IRREDUCIBLE REPRESENTATIONS FOR THE AFFINE-VIRASORO LIE ALGEBRA OF TYPE B

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1 Chin. Ann. Math. 5B:3004, IRREDUCIBLE REPRESENTATIONS FOR THE AFFINE-VIRASORO LIE ALGEBRA OF TYPE B l JIANG Cuipo YOU Hong Abstract An explicit construction of irreducible representations for the affine-virasoro Lie algebra of type B l, through the use of vertex operators and certain oscillator representations of the Virasoro algebra, is given. Keywords Affine-Virasoro Lie algebra, Vertex operator, Irreducible module 000 MR Subject Classification 17B65, 17B67 1. Introduction The theory of vertex operator representations of affine Lie algebras has many important and interesting applications in mathematics and physics. The first vertex construction of basic representations for affine Lie algebras, which is usually called the principal vertex representation, was discovered by Lepowsky and Wilson [13] and it has been generalized in [1] to all the affine Lie algebras of ADE type. Frenkel and Kac [8], and Segal [15], have given another construction of the same basic modules of all ADE type Lie algebras by using the vertex operators Xα, z which are very useful for the study of the dual resonance theory. A suitable chosen Heisenberg algebra or a Heisenberg system is crucial to these constructions. By enlarging the Fock spaces, one can construct vertex representations for other affine Lie algebras. These representations, however, are no longer irreducible. It was also observed in [8] that the basic given modules afford representations of the Virasoro algebra too. This makes the basic modules into irreducible modules of larger Lie algebras the semi-direct products of affine Lie algebras and the Virasoro algebra, which play an important role in the representation theory of affine and Virasoro algebras and in the quantum field theory. This paper is based on the work of [10], where vertex representations for the toroidal Lie algebras of type B l were constructed. A toroidal Lie algebras is the universal central extension of the iterated multi-loop algebra, G C[t ±1 0, t±1 1,, t±1 ν ] ν 1, where G is a simple finite-dimensional Lie algebra over the complex field C. Toroidal Lie algebras can be considered as the natural generalization of affine Lie algebras. Unlike the affine case, however, the universal central extension has infinite-dimensional center. By adding the full derivation algebra together with a -cocycle to the toroidal Lie algebra, we can get a larger Manuscript received April 4, 003. Department of Mathematics, Harbin Institute of Technology, Harbin , China. Department of Mathematics, Shanghai Jiaotong University, Shanghai 00030, China. cpjiang@sjtu.edu.cn Department of Mathematics, Harbin Institute of Technology, Harbin , China. Project supported by the National Natural Science Foundation of China No , No

2 360 JIANG, C. P. & YOU, H. toroidal Lie algebra which has finite-dimensional center. Over the past decade, the vertex representations of affine Lie algebras have been generalized to vertex representations of the toroidal Lie algebras e.g., see [14, 6, 16, 3]. For more results on toroidal Lie algebras, one can see [1,, 4, 5], etc. The modules constructed in [10] are not irreducible, since the corresponding modules of affine Lie algebra are not irreducible. In this paper, we extend the affine Lie algebra by the Virasoro algebra thus constructing a semi-direct product algebra containing the affine Lie algebra as an ideal and the Virasoro algebra as a subalgebra. By introducing Virasoro operators which, different from the previous ones, do not preserve the irreducible submodules of the affine Lie algebra, we prove that the given reducible module not only becomes a module of the larger algebra but also becomes irreducible. Through out the paper, we denote by Z + and Z the set of positive integers and the set of negative integers respectively.. Preliminaries Let g be the finite-dimensional simple Lie algebra of type B l over the complex field C, and the Killing form on g. Let h be a Cartan subalgebra of g, and the root system of g with respect to h. Let h be the dual space of h. Then there exists the normal orthogonal basis {e 1,, e l } of h such that the short root system is S = {±e i 1 i l} and the long root system is L = {±e i + e j, ±e i e j 1 i < j l}. The simple root system is = {e 1 e, e e 3,, e l 1 e l, e l }. Let α i = e i e i+1, 1 i l 1, α l = e l. Then the root lattice is { l } Q = k i α i k i Z, 1 i l = { l } k i e i k i Z, 1 i l. It is clear that α i α i =, 1 i l 1, α l α l = 1. Let α h be such that αα =. Define ɛ : Q Q {±1} by ɛe i, e j = 1, 1 i j l, ɛe j, e i = ɛe i, e j, i < j;.1 ɛa + b, c = ɛa, cɛb, c, ɛa, b + c = ɛa, bɛa, c.. Proposition.1. cf. [10] The Lie algebra g has a basis {e α, α i α, 1 i l} such that [α i, α j ] = 0, [α i, e α ] = αα i e α, α, 1 i l;.3 [e α, e α ] = ɛα, αα, α ;.4 [e α, e β ] = ɛα, βe α+β, α L, β, α + β ;.5 [e α, e β ] = ɛα, βe α+β, α, β S, α + β ;.6 [e α, e β ] = 0, α, β, 0 α + β /..7 Proposition.. {e αi, e αi, e αl, e αl 1 i l 1} are Chevalley generators of g, and {e α, e α, e β, e β, αi α + L, β + S, 1 i l} is a Chevalley basis of g. Proof. The results can be checked directly by Proposition.1 and the definition of ɛ. Let ĝ = g C[t, t 1 ] Cc be the associated affine Lie algebra with the following Lie bracket [x t m + λ 1 c, y t n + λ c] = [x, y] t m+n + mδ m+n,0 x yc,.8

3 IRREDUCIBLE REPRESENTATIONS 361 where x, y g, λ 1, λ C, m, n Z. For k Z, let h k be an isomorphic copy of h, and e l k + 1 an isomorphic copy of e l. Then H = h k Ce l k + 1 Cc 0 k Z k Z becomes a Lie algebra with the following Lie bracket [am, bn] = ma bδ m+n,0 c 0, [e l m + 1, e l n 1 ] = m + 1 δ m+n,0 c 0, [ H, [ c 0 ] = h k, e l k + 1 ] = 0, k Z where a, b h, m, n, k Z. It is easy to see that Ĥ = a Heisenberg Lie algebra. Let Ĥ = h k Ce l k + 1, k Z k Z h k Ce l k + 1 Cc 0 is k Z\{0} k Z and SĤ the symmetric algebra generated by Ĥ. We define the Fock space V = SĤ C[Q], where C[Q] = Ce α is the group algebra on the additive group Q. Then V has a α Q natural module structure for the Lie algebra H and the group algebra C[Q] with the actions defined by assigning c 0 acting as 1, a n and e l n + 1 acting as multiplications and an and e l n 1 acting as partial differential operators, for n > 0, a h, and a0 acting as a partial differential operator on C[Q] for which a0e r = a re r, where a h. Let z be a complex variable. We define elements of EndV [[z, z 1 ]] as follows: Y α, z = exp n=1 Y α, z = 1 exp exp exp n=0 ɛ a v e r = ɛa, rv e r, a Q,.9 z a v e r = z a r v e r, a Q,.10 z n n αn, α L,.11 z n n α n exp n=1 z n+1 n + 1 e l n=0 z n n α n exp n=1 n 1 Xα, z = Y α, zz α α e α z α ɛ α = n=1 z n n αn z n 1 n + 1 e l n + 1, α S,.1 n 1 Z X n αz n..13 For v e r = a 1 n 1 a k n k e l m e l m s + 1 er, where a i, r Q, n i, m j Z +, define k s degv e r = n i m j 1 1 r r..14

4 36 JIANG, C. P. & YOU, H. Let V 0 and V 1 be the subspaces of V spanned by {v er degv e r Z} and {v e r degv e r Z + 1 } respectively. Then V = V0 + V 1. Proposition.3. cf. [10] V 0 and V 1 Lie algebra ĝ. Particularly, we have where α, n Z. α t n v e r = are completely reducible modules of the affine α α αnv er,.15 e α t n v e r = X n αv e r,.16 cv e r = v e r,.17 by 3. Irreducible Modules of the Affine-Virasoro Lie Algebra ĝ vir Let d i = t i+1 d dt i Z be a derivation of C[t, t 1 ]. Extend it to the operators on ĝ d i x t m + λc = x d i t m = mx t m+i. 3.1 Then d i is a derivation of ĝ. Let d be the Lie algebra spanned by {d i i Z}. Then the universal central extension of d by a 1-dimensional center CK, is the Virasoro algebra, denoted by Vir, with the following Lie bracket [d m, d n ] = m nd m+n + m3 m δ m+n,0 K Now we consider the Lie algebra ĝ vir = g Vir with the Lie bracket defined by.8, 3.1, 3. and [ĝ vir, Cc CK] = 0. The goal of this section is to prove that V 0 and V 1 are irreducible ĝ vir -modules. Define operators on V as follows: L 0 = 1 L n = 1 l e i 0e i 0 + j>0 j Z l e i je i j + e l j 1 e l j , 3.3 j 0 e l j 1 e l j n, n Z\{0}. 3.4 l e i je i j + n + 1 j Z Lemma 3.1. For v e r V, we have L 0 v e r = degv e r + 1 v e r Lemma 3.. For m, n Z, α, we have Proof. First it is easy to see that [L m, X n α] = nx n+m α. 3.6 degx n αv e r = n + degv e r, 3.7 degl n v e r = n + degv e r. 3.8

5 If m = 0, then by 3.7 and 3.5 we have We can deduce 3.6 by using Lemma 3.3. For m, n Z, we have IRREDUCIBLE REPRESENTATIONS 363 [L 0, X n α] = nx n α. 3.9 [e i m, L n ] = me i m + n, 1 i l; 3.10 [e l m + 1 ], L n = m + 1 e l m + n Proof. It is straightforward to check the results. Lemma 3.4. For m, n Z, one has [L m, L n ] = m nl m+n + m3 m l + 1δ m+n, and From Lemmas and Proposition.3, we deduce Theorem 3.1. V 0 and V 1 are ĝ vir-modules with the action defined by d i v e r = L i v e r, i Z, 3.13 Kv e r = l + 1v e r Now, define a Hermitian form H, on V as follows: l H s i q i l s e i n ij k ij p e i m ij lij s = δ r1 r,0δ s,p δ nj,m j δ kj,l j l s i and for λ C, v, w V, k ij!n kij ij s e l n j 1 kj e r 1, e l m j 1 lj e r l s i δ si,q i δ nij,m ij δ kij,l ij k j! n j + 1 kj, λv, w = λv, w = v, λw, where we assume that n i1 < n i < < n isi, m i1 < m i < < m iqi, 1 i l, n 1 < n < < n s, m 1 < m < m p. Lemma 3.5. The Hermitian form H, is positive definite. Let e α α, α i 1 i l be the same as in Proposition.1. Let e ±α = ±e ±α, e ±β = e ±β, α + L, β + S. 3.15

6 364 JIANG, C. P. & YOU, H. Then by Proposition., {e α, αi α, 1 i l} is a Chevalley basis of g. Now define an antilinear operator ω 0 on ĝ vir by ω 0 λx = λω 0 x, x ĝ vir ; 3.16 ω 0 e α t m = e α t m, m Z, α ; 3.17 ω 0 α i t m = α i t m, m Z, 1 i l; 3.18 ω 0 c = c, ω 0 K = K, ω 0 d m = d m It is easy to see that ω 0 is an antilinear automorphism of ĝ vir. Lemma 3.6. For n Z, 1 i l and x = v e r1, y = w e r V, we have He i nx, y = Hx, e i ny; 3.0 H e l n + 1 x, y = H x, e l n 1 y. 3.1 Proof. We can assume that l s i s v = e i n ij kij w = l t i p e i m ij l ij e l n j 1 kj e r 1, e l m j 1 lj e r. For n Z +, let where e i nv 1 = e i nw 1 = 0. Then If n = 0, we have v = e i n k v 1, w = e i n q w 1, He i nv e r 1, w e r = knhe i n k 1 v 1 e r 1, e i n q w 1 e r = δ k 1,q k!n k Hv 1 e r 1, w 1 e r = He i n k v 1 e r 1, e i n q+1 w 1 e r = Hv e r 1, e i nw e r. He i 0v e r 1, w e r = r 1 e i Hv e r 1, w e r, Hv e r 1, e i 0w e r = r e i Hv e r 1, w e r. By the definition of H,, we have The proof of 3.1 is similar. Lemma 3.7. He i 0x, y = Hx, e i 0y. Let Y α, z be defined by.11 and.1, and let Y α, z = n 1 Z Y n αz n.

7 Then for v, w SĤ, r Q, n Z, we have IRREDUCIBLE REPRESENTATIONS 365 HY n αv e r, w e r = Hv e r, Y n αw e r, α L ; 3. HY n+ 1 αv er, w e r = Hv e r, Y n 1 αw er, α S. 3.3 Proof. Let E α, z = exp ± F e l, z = exp ± F e l, z = exp ± n=1 n=0 n=0 z ±n n α n = n Z E n αz n, z ±n+1 n + 1 e l z ±n+1 n + 1 e l n + 1 = Fn e l z n, n 1 Z n + 1 = Fn e l z n. n 1 Z Note that En α = E nα + = F ±e n 1 l = F + ±e n+ 1 l = 0 for n Z +. By Lemma 3.6, for any v 1, v SĤ, r Q, we have HE n αv 1 e r, v e r = Hv 1 e r, E ± n αv e r, α ; n Z, 3.4 HFn e l v 1 e r, v e r = Hv 1 e r, F n e ± l v e r, n 1 Z. 3.5 If α L, by the definition of H,, we can assume that deg v = deg w n = k k n. Then Therefore by 3.4 we have k Y n αv e r = E n i αe+ i αv e r, Y n αw e r = HY n αv e r, w e r = = k n E n i αe+ i αw e r. k HE n i αe+ i αv er, w e r k HE + i αv er, E + n+i αw er = H v e r, = H v e r, k k n = Hv e r, Y n αw e r. E i αe+ i n αw e r E n i αe+ i αw e r Therefore 3. holds. We can deduce 3.3 similarly, for β S, though we still point out that Y ±β, z = 1E ±β, ze + ±β, zf e l, zf + e l, z,

8 366 JIANG, C. P. & YOU, H. and i.e., H 1 Y n+ 1 βv er, w e r = 1 HY n+ 1 βv er, w e r = Hv e r, 1 Y n 1 βw er. Theorem 3.. H, is a contravariant Hermitian form on V with respect to ω 0, Hgx, y = Hx, ω 0 gy 3.6 for all g ĝ vir, x, y V. Therefore by Lemma 3.5, V 0 and V 1 are unitary modules of ĝ vir. Proof. By Lemma 3.6 and the definition of ω 0, 3.6 holds for all g h C[t, t 1 ] Cc CK, and it is easy to deduce by Lemma 3.6 that Hd n x, y = Hx, ω 0 d n y, n Z. Let x = v e r 1, y = w e r V. By 3.15, 3.17 and.16, we have to prove HX n αv e r1, w e r = Hv e r1, X n αw e r, α L ; 3.7 HX n βv e r1, w e r = Hv e r1, X n βw e r, β S. 3.8 We first prove 3.7. By the definition of H,, we can assume that r 1 + α = r. Then X n αv e r 1 = ɛα, r 1 Y n+α r1 +1αv e r, X n αw e r = ɛ α, r 1 + αy n α r1 1 αw e r 1. By.1 and., we know that ɛα, r 1 = ɛ α, r 1, ɛ α, α = 1. Therefore Then by 3. we have X n αw e r = ɛα, r 1 Y n α r1 1 αw e r 1. HX n αv e r 1, w e r = ɛα, r 1 Hv e r, Y n α r1 1 αw e r For β S, we can assume that r 1 + β = r. Then = ɛ α, r 1 Hv e r1, Y n α r1 1 αw e r1 = Hv e r1, X n αw e r. HX n βv e r 1, w e r = ɛβ, r 1 HY n+β r1+ 1 βv er, w e r = ɛβ, r 1 HY n+β r1 + 1 βv er 1, w e r 1, Hv e r1, X n βw e r = ɛβ, r 1 + βhv e r1, Y n β r1 1 βw er1. Now 3.8 follows from 3.3 and.1. Corollary 3.1. V 0 and V 1 are completely reducible ĝ vir-modules. Let h vir = h Cc Cd 0 CK, ĝ + vir and ĝ vir be the linear spans of {e α, d n α +, n Z + } and {e α, d n α, n Z } respectively.

9 IRREDUCIBLE REPRESENTATIONS 367 Definition 3.1. A highest weight representation of ĝ vir is a representation in a vector space M which admits a non-zero vector v, such that for given Λ h vir, hv = Λhv, h h vir, Uĝ + vir v = 0, M = Uĝ vir v, where Uĝ + vir and Uĝ vir are the universal evenloping algebras of ĝ+ vir and ĝ vir respectively. Remark. By Theorem 3. and the fact that V is graded, we know that any highest weight submodule of V is irreducible. Conversely, since V is a graded and restricted module, any irreducible submodule of V is a highest weight module. Therefore, as ĝ vir -module, V is a direct sum of highest weight modules. Theorem 3.3. V 0 and V 1 are irreducible ĝ vir-modules. Proof. We only prove V 0 is irreducible, since the proof for V 1 v e r V 0 be such that is quite similar. Let hv e r = λhv e r, h h vir, 3.9 Uĝ + vir v er = Then Uĝ + v e r = 0. Therefore r = 0 see [10]. Let S k be such that Se l, z = F e l, zf + e l, z = k 1 Z S k z k. Then similarly to the proof of 3.6, one can deduce that [L n, S m ] = m 1 n S m+n, m, n Z If v / C1 e 0, then we can assume that q v = a ki1,,k isi S ki1 S ki S kisi, where a ki1,,k isi C = C\{0}, k i1 + k i + + k isi = m 1 i q, k i1 > k i > > k isi 1, and k i1, k i,, k isi > k i+1,1, k i+1,,, k i+1,si+1. We say k i1, k i,, k isi > k j1, k j,, k jsj, if there exists p 1 such that k i1 = k j1,, k i,p 1 = k j,p 1, k ip > k jp. By 3.31, q, since L 1 v = 0. Let p 1 be such that k 1j = k j, 1 j p 1, k 1p > k p. Then by the fact that L 1 v = 0 and the use of 3.31, we have p = s 1, k p = k 1p 1 and k,p+1 = 1. Therefore k 1s1 3 and k 1s1 1 Note that L v = 0. If k 1s1 > 3, then a k11,,k 1s1 + 1 a k 1,,k s = k 1s1 1a k11,,k 1s1 S k11 S k1 S k1,s1 1S k1s1 + = 0. This means that a k11,,k 1s1 = 0, which contradicts the assumption. Therefore k 1s1 = 3 and From 3.3 and 3.33, we have a k11,,k 1s1 a k1,,k s = a k11,,k 1s1 = a k1,,k s = 0, which is also in contradiction with the assumption. Therefore v C1 e 0. This proves that V 0 is irreducible.

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