AC;&#x201C;559.

[49] Wong S.-C., The fiber of the iterated Freudenthal suspension., Mathematische Zeitschrift, 215 (1994), 377â&#x20AC;&#x201C;414.

Morfismos, Vol. 4, No. 2, 2000, pp. 61–76

Hipergrupos y ´algebras de Bose-Mesner∗ Isa´ıas L´opez

1

Resumen En este art´ıculo se prueba que toda ´algebra de Bose-Mesner es un hipergrupo con el producto usual y con el producto Hadamard de matrices. Adem´as, presentamos los diferentes tipos de isomorfismos entre ´algebras de Bose-Mesner y sus relaciones con los hipergrupos.

1991 Mathematics Subject Clasification: 05E45 Keywords and phrases: ´algebras de Bose-Mesner, hipergrupos

1

Introducci´ on

Existen varios objetos matem´aticos cuya esencia es el de un esquema de asociaci´on y que se conocen con varios nombres, pero esencialmente son el mismo concepto matem´atico; por ejemplo, ´algebras de adyacencia, ´algebras de Bose-Mesner, anillo centralizador, anillo de Hecke, anillo de Schur, ´algebra de caracteres, hipergrupos, grupos probabil´ısticos.

Los hipergrupos son una generalizaci´on de los grupos, en donde el producto de dos elementos est´a determinado por una funci´on de distribuci´on. Existe mucha investigacio´n en esta ´area, sobre todo para generalizar conceptos b´asicos de la teor´ıa de grupos, por ejemplo, en hipergrupos existe el teorema de Lagrange.

En el presente trabajo demostramos que toda ´algebra de Bose-Mesner, bajo cierta normalizaci´on, es un hipergrupo, tanto con el producto usual de matrices como con el producto de Hadamard (ver los Teoremas 3.3 y 3.4). Esto permite dar una visi´on distinta de conceptos tales como isomorfismos y dualidades en ´algebras de Bose-Mesner. ∗

Este trabajo se realiz´ o con el apoyo de CONACyT, a trav´es del proyecto No. 29275E. 1 Estudiante de doctorado del Departamento de Matem´ aticas, CINVESTAV-IPN. Becario de CONACyT.

61

´ ISA´IAS LOPEZ

62

2

Hipergrupos

La teor´ıa general de hipergrupos fue introducida por Dunkl [4], Jewett [9] y Spector [11] de manera independiente, y tratan a los hipergrupos como un caso especial.

Una interpretaci´on f´ısica es la siguiente: Un hipergrupo conmutativo finito es una colecci´on de part´ıculas, digamos {c0 , c1 , . . . , cn }, en las cuales est´a permitido la interacci´on por colisi´on entre ellas. Cuando dos part´ıculas colisionan forman una tercera part´ıcula. Si colisionamos ci con cj la probabilidad de que resulte la part´ıcula ck es nkij y este n´ umero es fijo. La part´ıcula c0 tiene la propiedad de ser absorbida en cualquier colisi´on; la llamamos un fot´on. Cada part´ıcula tiene una antipart´ıcula la cual est´a especificada por la siguiente regla: La colisi´on de dos part´ıculas tiene una probabilidad positiva de resultar en un fot´on si y s´olo si las dos part´ıculas son anti-part´ıculas una de la otra. La interacci´on de las part´ıculas son independientes del orden, del tiempo y de su posici´on en el espacio. La estructura del sistema est´a completamente determinado por las probabilidades nkij las cuales son invariantes bajo intercambio de todas las part´ıculas con sus anti-part´ıculas.

Definici´ on 1.1 Un hipergrupo generalizado es una pareja (H ⊂ A) donde H = {c0 , c1 , . . . , cn } y A es una ´ algebra asociativa con identidad c0 y con involuci´ on ⋆ sobre C, que satisface las siguientes condiciones: (A1) H es una base de A. (A2) H⋆ = H, que denotaremos por c⋆i = cσ(i) .

(A3) La estructura constante nkij ∈ C definida por: c i cj =

n !

nkij ck

k=0

satisface lo siguiente

c⋆i = cj ⇐⇒ n0ij > 0, c⋆i ̸= cj ⇐⇒ n0ij = 0.

En el resto de este trabajo H denotar´a a un hipergrupo generalizado.

Si A es conmutativo, entonces H es conmutativo. Adem´as, se dice que H es Hermitiano si c⋆i = ci ∀i; real si nkij ∈ R ∀i, j, k; positivo

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

63

si nkij ! 0 ∀i, j, k. Por otra parte, decimos que H es normalizado si satisface que (A4)

! k

nkij = 1 ∀i, j.

Un hipergrupo generalizado que es real y normalizado se dice que es hipergrupo signado. Un hipergrupo generalizado que es positivo y normalizado se llama simplemente un hipergrupo.

Definimos el peso de ci como (1.1)

w(ci ) = (n0iσ(i) )−1 > 0

y el peso de H como (1.2)

w(H) =

n !

w(ci ).

i=0

Consideremos la siguiente forma alternativa de (A4): !

(A4′ ) w(ci )−1 w(cj )−1 = nkij w(ck )−1 . k

Un hipergrupo que es real y satisface (A4′ ) se dice que es un ensamble.

Aunque la condici´on (A4) parezca m´as f´acil de probar que (A4′ ), esto no siempre es as´ı, es por eso la siguiente proposici´on.

Proposici´ on 1.2 Existe una correspondencia uno a uno entre ensambles e hipergrupos signados.

Demostraci´ on: Si H = {c0 , c1 , . . . , cn } es un hipergrupo signado, te¯ = {w(c0 )c0 , w(c1 )c1 , . . . , w(cn )cn } es un ensamble. nemos que H ¯ = {c0 , c1 , . . . , cn } un ensamble tenemos Reciprocamente, para H que H = {w(c0 )c0 , w(c1 )c1 , . . . , w(cn )cn } es un hipergrupo signado.

De aqu´ı en adelante, a menos que se diga lo contrario, s´olo consideraremos hipergrupos que son conmutativos.

Proposici´ on 1.3 La *-´ algebra es semisimple.

Demostraci´ on: Se sigue del hecho de que la *-´algebra no tiene elementos idempotentes.

Para a ∈ A, ad(a) ∈ End(A) denotar´a el operador multiplicaci´on por a.

´ ISA´IAS LOPEZ

64

Lema 1.4 El conjunto {ad(ci )} es linealmente independiente en End(A). !

Demostraci´ on: Si i ri ad(ci ) = 0, entonces multiplicando por c⋆j y considerando el coeficiente de c0 tenemos que rj = 0.

El ´algebra ad(A) ⊂ End(A) tiene dimensi´on n + 1 y es conmutativa y semisimple. Esto nos dice que End(A) es isomorfa al ´algebra de operadores diagonalizables en M (n + 1, C). Por lo tanto, podemos encontrar una base {eo , e1 , . . . , en } de A en la cual los operadores ad(ci ) son diagonales, esto es,

(1.3)

ci ej = χj (ci )ej ∀i, j

para alguna funci´on χj tal que

(1.4)

ej ek = δjk ej ,

donde δij es la delta de Kroenecker.

Si F (H) denota el espacio de todas las funciones de H en los complejos, entonces el conjunto de funciones {χi } es linealmente independiente en F(H) .

Definici´ on 1.5 Un car´ acter de H es cualquier χ ∈ F (H) que satisface (1.5)

χ(ci )χ(cj ) =

" k

nkij χ(ck ) ∀i, j.

Si la extensi´on lineal de χ en A se denota tambi´en por χ, tendremos la siguiente formulaci´ on equivalente.

(1.6)

χ(ci )χ(cj ) = χ(ci cj ) ∀i, j.

ˆ Al conjunto de todos los caracteres de H lo denotaremos por H. ˆ = {χ0 , χ1 , . . . , χn } y Proposici´ on 1.6 H (1.7)

χi (c⋆j ) = χi (cj ) ∀i, j.

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

65

para toda i, j, de donde Demostraci´ on: Tenemos que! ci ej = χj (ci )ej ! k χj (ci )χj (cs )ej = ci cs ej = nis ck ej = nkij χj (ck )ej , es decir, k k ! k nij χj (ck ). De aqu´ı se sigue la proposici´on porque A es χj (ci )χj (cs ) = k

isomorfa a la *-´algebra Cn+1 y esta tiene exactamente n + 1 caracteres que satisfacen las condiciones establecidas.

Claramente la funci´on id´enticamente 1 es un car´acter de H, el cual denotaremos por χ0 . ˆ es una base ortogonal de F (H). Para Ahora deseamos ver que H esto, para todo f, g ∈ F(H), definimos f ⋆ (ci ) = f (c⋆i ) e introducimos el producto interno

(1.8)

⟨f, g⟩ =

1 " w(ci )f (ci )g(ci ). w(H) i

Como H y {e0 , e1 , . . . , en } son bases para A podemos encontrar constantes αjk ∈ C tales que (1.9)

ej =

" k

αjk ck ∀j.

Multiplicando ambos lados por c⋆i y comparando los coeficientes de c0 podemos observar que αji = w(ci )χj (c⋆i )αj0 ,

(1.10) de donde

ej = αj0

(1.11)

"

w(ck )χj (c⋆k )ck .

k

Combinando esto con la ecuaci´on (1.4) y comparando los coeficientes de c0 obtenemos: (1.12)

δij = αj0

" k

w(ck )χi (ck )χj (c⋆k ) = αj0 w(H)⟨χi , χj ⟩.

De aqu´ı tenemos el siguiente lema:

´ ISA´IAS LOPEZ

66

ˆ = {χ0 , χ1 , . . . , χn } es una base ortogonal de F(H) con Lema 1.7 H respecto al producto interno definido en (1.8).

Como F(H) es una *-´algebra con unidad χ0 bajo la multiplicaci´on punto a punto y conjugaci´on compleja, podemos escribir (1.13)

χ i χj =

! k

mkij χk con mkij ∈ C.

Adem´as, si χ⋆i = χj podemos definir el peso de χi como w(χi ) = Entonces

(m0ij )−1 .

⟨χi , χi ⟩ = ⟨χi χi , χ0 ⟩

= ⟨χi χ⋆i , χ0 ⟩

= w(χi )−1 ⟨χ0 , χ0 ⟩ = w(χi )−1 .

(1.14)

De aqu´ı concluimos que w(χi ) ∈ R y w(χi ) > 0. Adem´as, usando la ecuaci´on (1.11) tenemos: Proposici´ on 1.8 (1.15)

ei =

w(χi ) ! w(ck )χ⋆i (ck )ck ∀i. w(H) k

En particular, (1.16)

e0 =

1 ! w(ck )ck . w(H) k

En conclusi´on, tenemos el siguiente resultado: ˆ Adem´ Teorema 1.9 Si H es un hipergrupo signado, tambi´en lo es H. as ˆ w(H) = w(H).

2

´ Algebras de Bose-Mesner

El concepto de esquema de asociaci´on es importante en ´algebra combinatoria. Dicho concepto aparece en el estudio de c´odigos, dise˜ nos,

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

67

gr´aficas de distancia regular, invariantes de nudos y en muchos otros temas. El estudio de los esquemas de asociaci´on lo inici´o Delsarte en 1973 (ver [3]). En el caso sim´etrico los esquemas de asociaci´on son esencialmente particiones de una gr´afica completa en subgr´aficas regulares que est´an relacionadas entre s´ı de alguna manera espec´ıfica. Para un an´alisis extensivo de este tema se pueden consultar [1] y [2].

Definici´ on 2.1 Un esquema de asociaci´ on de clase d es una familia de {0,1}-matrices {Ai |i = 0, . . . , d} de orden n que satisfacen lo siguiente: (B1) A0 = I (I es la matriz identidad).

(B2)

!d

i=0 Ai

= J (J es la matriz con todas sus entradas igual a uno).

(B3) Para todo i, t Ai = Aσ(i) , para alg´ un σ(i) ∈ {0, 1, . . . , d} (t A denota la traspuesta de A ).

(B4) Ai ◦ Aj = δij Ai (◦ denota el producto de Hadamard de matrices, entrada por entrada).

(B5) Ai Aj = Aj Ai =

d !

k=0

i, j, k ).

pkij Ak ( equivalentemente, pkij = pkji para todo

El ´algebra generada por {Ai |i = 0, 1, . . . , d} sobre C es una ´algebra conmutativa con el producto usual y con el producto de Hadamard de matrices y la familia {Ai |i = 0, 1, . . . , d} es una base para esta ´algebra, la cual es conocida como el ´ algebra de Bose-Mesner del esquema de asociaci´on de clase d. Esta ´algebra tiene la propiedad de ser cerrada bajo trasposici´on compleja.

En otro contexto, a este tipo de ´algebras tambi´en se les llama ´ algebras Doble-Frobenius (ver [10]).

En el caso en que t Ai = Ai para todo i, tenemos una ´algebra de Bose-Mesner sim´etrica.

Sea (2.1)

ni = poiσ(i) ,

en donde ni es el n´ umero de elementos en la diagonal de la matriz Ai Aσ(i) . El entero positivo ni se llama la valencia de Ai , y es claro que

´ ISA´IAS LOPEZ

68

n0 = 1, ni = nσ(i) , n = n0 + n1 + · · · + nd .

La siguiente proposici´on es importante para relacionar las ´algebras de Bose-Mesner con los hipergrupos. Su demostraci´on se puede ver en [1].

Proposici´ on 2.2 (i) pk0j = δjk , (ii) p0ij = ni δiσ(i) , σ(k)

(iii) pkij = pσ(i)σ(j) , (iv)

d !

j=0

pkij = ni ,

(v) nk pkij = nj pjσ(i)k = ni pikσ(j) , (vi)

d !

k=0

pkij pluk =

d !

s=0

psui plsj .

Como las matrices A0 , A1 , . . . , Ad son normales y conmutan por parejas, ellas son diagonalizables simult´aneamente por una matriz unitaria. As´ı, podemos encontrar una descomposici´on de Cn como suma directa de d + 1 eigenespacios de dimensi´on fj , 0 ≤ j ≤ d (ver [5] y [7]). Adem´as, como J pertenece al ´algebra, n es un eigenvalor de multiplicidad 1 y entonces podemos suponer que f0 = 1. Los fi se llaman las multiplicidades de los esquemas. Sea {E0 , E1 , . . . , Ed } la base de idempotentes ortogonales con el producto usual de matrices del ´algebra de Bose-Mesner, cada una de ellas corresponde a los proyectores de Cn en cada diferente eigenespacio, adem´as, fi = tr(Ei ), tenemos lo siguiente: (C1) E0 = n1 J, (C2) E0 + E1 + · · · + Ed = I, (C3) Ei Ej = δij Ei ,

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

un σ(i) ∈ {0, 1, . . . , d}, (C4) t Ei = Eσ(i) para alg´ (C5) Ei ◦ Ej = Ej ◦ Ei =

d !

k=0

kE . qij k

k son llamados los par´ Los coeficientes qij ametros de Krein del esquema de asociaci´on.

Como {E0 , E1 , . . . , Ed } es una base para el ´algebra de Bose-Mesner, tenemos

(2.2)

Ai =

d "

Pj (i)Ej .

j=0

An´alogamente d

(2.3)

Ei =

1" Qj (i)Ak . n j=0

Sea (2.4)

P = (Pj (i)),

cuya entrada (i, j) de la matriz es Pj (i), y sea (2.5)

Q = (Qj (i)),

cuya entrada (i, j) de la matriz es Qj (i). Las matrices P y Q se llaman la primera y segunda matriz del esquema de asociaci´on, respectivamente.

Ellas satisfacen lo siguiente: (2.6)

P Q = QP = nI.

Usando las ecuaciones (2.2) y (2.3) junto con las propiedades (A4) y (C3) obtenemos: (2.7) (2.8)

Ai Ej =Pj (i)Ej , 1 Ei ◦ Aj = Qj (i)Aj . n

69

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La ecuaci´on (2.7) muestra que cada vector columna de Ej es un vector propio de Ai asociado con el valor propio Pj (i). An´alogamente, la ecuaci´on (2.8) nos permite decir que cada vector columna de Aj es un vector propio de Ei asociado al valor propio n1 Qj (i) respecto al producto Hadamard.

Los par´ametros de Krein tienen las siguientes propiedades, similares a los de la Proposici´on 2.2. La demostraci´on de esta proposici´on se puede consultar en [1].

Proposici´ on 2.3 k =δ , (i) q0j jk 0 =f δ (ii) qij i iσ(j) , σ(k)

k =q (iii) qij σ(i)σ(j) ,

(iv)

d !

j=0

k =f , qij i

k = f qj i (v) fk qij j σ(i)k = fi qkσ(j) ,

(vi)

d !

k=0

k ql = qij uk

d !

s=0

s ql . qui sj

Finalmente, las entradas de la primera y segunda eigenmatrices P y Q satisfacen las siguientes relaciones: Proposici´ on 2.4 (i) Pj (i)Pk (i) =

d !

l=0

(ii) Qj (i)Qk (i) =

pljk Pl (i),

d !

l=0

3

l Q (i). qjk l

Hipergrupos y ´ algebras de Bose-Mesner

Consideremos una ´algebra de Bose-Mesner A = {A0 , A1 , . . . , Ad } de dimensi´on d + 1 sobre C. Recordemos que A es una ´algebra asociativa, conmutativa con respecto al producto usual de matrices, cuya identidad

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71

es I. Asimismo, A es una ´algebra asociativa, conmutativa con respecto al producto de Hadamard de matrices y con identidad J. Lema 3.1 A es un hipergrupo generalizado con el producto usual de matrices y base {A0 , A1 , . . . , Ad }.

Demostraci´ on: Sea {A0 , A1 , . . . , Ad } un esquema de asociaci´on y A = ⟨A0 , A1 , . . . , Ad ⟩ el ´algebra de Bose-Mesner generada por dicho esquema. En particular, A es una ´algebra asociativa, conmutativa, con identidad A0 y con involuci´on el mapeo traspuesta conjugada. Adem´as, claramente se tiene (A1) y (A2), de modo que s´olo resta probar (A3).

Para esto, observemos que de la Proposici´on 2.2(ii): !

0 si j ̸= σ(i), 0 Pij = ni si j = σ(i).

De aqu´ı se sigue (A3).

Por el lema 3.1 el peso de Ai es: #−1 " w(Ai ) = p0iσ(i) > 0.

(3.1)

Proposici´ on 3.2 A es un ensamble con el producto usual de matrices.

Demostraci´ on: Es suficiente probar el axioma (A4’). Usando la Proposici´on 2.2 tenemos: d $

=

=

=

k=0 d $

pkij w (Ak )−1 = pkij nk δkσ(k) =

k=0 d $ d $ k=0 r=0 d $ d $

pkij ptkr = pkij ptkr =

d $

pkij p0kσ(k)

k=0 d $

pkij

k=0 d d $$ r=0 k=0 d $ d $

%

d $ r=0

pkij ptkr psri ptsj

s=0 r=0 r=0 k=0 % d & d d $ $ $ t s = psj pri = ptsj ni s=0 r=0 s=0 0 0 =ni nj = piσ(i) pjσ(j) =w(Ai )−1 w(Aj )−1 .

ptkr

&

´ ISA´IAS LOPEZ

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De la correspondencia uno a uno entre hipergrupos signados y ensambles se deduce el siguiente resultado.

Teorema 3.3 A! = ⟨w(A0 )A0 , w(A1 )A1 , . . . , w(Ad )Ad ⟩ es un hipergrupo signado bajo el producto usual de matrices.

Finalmente, de la Proposici´on 2.3 se deduce el siguiente resultado, similar al teorema anterior.

Teorema 3.4 A genera un hipergrupo signado bajo el producto Hadamard de matrices y base {E0 , E1 , . . . , Ed }.

4

Isomorfismos en ´ algebras de Bose-Mesner

Algunos de los ejemplos cl´asicos de ´algebras de Bose-Mesner son los obtenidos a trav´es de gr´aficas fuertemente regulares y de distancia regular [2]. La clasificaci´on de gr´aficas fuertemente regulares y de distancia regular se realiza por medio del ´algebra de Bose-Mesner asociada.

Aunque la clasificaci´on de gr´aficas de distancia regular es un poco m´as compleja, ambas ocupan el concepto de BM-isomorfismo, que se define en esta secci´on. Otra de las aplicaciones de la teor´ıa de esquemas de asociaci´on es la clasificaci´on de modelos spin (invariantes de nudos) (ver [6]), en donde juega un papel importante el concepto de dualidad entre ´algebras de Bose-Mesner.

Sean A y B dos ´algebras de Bose-Mesner y ψ un isomorfismo de espacios vectoriales de A en B. Definici´ on 4.1 Se dice que ψ es un BM-isomorfismo de A en B si satisface: ψ(AB) = ψ(A)ψ(B) y ψ(A ◦ B) = ψ(A) ◦ ψ(B) para toda A, B ∈ A. Un ejemplo de un BM-isomorfismo se obtiene a trav´es de una matriz de permutaci´on. Es decir, si P es una matriz de permutaci´on, entonces ψ(A) = P −1 AP define un BM-isomorfismo. De hecho, este isomorfismo es conocido como isomorfismo combinatorial.

Si A tiene como base usual {Ai | i = 1, . . . , d} y su base de idempotentes ortogonales es {Ei | i = 1, . . . , d}, tenemos que {ψ(Ai ) | i =

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73

1, . . . , d} y {ψ(Ei ) | i = 1, . . . , d} son la base usual y la base de idempotentes ortogonales de B, respectivamente.

Usando la ecuaci´on (2.7) obtenemos la siguiente relaci´on ψ(Ai Ej ) = ψ(Pj (i)Ej ) = Pj (i)ψ(Ej ) = ψ(Ai )ψ(Ej ), de la cual se deduce que A y B tienen el mismo conjunto de valores propios y como ambas son diagonalizables simult´aneamente por una matriz unitaria obtenemos el siguiente resultado.

Teorema 4.2 Si ψ es un BM-isomorfismo de A en B, entonces existe una matriz unitaria U tal que ψ(A) = U ∗ AU para toda A ∈ A. La pregunta obvia, ya que toda matriz de permutaci´on es unitaria, es si la matriz unitaria que define el BM-isomorfismo del teorema anterior es una matriz de permutaci´on. Parece ser que la respuesta es negativa, como lo sugiere F. Jaeger [8], aunque hasta ahora no se cuenta ni con una demostraci´on ni con un contraejemplo.

Definici´ on 4.3 Decimos que ψ es una dualidad de A en B si satisface que 1 ψ(AB) = ψ(A) ◦ ψ(B) y ψ(A ◦ B) = ψ(A)ψ(B) n para toda A, B ∈ A. Es claro que si ψ es una dualidad de A en B, entonces n1 ψ −1 es una dualidad de B en A. Denotaremos por (A, B) a una pareja de ´algebras de Bose-Mesner en donde existe una dualidad, y en este caso diremos que (A, B) forman una pareja dual de ´algebras de Bose-Mesner. En el caso de que la dualidad est´e definida de A en s´ı misma, ψ es llamada una dualidad fuerte si satisface que ψ 2 = nτ , donde τ denota al mapeo trasposici´on. A esta ´algebra se le llama una ´algebra de Bose-Mesner autodual.

Es f´acil ver que la composici´on entre una dualidad y un BM-isomor– fismo es una dualidad. Adem´as, la composici´on entre dualidades, bajo cierta normalizaci´on, es un BM-isomorfismo. Por otra parte, salvo la composici´on por un BM-isomorfismo, las dualidades entre ´algebras de Bose-Mesner son u ´nicas (ver [6]).

Proposici´ on 4.4 [6]. Sea (A, B) una pareja dual de ´ algebras de BoseMesner. Entonces se cumple lo siguiente:

´ ISA´IAS LOPEZ

74

i) Los n´ umeros de intersecci´ on de A son iguales a los param´etros de Krein de B y viceversa. ii) La primera eigenmatriz de A es igual a la segunda eigenmatriz de B y viceversa. iii) Toda dualidad conmuta con el mapeo trasposici´ on.

De las ecuaciones (2.7) y (1.3) tenemos que el conjunto de caracteres de A es {χ0 , χ1 , . . . , χn }, donde χj est´a definida por Ai Ej = Pj (i)Ej = χj (Ai )Ej .

En este caso, a Aˆ la llamaremos la ´algebra primal de caracteres del ´algebra de Bose-Mesner A. De manera ´analoga, viendo a A como un hipergrupo con la base de idempotentes, los caracteres est´an determinados por la ecuaci´on (2.8) y en este caso diremos que Aˆ es la ´algebra dual de caracteres de A. Obs´ervese que el concepto de car´acter es consistente con la Proposici´on 2.4.

El siguiente corolario es consecuencia inmediata de la Proposici´on 4.4.

Corolario 4.5 Sea (A, B) una pareja dual de ´ algebras de Bose-Mesner.

Entonces el ´ algebra primal de caracteres de A es igual al ´ algebra dual de caracteres de B y viceversa. En particular, si A es una ´ algebra de BoseMesner autodual se tiene que las ´ algebras primal y dual de caracteres coinciden.

5

Conclusiones

En esencia, lo que hemos probado es que bajo cierta normalizaci´on de los elementos de la base de una ´algebra de Bose-Mesner, ´esta tiene el mismo comportamiento algebraico operacional de un hipergrupo. M´as a´ un, las ´algebras de Bose-Mesner est´an contenidas en un conjunto m´as grande, el conjunto de los hipergrupos. Por otro lado, viendo a una ´algebra de Bose-Mesner como un hipergrupo, los conceptos de isomorfismos est´an determinados por sus ´algebras de caracteres asociadas.

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

75

Agradecimientos Agradezco a los evaluadores las sugerencias y correcciones realizadas a este trabajo. Asimismo, al Dr. On´esimo Hern´andez-Lerma por la revisi´on ortogr´afica y sugerencias. Este trabajo se realiz´o bajo la supervisi´on del Dr. Isidoro Gitler.

Isa´ıas L´ opez Departamento de Matem´ aticas CINVESTAV-IPN A.P. 14 − 740 M´exico, D.F., C. P. 07000 M´exico. ilopez@math.cinvestav.mx

Referencias [1] E. Bannai and T Ito. Algebraic Combinatorics I. BenjaminCummings, Menlo Park, C. A, 1989.

[2] A. E. Brouwer, A. M. Cohen, A. Neumaier. Distance-Regular Graphs. Springer-Verlag, Berlin, Heidedelberg, New York 1989.

[3] P. Delsarte. An algebraic approach to the association achemes of coding theory, Phillips Research Reports Supplements 10 (1973).

[4] C.F Dunkl. The measure algebra of a locally compact hipergroup.

Trans. Amer. Math. Soc. 179,(1973), No. 32(1), 331-348.

[5] F. R. Gantmacher. The Theory of Matrices, Chelsea Publishing Company, New York, N. Y. 1960.

[6] I. Gitler and I. L´opez. Spin models, association schemes and ∆−Y transformations, Morfismos, Vol. 3, No. 2 (1999), 31-55.

[7] R. Horn and C. R. Johnson. Matrix Analysis, Cambridge University Press, 1991.

[8] F. Jaeger. On Spin models, triply regular association schemes, and duality, Journal of Algebraic Combinatorics 4 (1995), 103-144.

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[9] R. I. Jewett. Spaces with an abstract convolution of measures. Adv. Math. 18(1975), 1-101.

[10] M. Koppinen. On algebras with two multiplications, including Hopf algebras and Bose-Mesner algebras. J. Alg. 182 (1996), 256273.

[11] R. Spector. Measures invariantes sur les hipergroupes. Trans.

Amer. Math. Soc. 239(1978), 147-165.

[12] N. J. Wildberger. Lagrange’s theorem and integrality for finite commutative hipergroups with applications to strongly regular graphs. J. Alg. 182(1996), 1-37.

[13] N. J. Wildberger. Finite commutative hipergroups and aplication from group theory to conformal field theory. Contemporany Mathematics, Vol 183, Amer. Math. Soc., Providence(1995), 413-434.

Morfismos, Vol. 4, No. 2, 2000, pp.77–90

Sincronizaci´on de parejas de aut´omatas celulares J. Guillermo S´anchez Saint-Martin

1

Resumen Consideramos un sistema compuesto por dos aut´omatas celulares unidimensionales definidos en ZZN ın k a partir de la misma regla af´ R : ZZ3k → ZZk . El aut´omata gobernante evoluciona de forma aut´onoma. El aut´omata gobernado copia algunas de las entradas del gobernante despu´es de cada iteraci´on. La pareja de aut´omatas sincroniza si eventualmente el aut´omata gobernado evoluciona id´enticamente al aut´omata gobernante, esto para toda pareja de condiciones iniciales. En este caso, la diferencia entre gobernante y gobernado sigue un comportamiento lineal en ZZN k , de modo que el estudio de la sincronizaci´on equivale a realizar el estudio de la nilpotencia de una matriz en GL(N, ZZk ), la cual depende s´olo de las posiciones relativas de la entradas del gobernante que ser´an copiadas por el gobernado. En general estudiamos la relaci´on entre las propiedades aritm´eticas de k y la sincronizaci´on de los aut´omatas celulares. Tambi´en presentamos una herramienta para demostrar que no hay sincronizaci´on de parejas de aut´omatas celulares en ciertos alfabetos.

2000 Mathematics Subject Clasification: 37B15 Keywords and phrases: Aut´ omatas celulares, Nilpotencia de matrices.

1

Introducci´ on

Los aut´omatas celulares (AC) fueron introducidos por J. Von Neumann a finales de los an ˜os 40, para modelar sistemas biol´ogicos autoreproductivos.

1

Estudiante de doctorado, Ciencias Aplicadas, IICO-UASLP, Becario de CONACYT

77

78

´ GUILLERMO SANCHEZ

En la actualidad los aut´omatas celulares son aplicados a diversas ramas de la ciencia como lo son la f´ısica, la biolog´ıa, la computaci´on, la qu´ımica, etc.

La evoluci´on de un AC es sencilla de generar, y proporciona modelos simples para una gran variedad de fen´omenos [1]. Un AC es una idealizaci´on mat´ematica de sistemas f´ısicos en los cu´ales el espacio fase y el tiempo son discretos. El estado de un aut´omata celular est´a completamente determinado por los valores de todas sus celdas (c´elulas). El AC evoluciona de acuerdo a una regla local que depende de sus valores en una vecindad.

Por otro lado, la sincronizaci´on de sistemas acoplados ha despertado gran inter´es en estudios recientes [2], en particular cuando los sistemas que se acoplan son ca´oticos. Sistemas que muestran este comportamiento son temporalmente ca´oticos, pero espacialmente ordenados o coherentes. La coherencia presente es de un tipo particular: las din´amicas son las mismas o casi las mismas por per´ıodos largos de tiempo para todos los sistemas acoplados o en gran parte de ellos.

Para el acoplamiento unidireccional de dos sistemas din´amicos se necesita un sistema que evoluciona de manera aut´onoma, llamado gobernante y un sistema que evoluciona con cierto grado de libertad pero tambi´en es influido por medio del acoplamiento por el sistema gobernante, raz´on por la cual se le denomina sistema gobernado.

Los aut´omatas celulares (AC) son sistemas din´amicos discretos que evolucionan a pasos discretos en el tiempo. El espacio fase de un AC de tama˜ no N es el conjunto ZZN k de todas las secuencias de N celdas que toman valores en ZZ/kZZ, y la evoluci´on est´a definida por la repetida iteraci´on de la ley de evoluci´on F : ZZN ZN k →Z k . En este trabajo estudiamos algunos aspectos relativos al comportamiento de AC’s afines de rango 1 acoplados unidireccionalmente. En particular nos enfocamos al fen´omeno de la sincronizaci´on de aut´omatas acoplados unidireccionalmente.

En estos sistemas resulta que la sincronizaci´on est´a sujeta a condiciones aritm´eticas en los par´ametros que definen al sistema. Estas relaciones aritm´eticas son punto central de este trabajo.

2

Sincronizaci´ on de aut´ omatas celulares

Un Aut´ omata Celular (AC) es un sistema din´amico discreto definido a partir de:

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79

1. Un alfabeto A (conjunto finito de s´ımbolos) 2. Una regla local f : A2r+1 → A, donde el entero r es llamado rango del aut´omata.

La ley de evoluci´on F : AN → AN es tal que F (a0 a1 ...an−1 ) = b0 b1 ...bn−1 , donde bi = f (ai−r ...ai ...ai+r ) y la suma de ´ındices dentro del par´entesis debe entenderse m´odulo IN. Para este caso tomamos A = ZZk , k ∈ IN, por lo tanto el espacio fase es ZZN k , al cual dotamos con las operaciones usuales coordenada a coordenada en ZZk .

Un aut´omata celular es af´ın si su regla local es de la forma !

f (ai−r ...ai ...ai+r ) = rj=−r βj ai+j + c, con βj ∈ ZZk , donde a (ai−r ...ai ...ai+r ) le llamamos vecindad de ai de tama˜ no r.

2.1

Condiciones de frontera

Para generar un crecimiento con un n´ umero constante de celdas en cada tiempo, es necesario especificar condicones de frontera, en nuestro caso consideramos el estado del AC con sus celdas arregladas en forma de anillo, de tal manera que la u ´ltima celda es a su vez la celda anterior a la primera celda, a este arreglo le llamamos condiciones de frontera peri´ odicas.

2.2

Sincronizaci´ on en AC

Recordaremos c´omo se acoplan unidireccionalmente dos sistemas din´amicos. El sistema gobernante est´a relacionado con el gobernado por medio de una funci´on, la cual considera parte de la evoluci´on del sistema gobernante en el gobernado.

En el caso del AC, la informaci´on de cu´ales coordenadas son copiadas en el AC gobernado est´a dada por una secuencia de acoplamiento κ = (κ0 , κ1 , · · · , κN −1 ) de 0’s y 1’s. Las coordenadas de κ que son iguales a 1 son llamadas coordenadas de acoplamiento.

Si los estados al tiempo t del gobernante y el gobernado son xt y y t , respectivamente, entonces el estado al tiempo t + 1 del gobernante es xt+1 = F (xt ), y para el gobernado tenemos y t+1 = κ ∗ F (xt ) + (1 − κ) ∗ F (y t ) donde u∗v es el producto coordenada a coordenada de los vectores u y v. Aqu´ı 1 representa un vector formado de 1’s y la sincronizaci´on se

´ GUILLERMO SANCHEZ

80

da cuando la discrepancia al tiempo t entre el gobernado y el gobernante es nula. Esto es !

"

z t = xt − y t = (1 − κ) ∗ F (xt−1 ) − F (y t−1 ) = 0.

3

Nilpotencia de matrices y sincronizaci´ on

El caso en el cual se considera una regla local af´ın f que toma en cuenta s´olo a los primeros vecinos, es conocido como caso af´ın de rango 1 [4].

En este caso la ley de evoluci´on puede ser representada por una matriz, esto es: f (x) = Mf x + C, ⎛

con

y

⎜ ⎜ ⎜ Mf = ⎜ ⎜ ⎜ ⎝

β2 β3 0 β1 β2 β3 0 β1 β2 .. .

0 0 β3 .. .

0

0

β3

0

⎛

⎜ ⎜ ⎜ ⎜ ⎜ C=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

c c c .. .

⎞

0 0 0

⎞

· · · β1 ··· 0 ⎟ ⎟ ⎟ ··· 0 ⎟ .. ⎟ ⎟ . ⎠

· · · β1

(1)

β2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ c ⎟ ⎟ c ⎠

c

La discrepancia esta dada por: ) * z t = xt − y t = F (xt−1 ) − κ ∗ F (xt−1 ) + (1 − κ) ∗ F (y t−1 ) ) * = Mf xt−1 +C −κ(Mf xt−1 )−C −(1−κ)∗ Mf y t−1 + C = (1 − κ) ∗ Mf xt−1 − (1 − κ) ∗ Mf y t−1 = (1 − κ) ∗ Mf (xt−1 − y t−1 ) = (1 − κ) ∗ Mf z t−1 .

Esto es, la discrepancia al tiempo t depende s´olo de la discrepancia al tiempo t − 1, y por lo tanto la sincronizaci´on se da si existe un T ∈ IN tal que Mκ,f T = 0 para todo z ∈ ZZN k , donde Mκ,f ≡ (1 − κ) ∗ Mf .

La matriz Mκ,f tiene la forma:

´ DE AUTOMATAS ´ SINCRONIZACION

⎛

β2 ⎜ β1

Mκ,f

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

β3 β2

0 0 0 0 0 0 0 0 0

··· ··· ··· 0 0 0 ··· 0 0

β3

0

0 β3 .. . β1 0 0 ··· ··· ··· 0 ··· ··· .. .

0

... ...

0 0

β2 β1 0 0 0 0 0 0 0

β3 β2 ··· 0 0 0 ··· 0 0

0 0 .. .

0 0 0 0 0 0 0 0 0

···

0

0

0 0

··· ···

0 0

0 0 0 β2 β1 0 0 0 0 .. .

0

0 ··· ··· β3 β2 β1 0 0 0 0

81

⎞

0 0

··· ···

0 0

··· 0 0 0 β3 β2 0 0 0

0 0 .. .

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 β2 β1

0 0 0 0 ··· ··· 0 ··· β3

0 0 0 0 0 0 0 0 ···

0

0

0

··· ··· ··· ··· 0 0 ··· β3 β2 .. . ···

β1

β2

β1 0 ⎟

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Como los bloques de la matriz Mκ,f son independientes, para calcular la nilpotencia de esta matriz basta calcular la nilpotencia de sus bloques.

Utilizando el teorema de Hamilton-Cayley [5] podemos concluir que una matriz de rango ℓ es nilpotente si su polinomio caracter´ıstico es de la forma Pℓ (λ) = λℓ .

Definici´ on: Si una submatriz de Mk,f de tama˜ no ℓ×ℓ es nilpotente, entonces ℓ es una longitud de sincronizaci´on para el sistema acoplado definido por Mf y k.

Los polinomios de las submatrices de Mk,f obedecen la siguiente f´ormula recursiva: Pℓ (λ) = (λ − β2 )Pℓ−1 (λ) − β1 β3 Pℓ−2 (λ). donde Pℓ (λ) es el polinomio caracter´ıstico de una submatriz de tama˜ no ℓ × ℓ.

En conclusi´on, la sincronizaci´on de parejas de aut´omatas celulares afines de rango 1 se reduce al estudio de la nilpotencia de matrices del tipo Mκ,f . Como una matriz por bloques es nilpotente si sus bloques lo son, entonces basta estudiar la nilpotencia de bloques del tipo ⎛

β2 β1 .. .

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

0

β3 β2

0 ··· β3 · · · .. .

· · · β1 β 2 0 · · · β1

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ β3 ⎠

β2

82

4

´ GUILLERMO SANCHEZ

Clases de equivalencia y no sincronizaci´ on

Una vez fijo el alfabeto ZZk , sobre el cual definimos nuestros aut´omatas, las matrices que consideraremos est´an completamente determinadas por tripletas (β1 , β2 , β3 ) ∈ (ZZk )3 .

Definici´ on: Dos tripletas son equivalentes si y s´olo si los polinomios Pℓ que definen son iguales.

Definici´ on: La tripleta (β1 , β2 , β3 ) es similar a la tripleta (β˜1 , β˜2 , β˜3 ) (est´an en la misma clase), si β1 = β˜1 y β2 = β˜2 .

Dado que la nilpotencia s´olo depende del tipo de polinomio caracter´ıstico, podemos reducir el estudio de todas las tripletas al estudio de una tripleta representante por clase.

La tripleta (µ, β2 , ν) para las cuales µν ̸= 0 ser´a representada por (η, β2 , 1) donde µν = η.

Definici´ on: Las tripletas triviales son aquellas (η, 0, 0) y (0, 0, η).

Y son tales que Pℓ (λ) = λℓ ∀ ℓ.

Por lo anterior, para los anillos ZZk en lugar de estudiar las k 3 posibilidades para las tripletas (β1 , β2 , β3 ), s´olo se analizar´an k 2 − k.

M´as a´ un, se clasificar´an las tripletas (β1 , β2 , β3 ) de acuerdo al valor del producto β1 β3 y al valor de β2 dependiendo de si es igual a cero o no, de manera que tenemos 2(k − 1) clases para el anillo ZZk .

Denotaremos las clases de la forma siguiente: para el caso en que β2 =0 tenemos la clase (η, 0, 1), y si β2 ̸= 0 tenemos (η, ¯0, 1).

4.1

Reducci´ on de ZZk [λ]

Para reducir el anillo de polinomios ZZk [λ] a un anillo finito utilizaremos el siguiente hecho: Sea φi : ZZk [λ] → ZZk la transformaci´on definida por (a0 + a1 λ + ... + an λn )φi = (a0 + a1 i + ... + an in ) mod k la cual es un homomorfismo de ZZk [λ] en ZZk .

Consideremos ahora los ideales Ji = {p : pφi = 0} para i = 0, ..., k − 1. Entonces J = ∩k−1 en un ideal y por lo tanto el cociente i=1 Ji es tambi´ ZZk [λ]/J es un anillo [6].

Al polinomio de grado menor en cada clase le llamaremos representante can´ onico, el cual a lo m´as tiene grado k − 1. Denotaremos al

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representante can´onico de la clase [Pℓ (λ)] donde Pℓ (λ) es el polinomio de alguna tripleta, por Pℓ (λ) = aℓk−1 λk−1 + aℓk−2 λk−2 + ... + aℓ0 .

Es f´acil verificar que la suma de representantes can´onicos es tambi´en un representante can´onico. Por otro lado el representante can´onico de un producto de clases se puede calcular a partir de los representantes de cada clase como sigue: sea P (λ) = ak−1 λk−1 + ak−2 λk−2 + ... + a0 el representante de la clase [P ] y Q(λ) = bk−1 λk−1 + bk−2 λk−2 + ... + b0 el representante de la clase [Q], entonces el representante de la clase [F ] = [P ][Q] es f (λ) = ck−1 λk−1 + ck−2 λk−2 + ... + c0 donde los ci = !

Zk → ZZk es una funci´on que se puede s+t=τ (i) as bt mod k, donde τ : Z determinar y depende en general de k.

Para el caso de alfabetos cuyas cardinalidades son n´ umeros primos τ est´a dada por

τ (i) =

"

p−1 j

si s + t ≡ 0 mod p − 1 , si s + t ≡ j mod p − 1.

La relaci´on de recurrencia entre polinomios caracter´ısticos en ZZk [λ], induce una relaci´on de recurrencia entre los representantes can´onicos de las clases del anillo reducido ZZkk [λ] a trav´es de una relaci´on lineal entre los coeficientes: ℓ+1 ℓ (aℓ−1 i , ai ) → (ai ), para i = k − 1, ..., 0.

Ya que el objetivo es encontrar todos los polinomios para todas las longitudes de sincronizaci´on utilizando esta recurrencia, y dado que esta depende de una matriz de tama˜ no 2k×k que no se puede iterar, entonces nos valdremos de un desplazamiento para generar una recurrencia lineal del tipo: ℓ ℓ ℓ+1 (aℓ−1 i , ai ) → (ai , ai ), para i = k − 1, ..., 0.

Esta depende de la siguiente matriz de orden 2k×2k, que denotaremos por Hk y que s´ı puede ser iterada para encontrar los coeficientes de los polinomios:

´ GUILLERMO SANCHEZ

84

⎛

0 0

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ −β β 1 3 ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

0

5

0 0

··· ··· .. .

0 ··· 0 ··· 0 0 −β1 β3 0 .. .

0 0

0 0 0 0 ··· ···

0 0 ··· 0 0 0 0

··· · · · −β1 β3 0 ··· 0 −β1 β3

1 0

0 1 .. .

··· ···

0 0 ··· 0 0 ··· −β2 1 0 0 −β2 1 .. .

1 0

0 0

0 0

0 0

1 0 ··· ···

0 1 0 0

· · · −β2 1 ··· 0 −β2

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Resultados computacionales

De la relaci´on lineal entre los coeficientes de los representantes can´onicos podemos observar los siguientes 2 hechos: 1. La matriz es eventualmente peri´odica, es decir existe un k < l, tal que M k = M l , esto por la finitud del anillo de matrices M2k×2k [ZZk ].

2. Si Pℓ (λ) = λℓ en ZZk [λ], entonces Pℓ ∈ [Pℓ ] tiene la forma λχ(ℓ) , donde χ(ℓ) puede ser determinado.

El Hecho 1 implica que s´olo hay un n´ umero finito de casos que analizar. Si para una tripleta dada ninguno de los representantes can´onicos generados son del tipo λχ(ℓ) , entonces no puede haber longitudes de sincronizaci´on. no Consideremos la funci´on n #→ H n , donde H es la matriz de tama˜ 2k × 2k definida en la ecuaci´on anterior y donde las potencias son evaluadas en M2k×2k [ZZk ]. Esta funci´on es eventualmente peri´odica, es decir, existen T y P ∈ IN tales que H T = H T +P en ZZk .

En el caso en que k es primo, la funci´on n #→ H n es estrictamente peri´odica es decir, existe P ∈ IN tal que H = H P . En este caso decimos que la matriz H tiene per´ıodo P .

Para realizar el algoritmo computacional utilizamos el paquete scilab del sistema operativo UNIX que fue dise˜ nado en IRIA [3]. Este algoritmo nos indica para cada k, las tripletas y las longitudes para las cuales el representante can´onico de Pℓ es de la forma λχ(ℓ) , iterando 100,000

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85

polinomios. Se verific´o que la mitad de las k 2 − k matrices H posibles en ZZp , para p primo mayor que 2, tienen per´ıodo p2 − 1 y la otra mitad tiene per´ıodo 12 p(p2 − 1). Aun no podemos determinar a cuales de las matriz Hp le corresponde cada per´ıodo.

6

Resultados

Para las tripletas de la clase (η, 0, 1) no podemos aplicar el algoritmo antes mencionado. Para mostrar la sincronizaci´on enunciamos el siguiente teorema.

Teorema 6.0.1 Sea k ∈ N primo o potencia de primo, y sea C ∈ ZZk tal que (C, k) = 1.

Consideremos la familia Pℓ (λ) de polinomios en ZZk [λ], definidos por la recurrencia Pℓ (λ) = λPℓ−1 (λ) + CPℓ−2 (λ).

No existe ℓ ∈ N tal que Pℓ (λ) = λℓ .

Demostraci´ on: Al calcular los polinomios por inducci´on obtenemos P2m (λ) = λ2m + Xm (λ) + C m y P2m+1 (λ) = λ2m+1 + 2mCλ2m−1 + Ym (λ) + (m + 1)C m λ donde Xm (λ) y Ym (λ) son polinomios de grado 2m − 2 sin t´ermino independiente ni lineal.

Parte 1: P2m (λ) ̸= λ2m , ya que C m no es congruente con 0 mod k, pues (C,k)=1.

Parte 2: P2m+1 (λ) ̸= λ2m+1 .

Prueba de Parte 2: Si P2m+1 (λ) = λ2m+1 , entonces 2mC ≡ 0 y (m + 1)C m ≡ 0 m´odulo k, y como (C, k) = 1 esto implica que 2m ≡ 0 y m + 1 ≡ 0 m´odulo k.

Probaremos que esto no es posible.

Si k = pn , entonces 2m = βpn y m + 1 = αpn (con p primo).

Caso 1: Si p = 2 y n ≥ 2, entonces m = β2n−1 y m + 1 = α2n lo cual es una contradicci´on, pues hace a m par e impar a la vez.

Caso 2: Si p > 2 y n≥1, entonces 2m = βpn y m + 1 = αpn .

La primera de estas igualdades implica que m = β ′ pn , por lo tanto (α − β)pn = 1. Esto es imposible pues 2 no es divisible por p, de modo

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que 2 es divisible forzosamente por β, lo cual har´ıa que m y m+1 fueran simult´aneamente m´ ultiplos de k. ✷ Del teorema 6.0.1 se deriva el siguiente teorema: Teorema 6.0.2 La pareja de AC con evoluci´ on f (x) = M x + C f (y) = κM x + (1 − κ) ∗ M y + C donde M ∈ MN [ZZp ] tiene la forma de la ecuaci´ on (1), con (β1 , β2 , β3 ) ∈ (η, 0, 1) y con η ̸= 0, no sincronizan para κ ̸= 0.

Teorema 6.0.3 Si conocemos las tripletas y sus longitudes de sincronizaci´ on en alfabetos de la forma [ZZm × ZZn ], para (m, n) = 1, podemos predecir las tripletas y las longitudes de sincronizaci´ on en [ZZmn ].

Demostraci´ on: Sea ϕ:ZZmn → ZZm × ZZn , tal que (m, n) = 1. Sea ϕˆ : Mℓ [ZZmn ] → Mℓ [ZZm × ZZn ] el isomorfismo inducido por ϕ. ϕˆ es tal que ϕ(A)(i, ˆ j) = ϕ(A(i, j)).

Es f´acil ver que si M ∈ Mℓ [ZZmn ] es nilpotente, ϕ[M] ˆ ∈ Mℓ [ZZm × ZZn ] tambi´en es nilpotente para (m, n) = 1 y viceversa. ✷ Ejemplo Podemos predecir que la longitud de sincronizaci´on para la tripleta (5, 0, 1) en ZZ310 es ℓ = 2k − 1, pues sabemos que la tripleta (1, 0, 1) en ZZ32 tiene esta longitud de sincronizaci´on.

6.1

Resultados particulares

Las longitudes de sincronizaci´on para el sistema (ecuaci´on 1), en el caso en que M ∈ MN [ZZk ] y k no es primo, dependen fuertemente de la factorizaci´on de k. Del Tma. 6.0.3 se deduce f´acilmente lo que pasa cuando k = pq con p y q primos diferentes.

A continuaci´on describimos el comportamiento de las longitudes de sincronizaci´on, para algunos casos en que k es potencia de primo.

La f´ormula recursiva para las tripletas de la clase (0, 2, 0) en ZZ34 es de la siguiente forma: Pℓ (λ) = (2 − λ)ℓ Haciendo un c´alculo sencillo podemos observar que si ℓ es par sus polinomios caracter´ısticos tienen la forma Pℓ (λ) = λℓ , por lo tanto todo par es longitud de sincronizaci´on, lo mismo pasa para las tripletas de la clase (0, 4, 0) en ZZ38 . Procediendo de manera similar podemos demostrar que ℓ = 4m, m = 1, 2, 3, ... son longitudes de sincronizaci´on para las tripletas de las clases (0, 2, 0) y (0, 6, 0) en ZZ38 .

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Para la clase (2, 0, 1), en ZZ34 la f´ormula de recurrencia es: Pℓ+2 (λ) = λPℓ+1 (λ) − 2Pℓ (λ) y sustituyendo Pℓ+1 (λ) = λPℓ (λ) − 2Pℓ−1 (λ) obtenemos Pℓ+2 (λ) = λ(λPℓ − 2Pℓ−1 ) − 2Pℓ Pℓ+2 (λ) = λ2 Pℓ (λ) − 2λPℓ−1 (λ)) − 2Pℓ (λ), y sustituyendo λPℓ−1 (λ) = Pℓ (λ) + 2Pℓ−2 (λ), para ℓ ≥ 2, obtenemos Pℓ+2 (λ) = λ2 Pℓ (λ) − 2Pℓ (λ) − 4Pℓ−2 (λ) − 2Pℓ (λ).

Por lo anterior Pℓ+2 (λ) = λ2 Pℓ (λ) ∀ℓ ≥ 2, de modo que Pℓ+2 (λ) =

!

P2ℓ (λ) = λ2(ℓ−1) P2 (λ) si ℓ es par, P2ℓ+1 (λ) = λ2(ℓ1 ) P3 (λ) si ℓ es impar.

Dado que P0 (λ) = 1 y P1 (λ) = λ, entonces P2 (λ) = λ2 − 2, P3 (λ) = Por lo tanto, cuando ℓ es impar tenemos P2ℓ+1 (λ) = λ2ℓ+1 , por lo que todos los impares > 2 son longitudes de sincronizaci´on.

Para ℓ par tenemos

λ3 .

P2ℓ (λ) = λ2(ℓ)−1 (λ2 − 2), de modo que no hay sincronizaci´on para estas longitudes.

En ZZ38 , para la clase (4, 0, 1), podemos realizar un c´alculo similar y concluir que hay sincronizaci´on para ℓ impar.

Para ZZ39 las tripletas de la clase (0, 3, 0) tienen f´ormula recursiva de la forma: Pℓ (λ) = (λ + 3)ℓ .

Es f´acil ver que ℓ = 3m, es longitud de sincronizaci´on para m = 1, 2, 3, ... Para la clase (0, 6, 0), las longitudes de sincronizaci´on son las mismas. Por otro lado, para las tripletas de la clase (3, 0, 1), la f´ormula de recurrencia es: Pℓ (λ) = λPℓ−1 (λ) − 3Pℓ−2 (λ), de modo que P0 (λ) = 1, P1 (λ) = λ, P2 (λ) = λ2 − 3, y P3 (λ) = λ3 − 6λ.

Probaremos por inducci´on que

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P4+3m (λ) = λ4+3m P5+3m (λ) = λ5+3m − 3λ3+3m P6+3m (λ) = λ6+3m − 6λ4+3m para todo m = 0, 1, 2, ...

Si m = 0, se verifica directamente que P4 (λ) = λ4 , P5 (λ) = λ5 −3λ3 , y P6 (λ) = λ6 − 6λ4 .

Suponemos que la hip´otesis se cumple para un cierto m, entonces P4+3(m+1) (λ) = P7+3m (λ) = λP6+3m (λ) − 3P5+3m (λ) =λ(λ6+3m − 6λ4+3m ) - 3(λ5+3m − 3λ3+3m ) =λ7+3m − 6λ5+3m - 3λ5+3m + 9λ3+3m .

Como las operaciones se realizan mod 9, tenemos que P7+3m (λ) = λ7+3m .

Procediendo de manera similar tenemos que P8+3m (λ) = λ8+3m − 3λ6+3m , y P9+3m (λ) = λ9+3m − 6λ7+3m .

Por lo tanto, ℓ = 4 + 3m es longitud de sincronizaci´on m = 0, 1, ...

Finalmente, para la clase (6, 0, 1) podemos realizar c´alculos similares demostrando que hay sincronizaci´on para las mismas longitudes.

7

Comentarios finales

En este trabajo se analiz´o la sincronizaci´on de parejas de aut´omatas celulares afines de rango 1, en los anillos ZZk .

Despu´es de realizar el estudio computacional de los anillos ZZ4 , ZZ8 y ZZ9 , encontramos para qu´e tripletas existen longitudes de sincronizaci´on, as´ı como tambi´en que no hay otras tripletas que sincronicen en estos anillos.

Demostramos, ayudados por la computadora, que no hay sincronizaci´on, exceptuando las tripletas triviales, en los campos ZZp para p de 3 a 29.

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89

Para los alfabetos ZZk , k = pq con (p, q) = 1 y por el Teorema 6.0.3 podemos conocer las tripletas y sus longitudes de sincronizaci´on a partir de los resultados obtenidos en ZZp y ZZq .

En todos los campos ZZp , p > 2 que analizamos, la matriz Hp de recurrencia entre los representantes can´onicos es tal que Hpm = 1, donde en la mitad de las matrices posibles m = (p2 − 1) y en la otra mitad m = 12 p(p2 − 1).

A partir de los resultados anteriores podemos conjeturar que la forma de Hp es responsable de: • Per´ıodos de la forma (p2 − 1) y 12 p(p2 − 1)).

• Ausencia de representantes can´onicos de la forma λχ(ℓ) .

De aqu´ı se deducir´ıa que para los alfabetos ZZp con p primo > 2, las u ´nicas tripletas con longitudes de sincronizaci´on son las triviales. Esto se demostr´o para las tripletas del tipo (µ, 0, ν), µν ̸= 0.

Agradecimientos Agradezco a los Dres. Valentin Afraimovich, Gelasio Salazar y Jes´ us Urias por sus ense˜ nanzas, las cuales fueron de gran ayuda para la realizaci´ on de este trabajo.

Un agradecimiento especial al Dr. Edgardo Ugalde por sus invaluables consejos durante la realizaci´on de este trabajo.

J. Guillermo S´ anchez Saint-Martin IICO–UASLP, Av. Karakorum 1470, Lomas 4a.

San Luis Potos´ı, S.L.P. 78210 M´exico. jsmartin@cactus.iico.uaslp.mx

Referencias [1] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys. 55, 601 (1983).

[2] V. S. Afraimovich, N. N. Verichev, and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative systems, Izv. Vyssh. Uchebn. Zaved. Radiofiz 29, 1050 (1986) [Sov. Radiophys.

29, 795 (1986)].

[3] htpp://www.rocq.iria.fr/scilab/scilab.html

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[4] J. Urias, G. Salazar y E. Ugalde, Synchronization of cellular Automaton Pairs, CHAOS 8, 814 (1998).

[5] A. I. M´altsev, Fundamentos de algebra lineal, Editorial Mir Mosc´ u.

(1972).

[6] J. B. Fraleigh, Algebra abstracta, Addison-Wesley iberoamericana.

(1987).

MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matem´aticas del CINVESTAV, se termin´ o de imprimir en el mes de septiembre de 2000 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contenido

Algebraic K-theory and the η-invariant Jos´e Luis Cisneros Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Average optimal strategies in Markov games under a geometric drift condition Heinz-Uwe Ku ¨enle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Little cubes and homotopy theory Dai Tamaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Hipergrupos y a ´lgebras de Bose-Msner Isa´ıas L´ opez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Sincronizacio ´n de parejas de auto ´matas celulares J. Guillermo S´ anchez Saint-Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77