Morfismos, Vol 4, No 2, 2000 by Morfismos, Department of Mathematics, Cinvestav

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

MORFISMOS Comunicaciones Estudiantiles Departamento de Matemáticas Cinvestav Editores Responsables • Isidoro Gitler • Jes´ us González • Isa´ıas López

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Editores Asociados • Ricardo Berlanga • Samuel Gitler • Emilio Lluis Puebla • Guillermo Pastor • V´ıctor P´erez Abreu • Carlos Prieto • Carlos Renter´ıa • Luis Verde

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

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Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. MORFISMOS appears twice a year. Publication of papers is not restricted to students of CINVESTAV; we want to encourage students in Mexico and abroad to submit papers. Mathematics research reports or summaries of bachelor, master and Ph.D. theses will be considered for publication, as well as invited contributed papers by researchers. Papers submitted should be original, either in the results or in the methods. The Editors will assign as referees well–established mathematicians. Even though MORFISMOS is a refereed journal, the papers will be considered as preliminary versions which could later appear in other mathematical journals. If you have any suggestions about the journal, let the Editors know and we will gladly study the possibility of implementing them. We expect this journal to foster, as a preliminary experience, the development of a correct style of writing mathematics. 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

∗

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.

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.

´ LUIS CISNEROS MOLINA JOSE

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

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

(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)

E(R) =

EN (R).

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

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

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

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

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.

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

Cl1 = C

with basis

1, e1

Cl2 = H

with basis

1, e1 , e2 , e1 e2

´ 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

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.

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.

´ LUIS CISNEROS MOLINA JOSE

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

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.

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

´ LUIS CISNEROS MOLINA JOSE

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 ).

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

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).

´ 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

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

¨ 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.

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

— 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 ñ from σHn to σA [˜ ρn from σHn to σB ] exists such that πn (· | x0 , a0 , b0 , . . . , xn ) = π ñ (· | xn ) [ ρn (· | x0 , a0 , b0 , . . . , xn ) = ρñ (· | xn )] for all (x0 , a0 , b0 , . . . , xn ) ∈ Hn × A. (Notation: We identify πn as π ñ and ρn as ρñ .) 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

¨ 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

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

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

(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)

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

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. ✷

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

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

(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)

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 ,

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

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

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)

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

(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

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

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

[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

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 .

∗

Invited article

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:

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.

LITTLE CUBES AND HOMOTOPY THEORY

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

⑦

!∗

✯

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

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

⑦

!∗

✯

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

✲

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.

Basic Definitions and Fundamental Facts

Let us begin with a precise definition of little cube.

LITTLE CUBES AND HOMOTOPY THEORY

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.

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

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 ❅

❅

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

DAI TAMAKI

make the following diagram commutative Ωn X

E n✲

◗

Ωn Σn (Ωn X)

◗ ◗

◗ # ◗

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

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

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].

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

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

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.

LITTLE CUBES AND HOMOTOPY THEORY

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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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

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

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 .

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

N ✲ C (N ) × ∞ ΣN (π1 (B)/π1 (E))

❄

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

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

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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{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 )

˜ −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

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.

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

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) "

" ✒ "

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

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.

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.

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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LITTLE CUBES AND HOMOTOPY THEORY

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[33] Kato F., On the space realizing the intermediate stages of the Hurewicz homomorphism., Master’s thesis, Department of Mathematics, Shinshu University, March 1996. [34] Kuhn Nicholas J., The geometry of the James-Hopf maps., Pacific Journal of Mathematics, 102 (1982), 397–412. [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.

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

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

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

Introducci´ on

Existen varios objetos matemáticos cuya esencia es el de un esquema de asociación y que se conocen con varios nombres, pero esencialmente son el mismo concepto matemático; por ejemplo, álgebras de adyacencia, álgebras de Bose-Mesner, anillo centralizador, anillo de Hecke, anillo de Schur, álgebra de caracteres, hipergrupos, grupos probabil´ısticos. Los hipergrupos son una generalización de los grupos, en donde el producto de dos elementos está determinado por una función de distribución. Existe mucha investigacioń en esta área, sobre todo para generalizar conceptos básicos de la teor´ıa de grupos, por ejemplo, en hipergrupos existe el teorema de Lagrange. En el presente trabajo demostramos que toda álgebra de Bose-Mesner, bajo cierta normalización, 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 visión distinta de conceptos tales como isomorfismos y dualidades en álgebras 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.

´ ISA´IAS LOPEZ

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 interpretación f´ısica es la siguiente: Un hipergrupo conmutativo finito es una colección de part´ıculas, digamos {c0 , c1 , . . . , cn }, en las cuales está permitido la interacción por colisión 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 colisión; la llamamos un fotón. Cada part´ıcula tiene una antipart´ıcula la cual está especificada por la siguiente regla: La colisión de dos part´ıculas tiene una probabilidad positiva de resultar en un fotón si y sólo si las dos part´ıculas son anti-part´ıculas una de la otra. La interacción de las part´ıculas son independientes del orden, del tiempo y de su posición en el espacio. La estructura del sistema está 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

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 condición (A4) parezca más fácil de probar que (A4′ ), esto no siempre es as´ı, es por eso la siguiente proposición. 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, sólo consideraremos hipergrupos que son conmutativos. Proposici´ on 1.3 La *-´ algebra es semisimple. Demostraci´ on: Se sigue del hecho de que la *-álgebra no tiene elementos idempotentes. Para a ∈ A, ad(a) ∈ End(A) denotará el operador multiplicación por a.

´ ISA´IAS LOPEZ

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 álgebra ad(A) ⊂ End(A) tiene dimensión n + 1 y es conmutativa y semisimple. Esto nos dice que End(A) es isomorfa al álgebra 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

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 *-álgebra Cn+1 y esta tiene exactamente n + 1 caracteres que satisfacen las condiciones establecidas. Claramente la función idénticamente 1 es un carácter 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 .

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

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 *-álgebra con unidad χ0 bajo la multiplicación punto a punto y conjugación 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).

´ Algebras de Bose-Mesner

El concepto de esquema de asociación es importante en álgebra combinatoria. Dicho concepto aparece en el estudio de códigos, dise˜ nos,

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

gráficas de distancia regular, invariantes de nudos y en muchos otros temas. El estudio de los esquemas de asociación lo inició Delsarte en 1973 (ver [3]). En el caso simétrico los esquemas de asociación son esencialmente particiones de una gráfica completa en subgráficas regulares que están relacionadas entre s´ı de alguna manera espec´ıfica. Para un análisis 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)

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 álgebra generada por {Ai |i = 0, 1, . . . , d} sobre C es una álgebra 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 álgebra, la cual es conocida como el ´ algebra de Bose-Mesner del esquema de asociación de clase d. Esta álgebra tiene la propiedad de ser cerrada bajo trasposición compleja. En otro contexto, a este tipo de álgebras también se les llama ´ algebras Doble-Frobenius (ver [10]). En el caso en que t Ai = Ai para todo i, tenemos una álgebra de Bose-Mesner simétrica. 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

n0 = 1, ni = nσ(i) , n = n0 + n1 + · · · + nd . La siguiente proposición es importante para relacionar las álgebras de Bose-Mesner con los hipergrupos. Su demostración 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 simultáneamente por una matriz unitaria. As´ı, podemos encontrar una descomposición de Cn como suma directa de d + 1 eigenespacios de dimensión fj , 0 ≤ j ≤ d (ver [5] y [7]). Además, como J pertenece al álgebra, 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 álgebra de Bose-Mesner, cada una de ellas corresponde a los proyectores de Cn en cada diferente eigenespacio, además, 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

´ ISA´IAS LOPEZ

La ecuación (2.7) muestra que cada vector columna de Ej es un vector propio de Ai asociado con el valor propio Pj (i). Análogamente, la ecuación (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 parámetros de Krein tienen las siguientes propiedades, similares a los de la Proposición 2.2. La demostración de esta proposición 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

l Q (i). qjk l

Hipergrupos y ´ algebras de Bose-Mesner

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

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

es I. Asimismo, A es una álgebra 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 asociación y A = ⟨A0 , A1 , . . . , Ad ⟩ el álgebra de Bose-Mesner generada por dicho esquema. En particular, A es una álgebra asociativa, conmutativa, con identidad A0 y con involución el mapeo traspuesta conjugada. Además, claramente se tiene (A1) y (A2), de modo que sólo resta probar (A3). Para esto, observemos que de la Proposición 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

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 }.

Isomorfismos en ´ algebras de Bose-Mesner

Algunos de los ejemplos clásicos de álgebras de Bose-Mesner son los obtenidos a través de gráficas fuertemente regulares y de distancia regular [2]. La clasificación de gráficas fuertemente regulares y de distancia regular se realiza por medio del álgebra de Bose-Mesner asociada. Aunque la clasificación de gráficas de distancia regular es un poco más compleja, ambas ocupan el concepto de BM-isomorfismo, que se define en esta sección. Otra de las aplicaciones de la teor´ıa de esquemas de asociación es la clasificación de modelos spin (invariantes de nudos) (ver [6]), en donde juega un papel importante el concepto de dualidad entre álgebras de Bose-Mesner. Sean A y B dos álgebras 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 través de una matriz de permutación. Es decir, si P es una matriz de permutación, 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 =

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

1, . . . , d} y {ψ(Ei ) | i = 1, . . . , d} son la base usual y la base de idempotentes ortogonales de B, respectivamente. Usando la ecuación (2.7) obtenemos la siguiente relación 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 simultáneamente 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 permutación es unitaria, es si la matriz unitaria que define el BM-isomorfismo del teorema anterior es una matriz de permutación. Parece ser que la respuesta es negativa, como lo sugiere F. Jaeger [8], aunque hasta ahora no se cuenta ni con una demostración 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 álgebras de Bose-Mesner en donde existe una dualidad, y en este caso diremos que (A, B) forman una pareja dual de álgebras de Bose-Mesner. En el caso de que la dualidad esté definida de A en s´ı misma, ψ es llamada una dualidad fuerte si satisface que ψ 2 = nτ , donde τ denota al mapeo trasposición. A esta álgebra se le llama una álgebra de Bose-Mesner autodual. Es fácil ver que la composición entre una dualidad y un BM-isomor– fismo es una dualidad. Además, la composición entre dualidades, bajo cierta normalización, es un BM-isomorfismo. Por otra parte, salvo la composición por un BM-isomorfismo, las dualidades entre álgebras de Bose-Mesner son u ńicas (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

i) Los n´ umeros de intersecci´ on de A son iguales a los paramétros 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 está definida por Ai Ej = Pj (i)Ej = χj (Ai )Ej . En este caso, a Aˆ la llamaremos la álgebra primal de caracteres del álgebra de Bose-Mesner A. De manera ánaloga, viendo a A como un hipergrupo con la base de idempotentes, los caracteres están determinados por la ecuación (2.8) y en este caso diremos que Aˆ es la álgebra dual de caracteres de A. Obsérvese que el concepto de carácter es consistente con la Proposición 2.4. El siguiente corolario es consecuencia inmediata de la Proposición 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.

Conclusiones

En esencia, lo que hemos probado es que bajo cierta normalización de los elementos de la base de una álgebra de Bose-Mesner, ésta tiene el mismo comportamiento algebraico operacional de un hipergrupo. Más a´ un, las álgebras de Bose-Mesner están contenidas en un conjunto más grande, el conjunto de los hipergrupos. Por otro lado, viendo a una álgebra de Bose-Mesner como un hipergrupo, los conceptos de isomorfismos están determinados por sus álgebras de caracteres asociadas.

´ HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER

Agradecimientos Agradezco a los evaluadores las sugerencias y correcciones realizadas a este trabajo. Asimismo, al Dr. Onésimo Hernández-Lerma por la revisión ortográfica y sugerencias. Este trabajo se realizó bajo la supervisión 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.

´ ISA´IAS LOPEZ

[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

Sincronización de parejas de autómatas celulares J. Guillermo Sánchez Saint-Martin

Resumen Consideramos un sistema compuesto por dos autómatas celulares unidimensionales definidos en ZZN ın k a partir de la misma regla af´ R : ZZ3k → ZZk . El autómata gobernante evoluciona de forma autónoma. El autómata gobernado copia algunas de las entradas del gobernante después de cada iteración. La pareja de autómatas sincroniza si eventualmente el autómata gobernado evoluciona idénticamente al autómata 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 sincronización equivale a realizar el estudio de la nilpotencia de una matriz en GL(N, ZZk ), la cual depende sólo de las posiciones relativas de la entradas del gobernante que serán copiadas por el gobernado. En general estudiamos la relación entre las propiedades aritméticas de k y la sincronización de los autómatas celulares. También presentamos una herramienta para demostrar que no hay sincronización de parejas de autómatas celulares en ciertos alfabetos.

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

Introducci´ on

Los autómatas celulares (AC) fueron introducidos por J. Von Neumann a finales de los an õs 40, para modelar sistemas biológicos autoreproductivos. 1

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

´ GUILLERMO SANCHEZ

En la actualidad los autómatas celulares son aplicados a diversas ramas de la ciencia como lo son la f´ısica, la biolog´ıa, la computación, la qu´ımica, etc. La evolución de un AC es sencilla de generar, y proporciona modelos simples para una gran variedad de fenómenos [1]. Un AC es una idealización matématica de sistemas f´ısicos en los cuáles el espacio fase y el tiempo son discretos. El estado de un autómata celular está completamente determinado por los valores de todas sus celdas (células). El AC evoluciona de acuerdo a una regla local que depende de sus valores en una vecindad. Por otro lado, la sincronización de sistemas acoplados ha despertado gran interés en estudios recientes [2], en particular cuando los sistemas que se acoplan son caóticos. Sistemas que muestran este comportamiento son temporalmente caóticos, pero espacialmente ordenados o coherentes. La coherencia presente es de un tipo particular: las dinámicas 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 dinámicos se necesita un sistema que evoluciona de manera autónoma, llamado gobernante y un sistema que evoluciona con cierto grado de libertad pero también es influido por medio del acoplamiento por el sistema gobernante, razón por la cual se le denomina sistema gobernado. Los autómatas celulares (AC) son sistemas dinámicos 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 evolución está definida por la repetida iteración de la ley de evolución 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 fenómeno de la sincronización de autómatas acoplados unidireccionalmente. En estos sistemas resulta que la sincronización está sujeta a condiciones aritméticas en los parámetros que definen al sistema. Estas relaciones aritméticas son punto central de este trabajo.

Sincronizaci´ on de aut´ omatas celulares

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

´ DE AUTOMATAS ´ SINCRONIZACION

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 autómata. La ley de evolución 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 paréntesis debe entenderse módulo 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 autómata 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 cómo se acoplan unidireccionalmente dos sistemas dinámicos. El sistema gobernante está relacionado con el gobernado por medio de una función, la cual considera parte de la evolución del sistema gobernante en el gobernado. En el caso del AC, la información de cuáles coordenadas son copiadas en el AC gobernado está 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 sincronización se

´ GUILLERMO SANCHEZ

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.

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

⎜ ⎜ ⎜ Mf = ⎜ ⎜ ⎜ ⎝

β2 β3 0 β1 β2 β3 0 β1 β2 .. .

0 0 β3 .. .

β3

⎛

⎜ ⎜ ⎜ ⎜ ⎜ C=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

c c c .. .

⎞

0 0 0

⎞

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

· · · β1

(1)

β2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 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 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 β2 β1 0 0 0 0 .. . 0

0 ··· ··· β3 β2 β1 0 0 0 0

⎞

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 ··· 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 conclusión, la sincronización de parejas de autómatas 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

β3 β2

0 ··· β3 · · · .. .

· · · β1 β 2 0 · · · β1

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ β3 ⎠

β2

´ GUILLERMO SANCHEZ

Clases de equivalencia y no sincronizaci´ on

Una vez fijo el alfabeto ZZk , sobre el cual definimos nuestros autómatas, las matrices que consideraremos están completamente determinadas por tripletas (β1 , β2 , β3 ) ∈ (ZZk )3 . Definici´ on: Dos tripletas son equivalentes si y sólo 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 ) (están en la misma clase), si β1 = β˜1 y β2 = β˜2 . Dado que la nilpotencia sólo 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 será 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 ), sólo se analizarán k 2 − k. Más a´ un, se clasificarán 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

´ DE AUTOMATAS ´ SINCRONIZACION

representante canónico 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 fácil verificar que la suma de representantes canónicos es también un representante canónico. Por otro lado el representante canónico 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 función 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 está dada por

τ (i) =

p−1 j

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

La relación de recurrencia entre polinomios caracter´ısticos en ZZk [λ], induce una relación de recurrencia entre los representantes canónicos de las clases del anillo reducido ZZkk [λ] a través de una relación 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

⎛

0 0

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

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

1 0 ··· ···

0 1 0 0

· · · −β2 1 ··· 0 −β2

⎞

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

Resultados computacionales

De la relación lineal entre los coeficientes de los representantes canónicos podemos observar los siguientes 2 hechos: 1. La matriz es eventualmente periódica, 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 sólo hay un n´ umero finito de casos que analizar. Si para una tripleta dada ninguno de los representantes canónicos generados son del tipo λχ(ℓ) , entonces no puede haber longitudes de sincronización. no Consideremos la función n #→ H n , donde H es la matriz de tama˜ 2k × 2k definida en la ecuación anterior y donde las potencias son evaluadas en M2k×2k [ZZk ]. Esta función es eventualmente periódica, 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 función n #→ H n es estrictamente periódica 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 canónico de Pℓ es de la forma λχ(ℓ) , iterando 100,000

´ DE AUTOMATAS ´ SINCRONIZACION

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.

Resultados

Para las tripletas de la clase (η, 0, 1) no podemos aplicar el algoritmo antes mencionado. Para mostrar la sincronización 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 inducción 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 término 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 módulo k, y como (C, k) = 1 esto implica que 2m ≡ 0 y m + 1 ≡ 0 módulo 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 contradicción, 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

´ GUILLERMO SANCHEZ

que 2 es divisible forzosamente por β, lo cual har´ıa que m y m+1 fueran simultáneamente 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 fácil ver que si M ∈ Mℓ [ZZmn ] es nilpotente, ϕ[M] ˆ ∈ Mℓ [ZZm × ZZn ] también es nilpotente para (m, n) = 1 y viceversa. ✷ Ejemplo Podemos predecir que la longitud de sincronización 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 sincronización.

6.1

Resultados particulares

Las longitudes de sincronización para el sistema (ecuación 1), en el caso en que M ∈ MN [ZZk ] y k no es primo, dependen fuertemente de la factorización de k. Del Tma. 6.0.3 se deduce fácilmente lo que pasa cuando k = pq con p y q primos diferentes. A continuación describimos el comportamiento de las longitudes de sincronización, para algunos casos en que k es potencia de primo. La fórmula recursiva para las tripletas de la clase (0, 2, 0) en ZZ34 es de la siguiente forma: Pℓ (λ) = (2 − λ)ℓ Haciendo un cálculo 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 sincronización, 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 sincronización para las tripletas de las clases (0, 2, 0) y (0, 6, 0) en ZZ38 .

´ DE AUTOMATAS ´ SINCRONIZACION

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 sincronización para estas longitudes. En ZZ38 , para la clase (4, 0, 1), podemos realizar un cálculo similar y concluir que hay sincronización para ℓ impar. Para ZZ39 las tripletas de la clase (0, 3, 0) tienen fórmula recursiva de la forma: Pℓ (λ) = (λ + 3)ℓ . Es fácil ver que ℓ = 3m, es longitud de sincronización para m = 1, 2, 3, ... Para la clase (0, 6, 0), las longitudes de sincronización son las mismas. Por otro lado, para las tripletas de la clase (3, 0, 1), la fórmula 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 inducción que

´ GUILLERMO SANCHEZ

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 hipótesis 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 sincronización m = 0, 1, ... Finalmente, para la clase (6, 0, 1) podemos realizar cálculos similares demostrando que hay sincronización para las mismas longitudes.

Comentarios finales

En este trabajo se analizó la sincronización de parejas de autómatas celulares afines de rango 1, en los anillos ZZk . Después de realizar el estudio computacional de los anillos ZZ4 , ZZ8 y ZZ9 , encontramos para qué tripletas existen longitudes de sincronización, as´ı como también que no hay otras tripletas que sincronicen en estos anillos. Demostramos, ayudados por la computadora, que no hay sincronización, exceptuando las tripletas triviales, en los campos ZZp para p de 3 a 29.

´ DE AUTOMATAS ´ SINCRONIZACION

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 sincronización 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 canónicos 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 canónicos de la forma λχ(ℓ) . De aqu´ı se deducir´ıa que para los alfabetos ZZp con p primo > 2, las u ńicas tripletas con longitudes de sincronización son las triviales. Esto se demostró 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 realización de este trabajo. J. Guillermo S´ anchez Saint-Martin IICO–UASLP, Av. Karakorum 1470, Lomas 4a. San Luis Potos´ı, S.L.P. 78210 México. 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

´ GUILLERMO SANCHEZ

[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