Cmjv01i07p0865

J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

Analogue of The Schubert Submodules in Quantum Groups SARASIJA Department of Mathematics Noorul Islam University, Kumaracoil ABSTRACT In this paper, we give an analogue of the Schubert submodules in Quantum Groups Uqg, where g = (Sl(n+1),C). V as a Uqgmodule. If V is finite – dimensional, then V itself is a Schubert submodule. If V is infinite dimensional, then V as a union of Schubert submodules.

INTRODUCTION Quantum group Uqg , where q is non root of unity, is a certain deformation of the universal enveloping algebra U(g) of a complex simple finite dimensional Lie algebra g = (Sl(n+1),C) introduced by Drinfield4, Lusztig7,8, Jimbo6 and also given

by De Concin & Kac2.

Uqg = Uq- g ⊗ U q0 g ⊗ Uq+ g Uqb+ - subalgebra of Uqg generated by Ei, K i±1 ( 1≤ i≤ n), Rosso9. W is its Weyl group.

Schubert submodules have already begun to show their usefulness in the study of the representations of Kac-Moody algebras.

We will present here an analogue of Schubert submodules in Quantum group and investigate various aspects in this context.

1. PRELIMINARIES Let q , non root of unity ,be an indeterminate and let A = C[q,q-1] with the quotient field C(q).

For n ε Z, let [n] =

q n − q −n q − q −1

∈ A,

[n] [n − 1] … [1], and the binomial coefficient

[n] ! =

Gaussian

n  [n]! k  = [k ]![n − k ]! for k ε N.  

Let (aij ) be an Cartan matrix of type An . We define Uqg is the algebra over C(q) defined by the generators Ei, Fi, Ki, Ki-1 (1 ≤ i ≤ n) and the relation (1.1)-(1.5): Ki Ki-1 = Ki-1 Ki = 1,

KiKj = KjKi,

(1.1)

KiEj Ki-1 = qaij Ej, KiFj Ki-1 = q –aij Fj, (1.2) EiFj – FjEi = δ ij

K i − K i−1 , q − q −1

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

(1.3)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

866 1− aij

s 1 − a  1− a −s ∑ (−1) s  s ij  Ei ij Ej Ei = 0 if i ≠ j   s =0

1− aij

s 1 − a  1− a − s ∑ (−1) s  s ij  Fi ij Fj Fi   s =0

Then (α i | α j ) = a ij , so that we get

= 0 if i ≠ j

a symmetric Z-valued bilinear form on Q such that (α | α ) ∈ 2 Z .

Uq g is a Hopf algebra, called the quantum group associated to the matrix (aij) with co-multiplication ∆, antipode S and counit v defined by ∆Ei = Ei ⊗ 1 + Ki ⊗ Ei, ∆Fi = Fi ⊗ Ki-1 + 1 ⊗ Fi, ∆Ki = Ki ⊗ Ki , SEi = - Ki-1 Ei, SFi = -FiKi, SKi = Ki-1 vEi = 0, vFi = 0, vKi = 1.

(Uqg)i is the subalgebra of (Uqg) generated

by Ei, Fi, Ki±1 .Uq-g , Uq+g , and Uq0g are subalgebras of Uqg generated by Ei , Fi and Ki±1 ( 1 ≤ i ≤ n ) respectively.

Let P be a free abelian group with basis ω i , i = 1, 2, ...,n Define the following elements: n

i =1

i =1

ρ = ∑ ω i , α j = ∑ a ij ω i ( j = 1,…,n) Let

Q = ∑ Zα i , Q + = ∑ Z + α i . i

For β =

pairing

(1.4)

(1.5)

n

Define a bilinear P × Q → Z by (ω i | α j ) = δ ij .

Define automorphisms ri of P by ri

ω j = ω j — δ ij α i ( i,j = 1, 2, . . . , n).

Then riαj = αj— aijαi. Let W be the (finite) subgroup of GL(P) generated by r1, r2, …,rn . Then Q is W-invariant and the pairing P × Q → Z is W-invariant.

Let π = {α1, α2,…,αn}, R = Wπ, R+ = R ∩ Q+ Then, R is a root system corresponding to the cartan matrix (aij) . W is its Weyl group, R+ a set of positive roots.

Let ( V , π) be a representation of Uqg. Then V becomes a Uqg – module, via the action is x . v = π(x)(v) for x ε Uqg and v ε V. From Rosso9 ,V admits a weight space decomposition that is V = Vµ µε P Let Vµ be the weight space { v εV / (µ α i ) Ki.v = q v for all i }. An element of a weight space Vµ is called a weight vector of weight µ.

⊕ ∑

i

∑ k iα i ε Q i

Introduce a partial ordering on P by λ ≥ µ if λ - µ ε Q+.

Definition 1.6 An integrable representation is called a highest weight representation if there exists a non zero vector vλ such that Uq-g.vλ = (0) and vλ generates V as a Uqg – modules.

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

In particular, since π(Ei) and π(-Fi) are locally nilpotent on V [7], w can define and invertible liner transformation riΠ of V by riΠ = exp π(Ei) exp π(-Fi) exp π(Ei).

= q =

867

( µ α ) − 2( µ α )

i

i riΠ (v)

q ( ri ( µ )|α i ) riΠ (v)

This implies riΠ (v) ε

Vri ( µ ) .

Lemma 1.7

Hence riΠ( Vµ ) ⊂

Let π be an integrable representation of Uqg on a vector space V.

But from Rosso9 it follows that each weight space Vµ in V is finite dimensional

Then riΠ( Vµ ) =

and dim Vw( µ ) = dim Vµ for all w εW.

Vri ( µ )

for i = 1,2,….,n

From this we get riΠ( Vµ ) =

Proof :

(r )

Π −1

i

First we prove the identity,

( )

π(Ki) ri Π = π(Ki-1)

(1.8)

Since V is naturally an Uq(sl(2),C) submodule of V, the proof of (1.8) tantamount to check this identity on V as Uq(sl(2),C) module.

Then

(r )

Π −1

( )

π(Ki) ri Π = exp -π(Ei) expπ(Fi) exp -π(Ei).π(Ki) exp π(Ei) exp-π(Fi) exp π(Ei). = exp -π(Ei) expπ(Fi)(1- π(Ei)) π(Ki) (1+π(Ei)) exp-π(Fi) exp π(Ei). = exp -π(Ei) expπ(Fi) π(Ki) + (q-q-1) π(Ei) exp-π(Fi) exp π(Ei) = exp -π(Ei) π(Ki -1) + ( q - q-1 ) π(Ei) exp π(Ei) = π(Ki -1) - (q2- 1)q-1 π(Ei) + ( q – q-1) π(Ei) π(Ei) π(Ki -1) π(Ei) = π (Ki -1) i

Let v ε Vµ . Then (riΠ )-1Ki (riΠ ) (v) = Ki-1. v = q i.e., Ki riΠ (v) = q

(µ α )

− (µ α i ) Π ri (v)

Vri ( µ )

v.

Vri ( µ )

Corollary 1.8 Let j1,j2,…jt be a sequence of elements from {1,…n}. For any µ ε P,

r πj ,...r πj (v µ ) = v r j ...v r j ( µ ) 1 t 1 t Proof : By induction on ‘t’.

Lemma 2.0 Let v~ be a (Uqg)i-module of V. If v~ contains a non-zero element of weight µ, then it also contains a non-zero element of weight ri(µ).

Proof:

Since v~ is a (Uqg)i-module, so it is an invariant subspace for π(Ei) and π(Fi).

Thus riΠ( v~ ) ⊂ v~ Let v be a non zero element in ( v~ ∩ vµ).

Then riΠ (v) is a non zero element of riΠ( v~ ∩ vµ). i.e., riΠ (v) is non zero in v~ ∩ riΠ( vµ ).

From Lemma 1.7 riΠ (v) ε vri (µ).

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

868

2. SCHUBERT SUBMODULES QUANTUM GROUP

IN

Let (V, π ) be an integrable highest weight representation of Uqg with highest weight λ,1,3. For any w ε W proposition 6 in9 shows that the weight space Vw(λ) is one dimensional. Therefore, if we choose a non zero element vw ε Vw(λ), then any element of Vw(λ) will be a scalar multiple of vw.

Definition 2.1 For any w εW, define the Schubert submodule Vw to be the Uqb+ submodule of V generated by vw.

Lemma 2.2 For any wεW, the Schubert submodule Vw is finite dimensional.

Proof : Since Vwi is a Uqb+ - module, Vw is also an U

0 qg

– module. From V admits a

⊕ ∑

⊕

We claim that Vw =

( Vw ∩ Vµ )

µε P Any vεV can be written in the form V =

∑

i =1

v j , where v j ε Vµj and there exists (µ α i )

k such that q We have vε Vw

( j = 1,…m) are distinct.

m

Kn (v) =

∑ q n( µ |α ) v ε V j

i =1

( n = 0,1…m-1)

⊕

Let l(w) be a length function defined for w εW. If ( ri1 … rit ) be a reduced expression of w then l(w) = t.

Proposition 2.3

2

weight space decomposition that is V = From Rosso [9] ,V admits a weight space Vµ decomposition that is V = µε P

m

Hence all vj ε Vw. Therefore Vw = ( Vw ∩ Vµ ). µε P As in proposition 6 of Rosso9, each weight subspace Vµ is finite dimensional, so it suffices to show that Vw ∩ Vµ ≠ {0}for only a finite number of µ ε P. Now Vµ ≠{0} only when µ ≤ λ ;on the other hand , since Vw is the Uqb+ -module generated by vw and vw is of weight w(λ), Vw ∩ Vµ ≠ {0} only for µ ≥ w(λ). Thus Vw ∩ Vµ ≠ {0} implies w(λ) ≤ µ ≤ λ, which is satisfied by only a finite number of elements of µ of P.

w

Suppose that i ε { 1,…n} and wε W have the property that l(riw) > l(w). Then the Schubert submodule Vri w is a (Uqg)i – module.

Proof: Let ( ri1 … rit ) be a reduced expression of w, w = ( ri, ri1 … rit ) is a reduced expression of riw and thus vri w = riπ r πj ...r πj (vλ) generates 1 t Vri w . Since Vri w = Uqb+ . vri w and vri w is a weight vector, we have Vri w = Uq+g . vri w .

But Uq+g is generated by E1,…En, so vri w is spanned by { Ei1 ,....Eim .v ri w | m > 0, i1,….,im ε { 1,…n} } (2.4)

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

We can now show that vri w is closed under the action of Fi, which will be sufficient to prove that it is a (Uqg)i – module. From the spanning set (2.4), Vri w closed under the action of Fi is enough to prove that

Fi( Ei1 ,....Eim .vri w ) is in

Vri w for any m ≥ 0 and any i1,….,im ε { 1,…n }.

Which we do by induction on m.

For m = 0 Claim ; Fi . vri w = 0.

If -αi is a root and Fi ε (Uqg)-αi then there is a non zero Ei of (U q g ) ri ( −αi ) such that

π(Fi) riπ = riπ { - π(Ei)}. This is similar to the proof of lemma 1.7 Also by induction, π (Fi) riπ r πj ...r πj = riπ r πj ...r πj {-π(Ei)} 1 t 1 t Therefore Fi . vri w = π (Fi) riπ r πj ...r πj (vλ) = 1 t

riπ r πj ...r πj (-π(Ei.vλ)) = 0 since Eivλ = 0;

1 t claim holds.

So for m = 0, we have Fi. vri w = 0, which is certainly in Vri w . Now suppose the

result holds for some m and consider Fi.( Ei1 ,....Eim +1 .vri w ).

Fi.( Ei1 ,....Eim +1 .v ri w ) = Ei1 Fi ,....Eim +1 .v ri w + [Fi Ei1 ]Ei2 ...Eim +1 .v ri w By induction Fi.( Ei2 ,....Eim +1 .v ri w ) is in Vri w , But Vri w is a Uqb+- module,

V so. Ei Fi Ei ....Ei 1 2 m +1 is in ri w .

Also [Fi Ei1 ] = Fi Ei 1

k i − k i−1 q−q

−1

-

869

Ei1 Fi is either 0 or

and both are elements of Uq0g.

Hence Fi.( Ei1 ,....Eim +1 .v ri w ) ε Vri w .

Corollary: 2.4 If w εW is such that l(riw) > l(w), then Vw ⊆ Vri w Proof : Let vw ε Vw . But vw is in the weight space Vw(λ), that is vw ε Vw(λ) which is one dimensional. Therefore any nonzero element in the weight space w(λ) is a scalar multiple of vw.

To prove vw ε Vri w , it is enough to prove that Vri w contains a non zero element of weight w(λ). By proposition 2.3 Vri w is (Uqg)i – module. Vri w is a Schubert submodule generated by vri w . But vri w is in the weight space Vri w(λ ) .

Therefore Vri w contains a non zero element vri w of weight riw(λ), so by Lemma 2.0 Vri w contains a non zero element of weight ri{riw(λ)} = w(λ).

Lemma 2.5 From9 m ≥ 0 and 1 ≤ i,j ≤ n, the following identities hold in Uqg. (i)

 E F mi +1  = 0 for i ≠ j.   i j

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

870 m +1 (ii)  Ei Fi i   

Corollary 2.7



−m m (m +1) − q−(mi +1) ki q i − ki−1q i mi q i = Fi . −1 −1

q−q

(iii)

q −q

kj F mi +1 i = q

− aij ( mi +1)

m +1 Fi i kj

Proposition 2.6 Let w ε w and suppose i ε { 1,…n} is such that l(riw) > l(w). Then Vri w is the

For any w ε W and any i ε { 1, …n } the (Uqg)i – submodule of V generated by Vw is a Schubert submodule.

Proof : If l(riw) > l(w), then the (Uqg)i – submodule of V generated by Vw is Vri w by the Proposition 2.6 . If l(riw) < l(w) then the exchange condition for elements of w has a reduced expression of the form (ri, ri1 … rit ). By proposition 2.3 Vw itself is a

(Uqg)i – submodule of V generated by Vw.

Uqg)i –module. So the Uqg)i – submodule of V generated by Vw is just Vw.

Proof:

Definition 2.8

Let w be the (Uqg)i – submodule of V generated by Vw. Then by corollary 2.4 and proposition 2.3 w ⊆ Vri w . We have w

= (Uqg)i .vw and ([2] , proposition 1.7 ( c ) ) together with the fact that Vw is a Uqb+ module, shows that { Fim . v | m ≥ 0 and v ε Vw} spans w. The Lemma 3.5 shows that for any m ≥ 0 and any v ε Vw, Ej ( Fim . v) and ki ( Fim . v) are in w for j = 1,2…n . Since Uqb+ - module is generated by Ei , ki±1, w is a Uqb+ - module.

To show Vri w ⊂ w .

Let vr w be a non zero element in Vri w , so i vri w is the weight space Vri w(λ ) .

But Lemma 2.0 shows that Vri w also contains a non zero element of weight ri{riw(λ)} = w(λ). Hence it is enough to prove that w contains a non zero element of weight w(λ). But clearly vw is a non zero element of weight w(λ) in Vw and Vw ⊆ w ,and the result follows.

For w, w′ ε W we write w → w′ if w′ = riw for some i ε { 1,…n} and l( w′ ) > l (w). We write w ≤ w′ if w = w′ or if there is a sequence of elements w = w1 → w2 → …→ wn = w′ in W.

Lemma 2.9 If w and w′ are in W and w ≤ w′ then Vw ⊆ V w ′ .

Proof : This follows from the definition of the ordering and corollary 2.4 If V is infinite dimensional, then V cannot be a Schubert submodule, However, we can prove that V as a union of Schubert submodules.

Definition 2.10 Let D be a partially ordered set. We call D a directed set if for any d1, d2 ε D there is some d ε D with d1 ≤ d and d2 ≤ d.

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

871

Lemma 2.11

Proposition 2.12

Let ξ be the collection of all Schubert submodule of V, partially ordered by inclusion. Then ξ is a directed set.

Let ξ be the collection of all Schubert submodule of V. Then ∪ ξ = V.

Proof : Proof : Let Vw and Vw be two Schubert 1 2 submodule of V. Let ( rit … ri ) be a 1 reduced expression for w2 . We set W0 =

Vw1 and Wk = the (Uqg)jk – submodule of V generated by Wk −1 , for k = 1,…t. Clearly W0 ⊆ W1 ⊆ … ⊆ Wt .

From corollary 2.7 each Wk is a Schubert submodule.

Also S0 = 1, Sk = r j … r j for k = 1,..t k 1 and Sk = r j Sk-1. k We claim VS ⊆ Wk for k = 0, 1, k … t. induction on k For k = 0, we get VS = V1 we have 1 ≤ 0 w1, and by the Lemma 2.9 V1 ⊆ Vw = W0 .

1 Assume the result is true for k-1, that is , VS k −1 ⊆ Wk −1 . Since l ( r j k Sk-1) > l(Sk-1), VS k is contained in the (Uqg)jk – submodule of V generated by VS k −1 . Thus VS k is contained in the (Uqg)jk – submodule of V generated by Wk −1 . Which is Wk .

Hence the claim.

We then Vw = W0 ⊆ Wt and Vw = 1 1 VS t ⊆ Wt .

Hence the lemma.

~

Let W = ∪ ξ. Note that 1(λ) = λ, we take v1 to be vλ , a highest weight vector for V. From this it is clear that the Schubert submodule

~

V1 = Cvλ, in particular W

~

contains vλ , thus to show that W = V, it is

~

suffices to show that W is a Uqg – module.

Since each Schubert submodule is a Uqb+module and ξ forms a directed set under

~ ~ shall prove that W is closed under the

inclusion we have W is a Uqb+- module. We action of Fi’s.

~

Let 1 ≤ i ≤ n, and let v be in W . So that v is in Vw for some w ε W. then Fi . v is in the (Uqg)i – submodule of V generated by

~

Vw, by corollary 2.7 Fi . v is in W , which is

~

true for all i. Therefore W = V.

Theorem 2.13 If V is finite dimensional , then V is Schubert submodule.

Proof : Let v1, v2,…vn be a basis for V, since V = ∪ ξ there are elements w1,w2,…wn in W such that vi is in Vwi for i= 1,2,…n.

Since ξ is a directed set, there is a Schubert ~ with ~ for i= submodule V w Vwi ⊆ V w ~ for each I, which 1,2,…n. Then vi is in V w ~. gives that V = V w Hence the theorem.

Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)

Sarasija, J. Comp. & Math. Sci. Vol. 1(7), 865-872 (2010)

872

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Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)