NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), 11-15

A LOOP MODULE OF THE EXTENDED AFFINE LIE ALGEBRA OF TYPE A

M a r c u s E m m a n u e l B a r n e s

(Received May 2003)

Abstract. We shall construct a loop module for the extended affine Lie algebra over a quantum torus.

1. Introduction

Extended affine Lie algebras form a new class of infinite dimensional Lie algebras, which were first introduced by H0egh-Krohn and Torresani [9] as a generalization of the finite dimensional simple Lie algebras and the affine Kac -M oody Lie algebras, and systematically studied in the book [1 ].

In this note, we give some irreducible representations for the extended affine Lie algebra of type A coordinated by a quantum torus. This loop module-like was motivated by Chari’s work [5]. Representations for extended affine Lie algebras have been constructed by a number of people. The approach in this note is very elementary and straightforward in some sense.

This work was carried out in the summer of 2001 supported by a NSERC Under­graduate Student Research Award. I am grateful to my supervisor Professor Yun Gao for his support and guidance.

2. An Extended Affine Lie Algebra

In this section we present our Lie algebra obtained by using a quantum torus.Let Cg = Cq[x± 1 ,y ±1] be the quantum torus which is an associative unital

algebra over the complex field C with generators rr±1 and y±x, subject to the following relations:

x x ~ l — x ~ 1x = 1

y y ~ 1 = y ~ 1y = lyx = qxy.

Then C ^ a^ 1, y±x] = © m,nez(C:rmyn which we will write as Cq. We need some preliminary stuff before we define our algebra. Let dx and dy be the degree operators

2000 A M S Mathematics Subject Classification: 17B10.K ey words and phrases: extended affine Lie algebra, loop module, quantum torus, irreducible module.This work was supported by NSERC.

12 MARCUS EMMANUEL BARNES

-— that is,dxx rnyn = m xrnyn

dvx myn = nxmyn.

The degree operators dx and dy are derivations of Cq which can be lifted to be derivations of M n(Cq).

We now define a C-linear function e : Cq —> C by

£(xm n ) = h if m = n = 0,|0 otherwise.

Recall that for an associative algebra R , the matrix algebra M n( R ) is an asso­ciative algebra. Then the general linear Lie algebra over R is gln(R ) = M n(_R)~ where the Lie bracket is given by [A, B] = A B — B A for A, B € R. Consider the Lie algebra gln(Cq). Let cx and cy be symbols. Set

Sln(^g) = S^n(Cq) © C Cx © C Cy

where [A, B] = A B — B A + e (tr ((dxA )B )) cx + e (tr((dyA )6)) cy and

[cx,g ln(Cg)] = [gln(C9), cx\ = 0 [cy ,g ln{Cq)\ = [gln(Cqr), Cy] = 0

[C-XiCy] = [CyjCj;] = 0.

Claim. g\n(Cq) is a Lie algebra.

This can be verified by noting that e(dxa) — 0 = e(dya) = 0 for a G (C )q, e (tr (A B )) = e (tr (B A )), B , A e M n(C9), and the fact that dx , dy are derivations.

We will now form a semi-direct product of the Lie algebra gln(Cq) with the degree operators dx , dy .

= ® Cdjc © Cdy

= gln(Cq) © Ccx © C Cy © C d x © C d y

subject to the following brackets:\dx , cx\ — [dx, Cy\

[dy,Cx] - \ d y ,C y \

\dX 5 t/y] -- 0[dx ,A] = dxA

\dyi A\ — dyA.

The Lie algebra gln(Cq) is called an extended affine Lie algebra.

3. M n(Cq) — Modules

In this section we will introduce two modules of M n(Cq) which would naturally give modules for the extended affine Lie algebra gln(C9).

Let W be a Cg-module and V a M n(C)-module. We know that V <g) W is an M n(C1?)-module.

We will now construct our modules:

A LOOP MODULE 13

Construction. Let V — C71 be the natural module for M n(C) and let W\ — Cq with left multiplication as the module action. Then V ® W\ is a M n(Cq)-module.

Construction. Let V — Cn be the natural module for M n(C) and let W 2 — C[x,a;_1], with C g-module action defined by a: as left multiplication such that x mynf (x ) = x mf (q nx). Then V <g> W 2 is a M n(Cq)-module. Moreover, if q is not a root of unity, V ® W 2 is irreducible.

We will prove our claim in Construction 2, which will be a direct consequence of the following two lemmas:

L em m a 1. If V = Cn is the natural representation of M n(C) and W is irreducible as a Cq-module, then V (g> W 2 is an irreducible M n(Cq)-module.

Proof. Let N be a submodule of V (g> W . If U — £™=1e* 0 Wi ^ 0 G iV where {e j} is the standard basis of Cn then ek j{l)U G N. Recall that = Sjiekwhere Sji is the Kronecker delta and is the standard matrix unit. So ek j{l)U = S^=1efcjei <8> Wi — efc ® Wj. Hence <g> Wj G N. If U ^ 0 then ej <8> Wj G N , Wj ^ 0 for some j . Now eij(a )(ej ®Wj ) = e C q- irreducible,that N — V <g>W. Therefore V ® W is irreducible.

L em m a 2. W 2 = C [x ,x _1] is Cq-irreducible if q is not a root o f unity.

Proof. Let N be a submodule of W2. Let f ( x ) = T,2=_paix l / 0 G W 2 . We may assume that f ( x ) = ao + a\x + . . . + anx n G N. Now ymf ( x ) = f (q mx) G N for all m G Z. Consider the following n + 1 equations:

a0 + a\x + . . . + anx n = f ( x ) a0 + aiqx + . . . + anqnx n = f (qx)

a0 + a\q2x + . . . + anq2nx n = f {q 2x)

a0 + aiqnx + . . . + anqn2x n = f {q nx).

Consider the coefficient matrix which is a Vandermonde matrix of these n + 1 equations:

awj G N for all i and a G Cq. Since W is W C iV for all i. This implies that V 0 W C ]V , which implies

□

So

(1 1 .. 1 \1 Q .. qn

= 1 Q2 .. q2n

\1qn .. q"2)

/ a0 \ ( f ( x ) \a\x f{q x )

\anxn) \ f(qnx ) )

14 MARCUS EMMANUEL BARNES

Since q is not a root of unity, P is invertible. Hence

/ a° \ a\x

= P - 1

( f ( x ) \

f ( q x )

\anxn) \f(Qnx)JTherefore € N for al i i = 0 , 1 , . . . , n. We assumed at f { x ) 0 so a jxJ ^ 0 for some j . Hence a j 1x~ i ajx^ = l e J V . From this it follows that W 2 — N and hence W 2 is C9-irreducible. □

So from these two lemmas we immediately have:

Theorem 3. If V = Cn, the natural module for M n(C) and W = C[rr,£-1 ], a Cq-module, and a module action given by x myn f ( x ) = x rnf (q nx), then V is a M n(C9) — module if q is not a root of unity.

4. gln(Cq)— Modules

We now state and prove the main results of this paper.

Theorem 4. V <S> W\ is an gln(Cq)-module. Moreover, V ® W\ is irreducible.

Proof. The actions are defined in the natural way. For example, cx and cy act trivially. The proof follows from Lemma 1. □

Theorem 5. V <g> W 2 x C[y, y~ l ] is an g\n(Cq)-module. Moreover, if q is not a root of unity, V <8> W 2 is irreducible.

Proof. The actions are defined naturally. The actions cx and cy are trivial. The proof follows from Lemma 1 and Theorem 1. □

References

1. B.N. Allison, S. Azam, S. Berman, Y. Gao, and A. Pianzola, Extended affine Lie algebras and their root systems, Memoir. Amer. Math. Soc. 126 Number 605 (1997).

2. S. Berman, Y. Gao and Y. Krylyuk, Quantum tori and the structure of elliptic quasi-simple Lie algebras, J. Funct. Anal. 135 (1996), 339-389.

3. S. Berman, Y. Gao and S. Tan, A unified view o f some vertex operator con­structions, Israel J. Math, (to appear).

4. S. Berman and J. Szmigielski, Principal realization for extended affine Lie al­gebra o f type SI2 with coordinates in a simple quantum torus with two variables, Contemp. Math. 248 (1999), 39-67.

5. V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85 (1986), 317-335.

6. Y. Gao, Fermionic and bosonic representations of the extended affine Lie alge­bra glTl(C'g), Canad. Math. Bull, (to appear).

7. Y. Gao, Representations of extended affine Lie algebras coordinatized by certain quantum tori, Compositio Mathematica, 123 (2000), 1-25.

8. Y. Gao, Vertex operators arising from the homogeneous realization for gl^, Comm. Math. Phys. 211 (2000), 745-777.

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9. R. H0egh-Krohn and B. Torresani, Classification and construction of quasi­simple Lie algebras, J. Funct. Anal. 89 (1990), 106-136.

10. H.P. Jakobsen and V.G. Kac, A new class o f unitarizable highest weight rep­resentations of infinite-dimensional Lie algebras II, J. Funct. Anal. 82 (1989) 69-90.

Marcus Emmanuel Barnesc\o Professor Richard GanongDepartment of Mathematics and StatisticsYork UniversityN520 Ross Building4700 Keele Street,TorontoOntarioC A N A D A M3J 1P3 yu250248@mathstat.yorku.ca