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Iterated Loop Modules and a Filtration for Vertex Representation of Toroidal Lie Algebras Pacific Journal of Mathematics ITERATED LOOP MODULES AND A FILTRATION FOR VERTEX REPRESENTATION OF TOROIDAL LIE ALGEBRAS S. ESWARA RAO Volume 171 No. 2 December 1995 PACIFIC JOURNAL OF MATHEMATICS Vol. 171, No. 2, 1995 ITERATED LOOP MODULES AND A FILTERATION FOR VERTEX REPRESENTATION OF TOROIDAL LIE ALGEBRAS S. ESWARA RAO The purpose of this paper is two fold. The first one is to construct a continuous new family of irreducible (some of them are unitarizable) modules for Toroidal algebras. The second one is to describe the sub-quotients of the (integrable) modules constructed through the use of Vertex operators. Introduction. Toroidal algebras r[d] are defined for every d > 1 and when d — 1 they are precisely the untwisted affine Lie-algebras. Such an affine algebra Q can be realized as the universal central extension of the loop algebra Q ®C[t, t"1] where Q is simple finite dimensional Lie-algebra over C. It is well known that Q is a one dimensional central extension of Q ®C[ί, ί"1]. The Toroidal algebras ηd] are the universal central extensions of the iterated loop algebra Q ®C[tfλ, tJ1] which, for d > 2, turnout to be infinite central extension. These algebras are interesting because they are related to the Lie-algebra of Map (X, G), the infinite dimensional group of polynomial maps of X to the complex algebraic group G where X is a d-dimensional torus. For additional material on recent developments in the theory of Toroidal algebras one may consult [BC], [FM] and [MS]. In [MEY] and [EM] a countable family of modules (also integrable see [EMY]) are constructed for Toroidal algebras on Fock space through the use of Vertex Operators (Theorem 3.4, [EM]). However they are reducible and not completely reducible. In §5 we observe that the Fock space is a direct sum of certain b(λ)'s. In (5.10) to (5.12) we prove that each 6(λ) admits a filteration by an increasing sequence of modules such that the successive quotients are all isomorphic to V defined in §5. In (5.9) we prove that each V admits a filteration of decreasing sequence of modules such that the successive quotients are all irreducible. In our main Theorem 5.6 we will describe the irreducible modules as certain iterated loop modules twisted by an automorphism of τ^. 511 512 S. ESWARA RAO We now describe the contents of the paper. In §1 we construct an iterated loop algebra of Kac-Moody Lie-algebra Q and construct a family of completely reducible modules (Theorem (1.8)) using methods similar to [E]. In §2 we prove that some of the above modules are unitarizable (Proposition 2.3). In §3 we specialize these results for Q finite dimensional (Theorem 3.3) and for Q an aίfine Lie-algebra (Theorem 3.6) to get irreducible modules (some of them are unitarizable modules) for T[dj. The modules considered in Theorem 3.6 are the first examples of irreducible modules for T[d] where part of the centre acts non-trivially. It should be mentioned that it is the unitarizable modules (and also integrable modules) which lift to the group. In §4 we recall the construction of Vertex Operators and the Fock space. We also construct certain automorphisms of τy\ which are necessary in §5. 1. Let d and k be positive integers. Let Q be a Lie-algebra and let Qk = ®G 1 1 be k copies of Q. Let A — Ad = Cfίf , ίjj" ] be Laurent polynomial ring in d variables. Then QA = Q ® A is a Lie-algebra with Lie structure [X®a,Y®b] = [X,Y]®ab,X,Y eGa.be A. Let n = (ni, n2, nd) be a d—tupple of integers and let /n j.nιj.n2 j.rid i — τλ τ2 - ιd For 1 < i < k let a{ = (α<(l), • • α^d)) be a d—tupple of non-zero complex numbers. Let af = α^(l) a"d(d) be the product. Consider the Lie- algebra homomorphism Φ GA ->Gk It is elementary to check that φ is not surjective if and only Ίίai(£) = a,j(£) for some i ψ j and for all L First prove it for d = 1 and then using Vandermonde determinant for general d. Define derivations dι, d2 dd on QA by [di? X ® t~) = ΠiX ® t- and note [di, dj] — 0. Let D be the linear span of dλ, d2, dd and let QA = ^ ® A θD. For any Lie-algebra £, let ί7(^) denote the universal enveloping algebra. d Note that A,QA and U(QA) are obviously Z graded algebras. 1.1. From here onwards we will assume that Q is a Kac-Moody Lie-algebra. Fix a Cart an subalgebra hoίQ. d Let ψ : U(hA) —> A be a 7L graded homomorphism. Let Aψ be the image of φ which is a TLd graded subalgebra of A. FILTERATION OF A LOOP MODULES 513 can be treated as an hA— module by the ψ action, 1.2. Lemma. Aψ is an irreducible hΛ— module if and only if homogeneous elements of Aψ are inυertible in Aψ. Proof. Assume the homogeneous elements of Aψ are invertible in Aψ. To prove irreducibility it is sufficient to prove that given elements t- and t— there exists X in U(hA) such that X.t- = t—. By assumption t—~- belongs to Aψ. Since Aψ is the image of φ there exists X in U(hA) such that ψ(X) = t—~- and clearly X.t*- = ψ(X)t* = t^ . Now for the converse let t- belong to Aψ. First note that 1 belongs to Aψ. There exists X in U(hA) such that ψ(X).t- = 1 by irreducibility of Aψ. Then clearly ψ(X) — t~~- and we are done. D 1.3. The purpose of this section is to construct irreducible modules for QA. Let V(λi), V(Xk) be irreducible highest weight modules for Q with highest weights λ1,λ2, λΛ; respectively. Then clearly V — ®V{\i) is an irreducible module for Qk. 1.4. Also V can be treated as an irreducible ^A-module via the Lie-algebra homomorphism φ. But it is not a Zd— graded module. That is, it cannot be extended to QA. Consider VA :— V ® A which will be given QA module structure. (1.5) X ® t^{v ® t^) - φ(X ® <2L)V ® t^^ dτ(v ® t^) - mυ ®t^{X eg,ve V). We denote the QA module by (VA,π). 2 a d Let ψ : U(hA) -> A be /ι ® t - >-> ^λi(/ι)αfί be a Z - graded homomor- phism of algebras. 1.6. We believe that the conditions for φ to be surjectiυe are sufficient to prove that Aψ is irreducible, but we could not prove that. Instead we will give a continuous family of examples where Aψ — A. In particular for these examples Aψ is irreducible. 514 S. ESWARA RAO 1.7. Assume 1 < ί < d,ai(i)/aj(έ) is not a k—th root of unity. Then by Lemma 4.4 of [CP] the following Z—graded algebra homomorphism is surjective for all t, 3=1 Hence Aψ contains C\tι,tJι] for all ί. So that Aψ — A. From here on we will assume that Aψ is irreducible hA— module. We will also note φ and the module (VΛ, π) depends on the choice of λ's and For any υ G V let υ(n) — v®t-. We will now prove that (VA, π) as defined in (1.5) is a completely reducible G_A— module. 1.8. Theorem. Let G C Zd be such that {t^m G G} is a set of coset representatives of A/Aψ. Let υ = Vι <g> ® vk where each Vi is a highest weight vector of V(λi). Then (1) VA = ®meGU(v(rn)) as QA- module where U(v(m)) is the GA submod- ule generated by the vector v(rn). (2) Each U(v(m)) is an irreducible GA~ module. In particular {VA,K) is an irreducible GA -module whenever A — Aψ. Before we prove the theorem we prove some lemmas. 1.9. Lemma. Any non-zero GA~ submodule ofVΆ contains v(m) for some m. Proof. Consider the map S : VA —> V defined by S(w(m)) = w (extend linearly to VA). Then clearly S is a surjective GA- module map (it is not a ^-module map). Claim. S(W) — V for any non-zero GA module W of (VA,TΓ). Since V is an irreducible GA -module and S(W) is a submodule of V, to see the claim it is sufficient to prove that S(W) Φ 0. But that is clear. Since W is a 6U-module it therefore contains vector of the form w(m), w G V and S(w(m)) =wφQ. This proves the claim. Now let w in W be such that S(w) = υ where υ is the vector defined in the statement of the theorem. Since S is a h— module map, w and v are d the same weight. {υ(m),m G Z } are the only such weight vectors of VA and hence w = ^^Civ{rr£) for some complex numbers Cι and some rnι G Zd. i But W is a GA module so is (Zd-graded) and it follows that ^(m2) belongs to W. D FILTERATION OF A LOOP MODULES 515 1.10. Lemma. The following are true. R (1) υ(m) G U(hA)v(n) if and only ift^~ G Aφ. (2) v(m) G U{QA)v{n) if and only if t^~^ G Aψ.
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