Free field realizations of certain modules for affine Lie algebra slˆ(n,C)

For the affine Lie algebra slˆ(n,C) we study a realization in terms of infinite sums of partial differential operators of a family of representations introduced in [BBFK]. These representations generalize a construction of Imaginary Verma modules [F1]. The realization constructed in the paper exten...

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Datum:	2011
1. Verfasser:	Martins, R.A.
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spelling	irk-123456789-1547682019-06-17T01:30:41Z Free field realizations of certain modules for affine Lie algebra slˆ(n,C) Martins, R.A. For the affine Lie algebra slˆ(n,C) we study a realization in terms of infinite sums of partial differential operators of a family of representations introduced in [BBFK]. These representations generalize a construction of Imaginary Verma modules [F1]. The realization constructed in the paper extends the free field realization of Imaginary Verma modules constructed by B.Cox [С1]. 2011 Article Free field realizations of certain modules for affine Lie algebra slˆ(n,C) / R.A. Martins // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 28–52. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:17B67, 81R10 http://dspace.nbuv.gov.ua/handle/123456789/154768 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For the affine Lie algebra slˆ(n,C) we study a realization in terms of infinite sums of partial differential operators of a family of representations introduced in [BBFK]. These representations generalize a construction of Imaginary Verma modules [F1]. The realization constructed in the paper extends the free field realization of Imaginary Verma modules constructed by B.Cox [С1].
format	Article
author	Martins, R.A.
spellingShingle	Martins, R.A. Free field realizations of certain modules for affine Lie algebra slˆ(n,C) Algebra and Discrete Mathematics
author_facet	Martins, R.A.
author_sort	Martins, R.A.
title	Free field realizations of certain modules for affine Lie algebra slˆ(n,C)
title_short	Free field realizations of certain modules for affine Lie algebra slˆ(n,C)
title_full	Free field realizations of certain modules for affine Lie algebra slˆ(n,C)
title_fullStr	Free field realizations of certain modules for affine Lie algebra slˆ(n,C)
title_full_unstemmed	Free field realizations of certain modules for affine Lie algebra slˆ(n,C)
title_sort	free field realizations of certain modules for affine lie algebra slˆ(n,c)
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/154768
citation_txt	Free field realizations of certain modules for affine Lie algebra slˆ(n,C) / R.A. Martins // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 28–52. — Бібліогр.: 15 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT martinsra freefieldrealizationsofcertainmodulesforaffineliealgebraslˆnc
first_indexed	2025-07-14T06:52:37Z
last_indexed	2025-07-14T06:52:37Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 12 (2011). Number 1. pp. 28 – 52 c© Journal “Algebra and Discrete Mathematics” Free field realizations of certain modules for affine Lie algebra ŝl(n,C) Renato A. Martins Communicated by V. V. Kirichenko Abstract. For the affine Lie algebra ŝl(n,C) we study a realization in terms of infinite sums of partial differential operators of a family of representations introduced in [BBFK]. These repre- sentations generalize a construction of Imaginary Verma modules [F1]. The realization constructed in the paper extends the free field realization of Imaginary Verma modules constructed by B.Cox [C1]. 1. Introduction Representation theory of affine Lie algebras is a very rich subject with many applications in Mathematics and Physics. Representations of affine Lie algebras have some very distinct features which have no analogs in the finite dimensional case. For example, there exist modules for affine Lie algebras containing both finite and infinite-dimensional weight spaces. The simplest example of such modules is given by the so-called Imaginary Verma modules [F1]. These representations correspond to nonstandard partitions of the root system which are not equivalent under the Weyl group to the standard partition into positive and negative roots (for details see [DFG]). For affine Lie algebras, there are always only finitely many equivalence classes of such nonstandard partitions (see [F2]). These partitions give rise to Verma-type modules, which were first studied and classified by Jakobsen and Kac [JK], and by Futorny [F2, F3] (see also [C2, F1, FS, FK]). In a recent paper [BBFK] different Borel-type subalgebras that do not correspond to partitions of the root system of affine Lie algebra g were considered. They correspond to functions ϕ : N → {±} on the set N of positive integers, and give rise to a class of modules called ϕ-Imaginary 2000 Mathematics Subject Classification: 17B67, 81R10. R. A. Martins 29 Verma modules. These modules can be viewed as induced from ϕ-highest weight modules over the Heisenberg subalgebra of g. If ϕ(n) = + for all n ∈ N, then ϕ-Imaginary Verma modules are just Imaginary Verma modules. It was shown in [BBFK] that ϕ-Imaginary Verma module is irreducible if and only if its central charge is nonzero. In the case of Imaginary Verma modules for affine Lie algebra ŝl(n,C) their free field realization was constructed by B.Cox in [C1]. This realiza- tion generalizes the classical Wakimoto construction in terms of infinite sums of partial differential operators. Later this has been generalized by B.Cox and V.Futorny [CF] for other Verma type modules. The purpose of this paper is to extend the construction of [C1] to all ϕ-Imaginary Verma modules over the affine Lie algebra ŝl(n,C). The structure of the paper is as follows. In Section 3, we recall a construction of J-Imaginary Verma modules for affine Lie algebras follow- ing [BBFK]. In Section 4, we consider the case n = 2 and construct a realization ρJ of J-Imaginary Verma modules for affine sl(2). Finally, in Section 5, we construct a representation ρJ : ŝl(n,C) → gl(C[x, y]), for all J, using ϕJ and two auxiliary anti-automorphisms ρ1 : ŝl(n,C) → ŝl(n,C) and ρ2 : gl(C[x, y]) → gl(C[x, y]). Note that generically (when the central charge of the module is not zero) J-Imaginary Verma module is irreducible. Hence, our construction provides free field realization of a large family of irreducible modules for affine ŝl(n,C). 2. Preliminaries Let sl(n,C) be a complex Lie algebra of n×n-matrices with trace zero with the Killing form (X\|Y ) = tr(XY ). Denote by Eij the matrix units, i, j = 1, . . . , n and set Hi = Eii − Ei+1,i+1, i = 1, . . . , n − 1. Then Eij , i 6= j and Hi’s form a basis of sl(n,C). The affine Lie algebra ŝl(n,C) is the universal central extension with the 1-dimensional center Cc of the loop algebra sl(n,C)⊗ C[t, t−1]. Set X(m) := tm ⊗ X, for all X,Y ∈ sl(n,C) and m ∈ Z. Then ŝl(n,C) is generated by Ei(m), Fi(m) and Hi(m), with m ∈ Z and 1 6 i 6 n, and the central element c. Fix a symbol A and a sequence {A(m)}m∈Z. Define: A+(z) := ∑ m∈Z>0 A(m)z−m, (2.1) A−(z) := ∑ m∈Z60 A(m)z−m, (2.2) A(z) := A+(z) +A−(z) = ∑ m∈Z A(m)z−m, (2.3) 30 Free field realizations of certain modules Ȧ(z) := Ȧ+(z) + Ȧ−(z) = ∑ m∈Z mA(m)z−m, (2.4) where, for example, Ḣ+ i (z) = ∑ m∈Z>0mHi(m)z−m. Then we have the following defining relations in ŝl(n,C): (R1) [Hi(z), Hj(w)] = −(Hi\|Hj)cδ̇(z − w), (R2) [Hi(z), Ej(w)] = Cij ∑ nEj(z)z nw−n = CijEj(z)δ(z − w), (R3) [Hi(z), Fj(w)] = −Cij ∑ n Fj(z)z nw−n = −CijFj(z)δ(z − w), (R4) [Ei(z), Fj(w)] = δij(Hi(z)δ(z − w)− cδ̇(z − w)), (R5) [Ei(z), Ej(w)] = [Fi(z), Fj(w)] = 0 if Cij 6= −1, (R6) [Fi(z1), Fi(z2), Fj(w)] = [Ei(z1), Ei(z2), Ej(w)] = 0 if Cij = −1, where C = (Cij) denotes the Cartan matrix of type An and [X,Y, Z] := [X, [Y, Z]]. ElementsHi, i = 1, . . . , n−1 and c span a Cartan subalgebra of ŝl(n,C), while Hi(m), m ∈ Z, 1 6 i 6 n, and c span a Heisenberg subalgebra of ŝl(n,C). The central element c acts a scalar on any irreducible module V . This scalar is called the central charge of V . Now fix γ ∈ C∗ and for all 1 6 i 6 n, fix λi ∈ C. Let 2c = γ2. Then: C[x] := C[xij(m) \| i, j,m ∈ Z, 1 6 i, j 6 n] (2.5) C[y] := C[yi(m) \| i,m ∈ N ∗, 1 6 i 6 n] (2.6) are the algebras generated over C by xij(m) and yi(m), respectively. In C[x], define operators: aij := −xij(m); (2.7) a∗ij(m) := ∂ ∂xij(−m) . (2.8) where [aij(m), a∗kl(p)] = δikδjlδm,−p. Fix an arbitrary J ⊆ N ∗. In C[y] R. A. Martins 31 define operators bi(m), with m ∈ Z and 1 6 i 6 n as follows: bi(m) :=    −γ−1λi if m = 0; −γm ∂ ∂yi(m) if m ∈ N ∗ \ J; −γ−1yi(m) if m ∈ J; γm ∂ ∂yi(−m) if −m ∈ J; −γ−1yi(−m) if −m ∈ N ∗ \ J. (2.9) where we have [bi(m), bj(p)] = mδijδm,−p. Let C[x, y] := C[x]⊗C C[y]. 3. J-Imaginary Verma modules for g Fix J ⊆ N ∗ and let ψJ : Z → {0, 1} be the function given by: ψJ(m) := { 1 if m ∈ J or −m ∈ N ∗ \ J; 0 if m ∈ N ∗ \ J or −m ∈ J. (3.1) Let g = ŝl(n,C) and consider SψJ = {α + nδ \| α ∈ ∆̇+, n ∈ Z} ∪ {nδ \| n ∈ N, ϕJ(n) = 1} ∪ {−mδ \| m ∈ N, ϕJ(m) = 0}. Then the spaces gSψJ = ⊕ α∈SψJ gα and g−SψJ = ⊕ α∈−SψJ gα are subalgebras of g such that g = g−SψJ ⊕ h⊕ gSψJ of g. Let λ ∈ h∗ and suppose λ(c) = a, a ∈ C. Let bψJ = h ⊕ gSψJ be the Borel subalgebra corresponding to SψJ , and note that bψJ ⊃ Cc⊕ L+ψJ . Let Cvλ be a module of dimension one for bψJ such that for all h ∈ h, gSψJvλ = 0 and hvλ = λ(h)vλ. Remark 3.1. We call the g-module MψJ (λ) := MbψJ (λ) = U(g)⊗U(bψJ ) Cvλ a J-Imaginary Verma module. Identifying 1 ⊗ vλ with vλ, the U(L)- submodule of MψJ (λ) generated by vλ is isomorphic to MψJ (a). If ψJ(n) = 1 for all n, then MψJ (λ) coincides with the imaginary Verma module MS(λ) where S = ∆+. Here we present some basic properties of MψJ (λ). Proposition 3.2. [BBFK, Proposition 3.4] Let λ ∈ h∗ be such that λ(c) = a. If a 6= 0, then the following statements are true for MψJ (λ). • MψJ (λ) is a free U(g−SψJ )-module of rank 1. 32 Free field realizations of certain modules • MψJ (λ) has a unique maximal submodule and hence a unique irre- ducible quotient. • supp ( MψJ (λ) ) = ⋃ β∈Q̇+ {λ− β + nδ \| n ∈ Z} where Q̇+ is the free abelian monoid generated by all the simple roots in ∆̇+. • If ψJ(k) 6= ψJ(ℓ) for some k, ℓ ∈ N, then dimMψJ (λ)µ = ∞ for any µ ∈ supp ( MψJ (λ) ) . We have the following irreducibility criterion for modules MψJ (λ). Theorem 3.3. [BBFK, Theorem 3.5] Let λ ∈ h∗, λ(c) = a. Then MψJ (λ) is irreducible if and only if a 6= 0. 4. The case n = 2 We denote E(m) instead of E1(m), am instead of a11(m), etc. Definition 4.1. Let ϕJ : ŝl(2,C) → gl(C[x, y]) be the function given by: ϕJ(E(m)) := − ∂ ∂xm ; ϕJ(F (m)) := ∑ l,j xlxj−l−m ∂ ∂xj + ∑ j∈J xj−m ∂ ∂yj + 2K ∑ i∈J jyjx−j−m + Kmx−m − 2Jx−m; ϕJ(H(m)) := −2 ∑ j xj−m ∂ ∂xj −ψJ(m) ∂ ∂ym +ψJ(−m)2mKy−m+2δm,0J. Lemma 4.2. We have the following: (a) [ϕJ(H(m)), ϕJ(E(n))] = 2ϕJ(E(m+ n)); (b) [ϕJ(H(m)), ϕJ(H(n))] = 2δm,−nmK; (c) [ϕJ(H(m)), ϕJ(F (n))] = −2ϕJ(F (m+ n)); (d) [ϕJ(E(m)), ϕJ(F (n))] = ϕJ(H(m+ n)) +mδm,−nK. Proof. We have: (a) [ϕJ(H(m)), ϕJ(E(n))] = [−2 ∑ j xj−m ∂ ∂xj − ψJ(m) ∂ ∂ym + ψJ(−m)2mKy−m + 2δm,0J,− ∂ ∂xn ] = [−2 ∑ j xj−m ∂ ∂xj ,− ∂ ∂xn ] = 2 ∂ ∂xm+n = 2ϕJ(E(m+ n)). R. A. Martins 33 (b) [ϕJ(H(m)), ϕJ(H(n))] = [−2 ∑ j xj−m ∂ ∂xj − ψJ(m) ∂ ∂ym + ψJ(−m)2mKy−m + 2δm,0J, − 2 ∑ j xj−n ∂ ∂xj − ψJ(n) ∂ ∂yn + ψJ(−n)2nKy−n + 2δn,0J ] = 4 ∑ j xj−m ∂ ∂xj+n − 4 ∑ j xj−n ∂ ∂xj+m − δm,−nψJ(m)ψJ(−n)2nK + 2δm,−nψJ(n)ψJ(−m)Km+ 0 = 2δm,−nmK(ψJ(m) + ψJ(−m)) = 2δm,−nmK. (c) [ϕJ(H(m)), ϕJ(F (n))] = [ − 2 ∑ j xj−m ∂ ∂xj − ψJ(m) ∂ ∂ym + ψJ(−m)2mKy−m + 2δm,0J, ∑ l,i xlxi−l−n ∂ ∂xi + ∑ i∈J xi−n ∂ ∂yi + 2K ∑ i∈J iyix−i−n +Knx−n − 2Jx−n ] = −2 (∑ j xj−m (∑ i xi−j−n ∂ ∂xi + ∑ l xl ∂ ∂xj+l+n ) − ∑ i,l xlxi−l−n ∂ ∂xi+m ) − 2 ∑ j xj−mψJ(j + n) ∂ ∂yj+n − 4K ∑ j xj−mψJ(−j − n)(−j − n)y−j−n − 2Knx−n−m + 4JX−n−m − 2ψJ(m)KψJ(m)mx−m−n − ψJ(−m)2KmψJ(−m)x−m−n = −4 ∑ j,i xjxi−j−m−n ∂ ∂xi + 2 ∑ l,i xlxi−l−m−n ∂ ∂xi − 2 ∑ j ψJ(j)xj−m−n ∂ ∂yj − 4K ∑ j ψJ(−j)(−j)y−jxj−m−n − 2Knx−m−n + 4Jx−m−n − 2Kmx−m−n 34 Free field realizations of certain modules = −2 ∑ j,i xjxi−j−m−n ∂ ∂xi − 2 ∑ j∈J xj−m−n ∂ ∂xj − 4K ∑ j∈J jyjx−j−m−n − 2K(m+ n)x−m−n + 4Jx−m−n = −2ϕJ(F (m+ n)). (d) [ϕJ(E(m)), ϕJ(F (n))] = [ − ∂ ∂xm , ∑ l,j xlxj−l−n ∂ ∂xj + ∑ j∈J xj−n ∂ ∂yj + 2K ∑ j∈J jyjx−j−n +Knx−n − 2Jx−n ] = − (∑ j xj−m−n ∂ ∂xj + ∑ l xl ∂ ∂xl+m+n + ψJ(m+ n) ∂ ∂ym+n − 2KψJ(−m− n)(m+ n)Y−m−n − δm,−n(Km+ 2J) ) = −2 ∑ j Xj−m−n ∂ ∂xj − ψJ(m+ n) ∂ ∂ym+n + 2ψJ(−m− n)(m+ n)Ky−m−n + 2δm,−nJ +Kmδm,−n = ϕJ(H(m+ n)) +mδm,−nK. Now let ρ1 : ŝl(2,C) → ŝl(2,C) be the function given by ρ1(E(m)) = −F (−m), ρ1(F (m)) = −E(−m), ρ1(H(m)) = H(−m) and ρ1(K) = K. Let ρ2 : gl(C[x, y]) → gl(C[x, y]) be the function given by ρ2(x−m) = ∂ ∂xm , ρ2( ∂ ∂xm ) = x−m, ρ2(yk) = − ∂ ∂yk and ρ2( ∂ ∂yk ) = −yk. Note that ρ1 and ρ2 are anti-automorphisms of ŝl(2,C) and gl(C[x, y]), respectively. Definition 4.3. Let ρJ := ρ2 ◦ ϕJ ◦ ρ1. We have: (a) ρJ(F (m)) = ρ2 ◦ ϕJ ◦ ρ1(F (m)) = ρ2 ◦ ϕJ(−E(−m)) = ρ2 ( ∂ ∂x−m ) = xm. (b) ρJ(E(m)) = ρ2 ◦ ϕJ ◦ ρ1(E(m)) = ρ2 ◦ ϕJ(−F (−m)) R. A. Martins 35 = ρ2 ( − ∑ l,j xlxj−l+m ∂ ∂xj − ∑ j∈J xj+m ∂ ∂yj − 2K ∑ j∈J jyjx−j+m −K(−m)xm + 2Jxm ) = − ∑ l,j x−j ∂ ∂x−l ∂ ∂x−j+l−m − ∑ j∈J (−yj) ∂ ∂x−j−m − 2K ∑ j∈J j(− ∂ ∂yj ) ∂ ∂xj−m −K(−m) ∂ ∂x−m + 2J ∂ ∂x−m = − ∑ l,j xj ∂ ∂xl ∂ ∂xj−l−m + ∑ j∈J yj ∂ ∂x−j−m + 2K ∑ j∈J j ∂ ∂yj ∂ ∂xj−m + (Km+ 2J) ∂ ∂x−m = − ∑ l,j xj+l+m ∂ ∂xl ∂ ∂xj + ∑ j∈J yj ∂ ∂x−j−m + 2K ∑ j∈J j ∂ ∂yj ∂ ∂xj−m + (Km+ 2J) ∂ ∂x−m . (c) ρJ(H(m)) = ρ2 ◦ ϕJ ◦ ρ1(H(m)) = ρ2 ◦ ϕJ(H(−m)) = ρ2 ( − 2 ∑ j xj+m ∂ ∂xj − ψJ(−m) ∂ ∂y−m + ψJ(m)2(−m)Kym + 2δm,0J ) = −2 ∑ j x−j ∂ ∂x−j−m + ψJ(−m)y−m + ψJ(m)2mK ∂ ∂ym + 2δm,0J = −2 ∑ j xj+m ∂ ∂xj + ψJ(−m)y−m + ψJ(m)2mK ∂ ∂ym + 2δm,0J. The function ρJ satisfy (R1) - (R6) because ϕJ satisfy these relations and ρ1 and ρ2 are anti-automorphisms. Hence, it defines a representation of ŝl(2,C). Now we can start to study a candidate ρJ when n is arbitrary. 5. Free field realization of ŝl(n,C) Let ρJ be the function given by: (a) ρJ(Fr)(w) := −ar,r+1(w) + r−1∑ j=1 aj,r+1(w)a ∗ jr(w); 36 Free field realizations of certain modules (b) ρJ(Hr)(w) := 2ar,r+1(w)a ∗ r,r+1(w) + r−1∑ i=1 ( ai,r+1(w)a ∗ i,r+1(w)− air(w)a ∗ ir(w) ) + n∑ j=r+2 ( arj(w)a ∗ rj(w)− ar+1,j(w)a ∗ r+1,j(w) ) − γbr(w) + γ 2 ( b+r−1(w) + b+r+1(w) ) ; (c) ρJ(Er)(w) := ar,r+1(w)a ∗ r,r+1(w) 2 − n∑ j=r+2 ar+1,j(w)a ∗ r+1,j(w) + r−1∑ j=1 ajr(w)a ∗ j,r+1(w) + n∑ j=r+2 ( arj(w)a ∗ rj(w)− ar+1,j(w)a ∗ r+1,j(w) ) a∗r,r+1(w) − γa∗r,r+1(w)br(w) + γ 2 a∗r,r+1(w) ( b+r−1(w) + b+r+1(w) ) − γ 2 a∗r,r+1(w). Theorem 5.1. The function ρJ : ŝl(n,C) → gl(C[x, y]) is a representation of ŝl(n,C). To prove that ρJ defines a representation of ŝl(n,C) we need only verify the relations (R1)− (R6) but to do it, we need the following lemmas: Lemma 5.2. We have the following: [aij(z), a ∗ kl(w)] = δikδjlδ(z − w); (5.1) [aij(z)a ∗ kl(z), amn(w)a ∗ pq(w)] = δpiδqjamn(w)a ∗ kl(z)δ(z − w)− δkmδlnaij(z)a ∗ pq(w)δ(w − z); (5.2) [aij(z), ȧ ∗ kl(w)] = δikδjlδ̇(z − w); (5.3) ȧ∗ij(z)δ(w − z) = a∗ij(z)δ̇(w − z)− a∗ij(w)δ̇(w − z); (5.4) [bi(z), b ± j (w)] = [b∓i (z), b ± j (w)] = δij δ̇ ∓(w − z); (5.5) [bi(z), bj(w)] = [b+i (z), b − j (w)] + [b−i (z), b + j (w)] = δij δ̇(w − z); (5.6) [ar,r+1(z)a ∗ r,r+1(z), ar,r+1(w)a ∗ r,r+1(w)a ∗ r,r+1(w)] = ar,r+1(w)a ∗ r,r+1(z)a ∗ r,r+1(w)δ(w − z); (5.7) [arj(z)a ∗ rj(z), f(w)a ∗ kl(w)] = f(w)δrkδjla ∗ rj(z)δ(w − z), (5.8) where f(w) comutes with arj(z), a ∗ rj(z) and a∗kl(w). R. A. Martins 37 Proof. We have: (5.1) [aij(z), a ∗ kl(w)] = ∑ m,n∈Z [aij(m), a∗kl(n)]z −mw−n = ∑ m,n∈Z δikδjlδm,−nz −mw−n = ∑ n∈Z δikδjlz nw−n = δikδjlδ(z − w). (5.2) [aij(z)a ∗ kl(z), amn(w)a ∗ pq(w)] = aij(z)[a ∗ kl(z), amn(w)]a ∗ pq(w) + aij(z)amn(w)a ∗ kl(z)a ∗ pq(w) − amn(w)[a ∗ pq(w), aij(z)]a ∗ kl(z)− amn(w)aij(z)a ∗ pq(w)a ∗ kl(z) = δpiδqjamn(w)a ∗ kl(z)δ(z − w)− δkmδlnaij(z)a ∗ pq(w)δ(w − z). (5.3) [aij(z), ȧ ∗ kl(w)] = ∑ m,n∈Z [aij(m), na∗kl(n)]z −mw−n = ∑ m,n∈Z nδikδjlδm,−nz −mw−n = ∑ n∈Z δikδjlnz nw−n = δikδjlδ̇(z − w). (5.4) ȧ∗ij(z)δ(w − z) = ( ∑ m∈Z (−m)a∗ij(m)z−m)( ∑ n∈Z z−nwn) = ∑ m,n∈Z (−m)a∗ij(m)z−m−nwn = ∑ m,n∈Z −(m+ n)a∗ij(m)z−m−nw(m+n)−m + ∑ m,n∈Z na∗ij(m)z−m−nwn = ∑ m,n∈Z −na∗ij(m)z−nwn−m + ∑ m,n∈Z na∗ij(m)z−mwnz−n = −( ∑ m∈Z a∗ij(m)w−m)δ̇(w − z) + ∑ m∈Z a∗ij(m)z−mδ̇(w − z) = a∗ij(z)δ̇(w − z)− a∗ij(w)δ̇(w − z). (5.5) [bi(z), b − j (w)] = ∑ m∈Z n∈Z∗ − [bi(m), bj(n)]z −mw−n = ∑ m∈Z n∈Z∗ − mδijδm,−nz −mw−n = ∑ m∈Z∗+ n∈Z∗ − mδijδm,−nz −mw−n = [b+i (z), b − j (w)] 38 Free field realizations of certain modules = ∑ n∈Z− (−n)δijznw−n = δij δ̇ +(w − z). and similarly for the other case. (5.6) [bi(z), bj(w)] = [b+i (z), b − j (w)] + [b−i (z), b + j (w)] = ∑ m,n∈Z mδijδm,−nz −mw−n = δij ∑ n∈Z (−n)znw−n = δij δ̇(w − z). (5.7) [ar,r+1(z)a ∗ r,r+1(z), ar,r+1(w)a ∗ r,r+1(w)a ∗ r,r+1(w)] = ar,r+1(z)[a ∗ r,r+1(z), ar,r+1(w)]a ∗ r,r+1(w)a ∗ r,r+1(w) − ar,r+1(w)a ∗ r,r+1(w)[a ∗ r,r+1(w), ar,r+1(z)]a ∗ r,r+1(z) + ar,r+1(z)ar,r+1(w)a ∗ r,r+1(z)a ∗ r,r+1(w)a ∗ r,r+1(w) − ar,r+1(w)a ∗ r,r+1(w)ar,r+1(z)a ∗ r,r+1(w)a ∗ r,r+1(z) = −ar,r+1(z)δ(w − z)a∗r,r+1(w)a ∗ r,r+1(w) + ar,r+1(w)a ∗ r,r+1(w)δ(w − z)a∗r,r+1(z) + ar,r+1(w)[ar,r+1(z), a ∗ r,r+1(w)]a ∗ r,r+1(z)a ∗ r,r+1(w) = ar,r+1(w)a ∗ r,r+1(z)a ∗ r,r+1(w)δ(w − z). (5.8) [arj(z)a ∗ rj(z), f(w)a ∗ kl(w)] = f(w)(arj(z)a ∗ rj(z)a ∗ kl(w)− a∗kl(w)arj(z)a ∗ rj(z)) = f(w)[arj(z), a ∗ kl(w)]a ∗ rj(z) = f(w)δrkδjla ∗ rj(z)δ(w − z). Lemma 5.3. We have the following: n∑ j=r+2 s−1∑ k=1 [aks(w)a ∗ k,s+1(w), arj(z)a ∗ rj(z)] = −δs,r+1ar,r+1(w)a ∗ r,r+2(z)δ(w − z); (5.9) r−1∑ j=1 n∑ k=s+2 [as+1,k(w)a ∗ sk(w), ajr(z)a ∗ j,r+1(z)] = 0; (5.10) n∑ j=r+2 s−1∑ k=1 [aks(w)a ∗ k,s+1(w), ar+1,j(z)a ∗ r+1,j(z)] = 0; (5.11) n∑ j=r+2 n∑ k=s+2 [as+1,k(w)a ∗ sk(w), arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z)] = −2δrsδ(w − z) n∑ j=r+2 ar+1,j(w)a ∗ rj(z) R. A. Martins 39 + δr,s+1δ(w − z) n∑ j=r+2 arj(z)a ∗ r−1,j(w) + δs,r+1δ(w − z) n∑ j=r+3 ar+2,j(w)a ∗ r+1,j(z); (5.12) r−1∑ j=1 [a∗s,s+1(w), ajr(z)a ∗ j,r+1(z)] = −δr,s+1a ∗ r−1,r+1(z)δ(w − z); (5.13) n∑ j=r+2 [a∗s,s+1(w), ar+1,j(z)a ∗ rj(z)] = −δr+1,sa ∗ r,r+2(z)δ(w − z). (5.14) Now we are able to verify (R1) - (R6). Lemma 5.4. (R1) [ρJ(Hr)(z), ρJ(Hs)(w)] = −(Hr\|Hs)cδ̇(z − w). Proof. Observe that if \|r−s\|> 1, then [ρJ(Hr)(z), ρJ(Hs)(w)] = 0 because all summations are equal to zero. So, writing in terms of δs,f(r), we have the following: [ρJ(Hr)(z), ρJ(Hs)(w)] = δrs([ρJ(Hr)(z), ρJ(Hr)(w)]) + δr,s+1([ρJ(Hr)(z), ρJ(Hr−1)(w)]) + δr,s−1([ρJ(Hr)(z), ρJ(Hr+1)(w)]) = δrs ( [2ar,r+1(z)a ∗ r,r+1(z), ρJ(Hr)(w)] + r−1∑ i=1 [ai,r+1(z)a ∗ r,r+1(z)− air(z)a ∗ ir(z), ρJ(Hr)(w)] + n∑ j=r+2 [arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z), ρJ(Hr)(w)] + [−γbr(z) + γ 2 (b+r−1(z) + b+r+1(z)), ρJ(Hr)(w)] ) + δr,s+1 ( [2ar,r+1(z)a ∗ r,r+1(z), ρJ(Hr−1)(w)] + r−1∑ i=1 [ai,r+1(z)a ∗ i,r+1(z)− air(z)a ∗ ir(z), ρJ(Hr−1)(w)] + n∑ j=r+2 [arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z), ρJ(Hr−1)(w)] + [−γbr(z) + γ 2 (b+r−1(z) + b+r+1(z)), ρJ(Hr−1)(w)] ) + δr,s−1 ( [ρJ(Hr)(z), ρJ(Hr+1)(w)] ) 40 Free field realizations of certain modules = δrs ( 4[ar,r+1(z)a ∗ r,r+1(z), ar,r+1(w)a ∗ r,r+1(w)] + r−1∑ i=1 [ai,r+1(z)a ∗ r,r+1(z), ai,r+1(w)a ∗ i,r+1(w)] + r−1∑ i=1 [air(z)a ∗ ir(z), air(w)a ∗ ir(w)] + 0 + γ2[br(z), br(w)] + γ2 4 [b+r−1(z), b + r−1(w)] + γ2 4 [b+r+1(z), b + r+1(w)] ) + δr,s+1 ( [2ar,r+1(z)a ∗ r,r+1(z), n∑ j=r+1 −a(r−1)+1,j(w)a ∗ (r−1)+1,j(w)] + r−1∑ i=1 [−air(z)a ∗ ir(z), r−2∑ j=1 −ajr(w)a ∗ jr(w)] + n∑ j=r+2 [arj(z)a ∗ rj(z), n∑ i=r+1 −ari(w)a ∗ ri(w)] + [−γbr(z), γ 2 b+(r−1)+1(w)] + [ γ 2 b+r−1(z),−γbr−1(w)] ) + δr,s−1 ( [ρJ(Hr)(z), ρJ(Hr+1)(w)] ) = δrs(0 + 0 + 0 + 0 + 2cδ̇(w − z) + 0 + 0) + δr,s+1(−2[ar,r+1(z)a ∗ r,r+1(z), ar,r+1(w)a ∗ r,r+1(w)] − r−2∑ i=1 [air(z)a ∗ ir(z), air(w)a ∗ ir(w)] + n∑ j=r+2 [arj(z)a ∗ rj(z),−arj(w)a ∗ rj(w)] − cδ̇−(w − z) + cδ̇−(z − w)) + δr,s−1([ρJ(Hr)(z), ρJ(Hr+1)(w)]) + δrs(2cδ̇(w − z)) + δr,s+1(0 + 0 + 0 +−cδ̇−(w − z)− cδ̇+(w − z)) + δr,s−1([ρJ(Hr)(z), ρJ(Hr+1)(w)]) = δrs(2cδ̇(w − z)) + δr,s+1(−cδ̇(w − z)) + δr,s−1(−cδ̇(w − z)), where in the last equality we have: [ρJ(Hr)(z), ρJ(Hr+1)(w)] = − [ρJ(Hr+1)(w), ρJ(Hr)(z)] = cδ̇(z − w) = −cδ̇(w − z). Then the relation (R1) is verified. R. A. Martins 41 Lemma 5.5. (R2)[ρJ(Hr)(z), ρJ(Es)(w)] = CrsρJ(Es)(z)δ(z − w). Proof. We have the following: (a) 2[ar,r+1(z)a ∗ r,r+1(z), ρJ(Es)(w)] = δsr(2[ar,r+1(z)a ∗ r,r+1(z), ar,r+1(w)a ∗ r,r+1(w)a ∗ r,r+1(w)] + 2[ar,r+1(z)a ∗ r,r+1(z), ( n∑ j=r+2 arj(w)a ∗ rj(w)− ar+1,j(w)a ∗ r+1,j(w))a ∗ r,r+1(w)] + 2[ar,r+1(z)a ∗ r,r+1(z), − γa∗r,r+1(w)( 1 2 + br(w)− 1 2 b+r−1(w)− 1 2 b+r+1(w))]) + δs,r+1(2[ar,r+1(z)a ∗ r,r+1(z), ar,r+1(w)a ∗ r,r+2(w)]) + δs,r−1(−2a∗r−1,r+1(z)ar,r+1(w)δ(w − z)) (5.8) (5.7) = δsr(2ar,r+1(w)a ∗ r,r+1(z)a ∗ r,r+1(w)δ(w − z) + 2 n∑ j=r+2 (arj(w)a ∗ rj(w)− ar+1,j(w)a ∗ r+1,j(w))a ∗ r,r+1(z)δ(w − z) − 2γa∗r,r+1br(w)δ(w − z) + γa∗r,r+1(z)(b + r−1(w) + b+r+1(w))δ(w − z) − γ2a∗r,r+1(z)δ̇(w − z)) + δs,r+1(2ar,r+1(z)a ∗ r,r+2(w)δ(w − z)) + δs,r−1(−2a∗r−1,r+1(z)ar,r+1(w)δ(w − z)). (b) r−1∑ i=1 [ai,r+1(z)a ∗ i,r+1(z)− air(z)a ∗ ir(z), ρJ(Es)(w)] = δsr( r−1∑ i=1 [ai,r+1(z)a ∗ i,r+1(z)− air(z)a ∗ ir(z), r−1∑ j=1 ajr(w)a ∗ j,r+1(w)]) + δs,r+1( r−1∑ i=1 [ai,r+1(z)a ∗ i,r+1(z), ai,r+1(w)a ∗ i,r+2(w)]) + δs,r−1(−(ar−1,r(w)a ∗ r−1,r(z)a ∗ r−1,r(w) + n∑ j=r+1 (ar−1,j(w)a ∗ r−1,j(w) − arj(w)a ∗ rj(w))a ∗ r−1,r(z) + r−2∑ j=1 aj,r−1(w)a ∗ jr(z) − ar,r+1(w)a ∗ r−1,r+1(z)− γa∗r−1(w)br−1(w) + γ 2 (a∗r−1,r(w)b + r−2(w)− a∗r−1,r(w)b + r (w))δ(w − z) 42 Free field realizations of certain modules + γ2 2 a∗r−1,r(z)δ̇(w − z)) (5.2) = δsr(2 r−1∑ i=1 air(w)a ∗ i,r+1(w)δ(w − z)) + δs,r+1(− r−1∑ i=1 ai,r+1(z)a ∗ i,r+2(w)δ(w − z)) + δs,r−1(−(ar−1,r(w)a ∗ r−1,r(z)a ∗ r−1,r(w) + n∑ j=r+1 (ar−1,j(w)a ∗ r−1,j(w) − arj(w)a ∗ rj(w))a ∗ r−1,r(z) + r−2∑ j=1 aj,r−1(w)a ∗ jr(z) − ar,r+1(w)a ∗ r−1,r+1(z)− γa∗r−1(w)br−1(w) + γ 2 (a∗r−1,r(w)b + r−2(w)− a∗r−1,r(w)b + r (w))δ(w − z) + γ2 2 a∗r−1,r(z)δ̇(w − z)). (c) n∑ j=r+2 [arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z), ρJ(Es)(w)] = δsr( n∑ j=r+2 [−ar+1,j(z)a ∗ r+1,j(z),−ar+1,j(w)a ∗ r+1,j(w)] + n∑ j=r+2 [arj(z)a ∗ rj(z), arj(w)a ∗ rj(w)] + n∑ j=r+2 [arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z), (arj(w)a ∗ rj(w)− ar+1,j(w)a ∗ r+1,j(w))a ∗ r,r+1(w)]) + δs,r+1(−(ar+1,r+2(w)a ∗ r+1,r+2(z)a ∗ r+1,r+2(w) + n∑ j=r+3 (ar+1,j(w)a ∗ r+1,j(w) − ar+2,j(w)a ∗ r+2,j(w))a ∗ r+1,r+2(z)− ar,r+1(w)a ∗ r,r+2(z) − n∑ j=r+3 ar+2,j(w)a ∗ r+1,j(w) − γa∗r+1,r+2(w)br+1(w) + γ 2 a∗r+1,r+2(w)(b + r (w) + b+r+2(w)))δ(w − z) R. A. Martins 43 + γ2 2 a∗r+1,r+2(z)δ̇(w − z)) + δs,r−1( n∑ j=r+2 arj(w)a ∗ r−1,j(w)δ(w − z)) (5.2) (5.7) = δsr(0 + 0 + 2 n∑ j=r+2 ar+1,j(w)a ∗ rj(z)δ(w − z)) + δs,r+1(−(ar+1,r+2(w)a ∗ r+1,r+2(z)a ∗ r+1,r+2(w) + n∑ j=r+3 (ar+1,j(w)a ∗ r+1,j(w) − ar+2,j(w)a ∗ r+2,j(w))a ∗ r+1,r+2(z)− ar,r+1(w)a ∗ r,r+2(z) − n∑ j=r+3 ar+2,j(w)a ∗ r+1,j(w) − γa∗r+1,r+2(w)br+1(w) + γ 2 a∗r+1,r+2(w)(b + r (w) + b+r+2(w)))δ(w − z) + γ2 2 a∗r+1,r+2(z)δ̇(w − z)) + δs,r−1( n∑ j=r+2 arj(w)a ∗ r−1,j(w)δ(w − z)). Finally, we have: (d) [−γbr(z) + γ 2 (b+r−1(z) + b+r+1(z)), ρJ(Es)(w)] = δsr(γ 2[br(z), a ∗ r,r+1(w)br(w)]) + δs,r+1(− γ2 2 a∗r+1,r+2(w)([br(z), b + r (w)] + [b+r+1(z), br+1(w)])) + δs,r−1( γ2 2 a∗r−1,r(w)([br(z), b + r (w)] + [b+r−1(z), br−1(w)])) = δsr(γ 2a∗r,r+1(w)δ̇(w − z)) + δs,r+1(− γ2 2 a∗r+1,r+2(w)δ̇(w − z)) + δs,r−1( γ2 2 a∗r−1,r(w)δ̇(w − z)). Adding the four equations up we get: [ρJ(Hr)(z), ρJ(Es)(w)] = δsr2ρJ(Er)(z)δ(w − z) + δs,r+1(−ρJ(Er+1)(z)δ(w − z)) + δs,r−1(−ρJ(Er−1)(z)δ(w − z)), proving (R2). Lemma 5.6. (R3)[ρJ(Hr)(z), ρJ(Fs)(w)] = −CrsρJ(Fs)(z)δ(z − w). 44 Free field realizations of certain modules Proof. We have: (a) 2[ar,r+1(z)a ∗ r,r+1(z), ρJ(Fs)(w)] = δsr(2ar,r+2(z)δ(z − w)) + δs,r+1(2[ar,r+1(z)a ∗ r,r+1(z), ar,r+2(w)a ∗ r,r+1(w)]) + δs,r−10 = δsr(2ar,r+2(z)δ(z − w)) + δs,r+1(2ar,r+2(w)a ∗ r,r+1(z)δ(w − z)). (b) r−1∑ i=1 [ai,r+1(z)a ∗ i,r+1(z)− air(z)a ∗ ir(z), ρJ(Fs)(w)] = δsr( r−1∑ i=1 [ai,r+1(z)a ∗ i,r+1(z), ai,r+1(w)a ∗ ir(w)] − r−1∑ i=1 [air(z)a ∗ ir(z), ai,r+1(w)a ∗ ir(w)]) + δs,r+1( r∑ j=1 [ai,r+1(z)a ∗ i,r+1(z), aj,r+2(w)a ∗ j,r+1(w)]) + δs,r−1([ar−1,r(z)a ∗ r−1,r(z), ar−1,r(w)] − r−2∑ i=1 air(z)[a ∗ ir(z), air(w)]a ∗ i,r−1(w)) = δsr(−2 r−1∑ i=1 ai,r+1(z)a ∗ ir(w)δ(z − w)) + δs,r+1( r−1∑ i=1 ai,r+2(w)a ∗ i,r+1(z)δ(z − w)) + δs,r−1((−ar−1,r(z) + r−2∑ i=1 air(z)a ∗ i,r−1(w))δ(z − w)). (c) n∑ j=r+2 [arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z), ρJ(Es)(w)] = δsr0 + δs,r+1( n∑ j=r+2 r∑ k=1 [arj(z)a ∗ rj(z), ak,r+2(w)a ∗ k,r+1(w)] + n∑ j=r+2 [ar+1,j(z)a ∗ r+1,j(z), ar+1,r+2(w)]) + δs,r−10 = δs,r+1(−ar,r+2(z)a ∗ r,r+1(w)δ(z − w)− ar+1,r+2(z)δ(z − w)). (d) [−γbr(z) + γ 2 (b+r−1(z) + b+r+1(z)), ρJ(Fs)(w)] = 0. R. A. Martins 45 Adding the four equations up, we have the result. Lemma 5.7. (R4) [ρJ(Er)(z), ρJ(Fs)(w)] = δsr(ρJ(Hr)(z)δ(z − w) − cδ̇(z − w)). Proof. Consider the case \|s− r\| > 1. We have [ρ(Er)(z),−ar,r+1(w)] = 0 and ∑r−1 j=1 ∑s−1 k=1[akr(z)a ∗ k,r+1(z), aj,s+1(w)a ∗ js(w)] = 0, because r 6= s. Finally: s−1∑ j=1 [ n∑ k=r+2 (ark(z)a ∗ rk(z)− ar+1,k(z)a ∗ r+1,k(z))a ∗ r,r+1(z) + n∑ k=r+2 ar+1,k(z)a ∗ rk, aj,s+1(w)a ∗ js(w)] = s−1∑ j=1 n∑ k=r+2 [ark(z)a ∗ rk(z)− ar+1,k(z)a ∗ r+1,k(z), aj,s+1(w)a ∗ js(w)]a ∗ r,r+1(z) − s−1∑ j=1 n∑ k=r+2 [ar+1,k(z)a ∗ rk(z), aj,s+1(w)a ∗ js(w)] = s−1∑ j=1 n∑ k=r+2 (aj,s+1(w)a ∗ rk(z)δskδrj − ark(z)a ∗ js(w)δs+1,kδjr − aj,s+1(w)a ∗ r+1,k(z)δr+1,jδks + ar+1,k(z)a ∗ js(w)δr+1,jδk,s+1)a ∗ r,r+1(z)δ(z − w) + s−1∑ j=1 n∑ k=r+2 (ar+1,k(z)a ∗ js(w)δrjδk,s+1 − aj,s+1(w)a ∗ rk(z)δr+1,jδks)δ(z − w) = ⊛ If s + 1 > n then aj,s+1(w) = 0 = δk,s+1 and ⊛ = 0. Now suppose s+ 1 6 n. If s− 1 < r then j 6 s− 1 < r, δjr = δj,r+1 = 0 and ⊛ = 0. If s−1 > r we have s−1 > r because \|s−r\|> 1. Then s > r+2 and for this reason the terms j = r, r + 1 and k = s, s+ 1 appears in the expression. Then: ⊛ =(ar,s+1(w)a ∗ rs(z)− ar,s+1(z)a ∗ rs(w)− ar+1,s+1(w)a ∗ r+1,s(z) + ar+1,s+1(z)a ∗ r+1,s(w))a ∗ r,r+1(z)δ(z − w) + (ar+1,s+1(z)a ∗ rs(w)− ar+1,s+1(w)a ∗ rs(z))δ(z − w) = 0. So we only have nontrivial expressions when r = s, r = s − 1 or r = s+ 1. Then: [ρJ(Er)(z), ρJ(Fs)(w)] = δrs([ρJ(Er(z)),−ar,r+1(w)] 46 Free field realizations of certain modules + [ρJ(Er(z)), r−1∑ j=1 (aj,r+1(w)a ∗ jr(w)]) + δr,s−1([ρJ(Es−1(z)),−as,s+1(w)] + [ρJ(Es−1(z)), s−1∑ j=1 (aj,s+1(w)a ∗ js(w)]) + δr,s+1([ρJ(Es+1)(z),−as,s+1(w)] + [ρJ(Es+1)(z), s−1∑ j=1 aj,s+1(w)a ∗ js(w)]) = δrs ( (2ar,r+1(z)a ∗ r,r+1(z) + n∑ j=r+2 (a∗rj(z)arj(z)− a∗r+1,j(z)ar+1,j(z))− γbr(z) + γ 2 (b+r−1(z) + b+r+1(z)))δ(z − w)− γ2 2 δ̇(z − w) + r−1∑ j=1 [ajr(z)a ∗ j,r+1(z), aj,r+1(w)a ∗ jr(w)] ) + δr,s+1(0 + 0) + δr,s−1(− n∑ j=s+1 [as,s+1(w), asj(z)a ∗ s−1,s(z)a ∗ sj(z)] + as,s+1(z)a ∗ s−1,s(z)δ(w − z)) = δrs((2ar,r+1(z)a ∗ r,r+1(z) + n∑ j=r+2 (a∗rj(z)arj(z)− a∗r+1,j(z)ar+1,j(z))− γbr(z) + γ 2 (b+r−1(z) + b+r+1(z)))δ(z − w) − γ2 2 δ̇(z − w) + r−1∑ j=1 (aj,r+1(z)a ∗ j,r+1(z) − ajr(z)a ∗ jr(z))δ(z − w)) + δr,s−1(−as,s+1(z)a ∗ s−1,s(z)δ(w − z) + as,s+1(z)a ∗ s−1,s(z)δ(w − z)) = δsr(ρJ(Hr)(z)δ(z − w)− cδ̇(z − w)). Then the lemma is verified. Lemma 5.8. We have the following: (R5\|R6) [ρJ(Fr)(z), ρJ(Fs)(w)] = [ρJ(Er)(z), ρJ(Es)(w)] = 0 if Crs 6= −1; R. A. Martins 47 [ρJ(Fr)(z1), ρJ(Fr)(z2), ρJ(Fs)(w)] = [ρJ(Er)(z1), ρJ(Er)(z2), ρJ(Es)(w)] = 0 if Crs = −1. Proof. We have: [ρJ(Fr)(z), ρJ(Fs)(w)] = [−ar,r+1(z), ρJ(Fs)(w)] + [ r−1∑ j=1 aj,r+1(z)a ∗ jr(z), ρJ(Fs)(w)] = δs,r+1ar,r+2(w)δ(w − z) + δr,s+1ar−1,r+1(w)δ(w − z) − δr,s+1 s−1∑ j=1 aj,r+1(w)a ∗ js(w)δ(w − z) + δs,r+1 r−1∑ j=1 aj,r+2(w)a ∗ jr(w)δ(w − z) = (δs,r+1(ar,r+2(w) + r−1∑ j=1 aj,r+2(w)a ∗ jr(w)) + δr,s+1(ar−1,r+1(w) − r−2∑ j=1 aj,r+1(w)a ∗ j,r−1(w)))δ(w − z). Then [ρJ(Fr)(z), ρJ(Fs)(w)] = 0 if \|r − s\|6= 1 (or equivalently, if Crs 6= −1). Now: [ρJ(Er)(z), ρJ(Es)(w)] = [ar,r+1(z)a ∗ r,r+1(z) 2, ρJ(Es)(w)] + [ n∑ j=r+2 (arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z))a ∗ r,r+1(z), ρJ(Es)(w)] + [ r−1∑ j=1 ajr(z)a ∗ j,r+1(z)− n∑ j=r+2 ar+1,j(z)a ∗ rj(z), ρJ(Es)(w)] + [−γa∗r,r+1(z)(br(z)− 1 2 a∗r,r+1(z)(b + r−1(z) + b+r+1(z))) − γ2 2 ȧ∗r,r+1(z), ρJ(Es)(w)] = (−δrs n∑ j=r+2 (arj(w)a ∗ rj(w)− ar+1,j(w)a ∗ r+1,j(w))a ∗ r,r+1(z) 2δ(w − z) − δr,s+1ar,r+1(w)a ∗ r−1,r(w)a ∗ r,r+1(z) 2δ(w − z) + 2δr,s−1ar,r+1(z)a ∗ r,r+1(z)a ∗ r,r+2(w)δ(w − z) − 2δr,s+1ar,r+1(z)a ∗ r,r+1(z)a ∗ r−1,r+1(w)δ(w − z) 48 Free field realizations of certain modules − δrsγa ∗ r,r+1(z) 2(br(w)− 1 2 (b+r−1(w) + b+r+1(w)))δ(w − z) − δrs γ2 2 a∗r,r+1(z) 2δ̇(w − z)) + (δrs n∑ j=r+2 (arj(w)a ∗ rj(w) − ar+1,j(w)a ∗ r+1,j(w))a ∗ r,r+1(z) 2δ(w − z) + δs,r+1as,s+1(w)a ∗ s,s+1(z) 2a∗s,s+1(w)δ(w − z) + δs,r+1 n∑ k=r+3 (ar+1,k(w)a ∗ r+1,k(w) − ar+2,k(w)a ∗ r+2,j(w))a ∗ r,r+1(z)a ∗ r+1,r+2(z)δ(w − z) − δr,s+1 n∑ j=r+2 (arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z)) a∗r−1,r(w)a ∗ r,r+1(w)δ(w − z) + n∑ j=r+2 (arj(z)a ∗ rj(z)− ar+1,j(z)a ∗ r+1,j(z)) (δr,s+1a ∗ r,r+2(w)− δr,s+1a ∗ r−1,r+1(w))δ(w − z) − δs,r+1(ar,r+1(w)a ∗ r,r+2(z) + n∑ j=r+3 ar+2,k(w)a ∗ r+1,j(z))a ∗ r,r+1(z)δ(w − z) + 2δrs n∑ j=r+2 ar+1,j(w)a ∗ rj(z)a ∗ r,r+1(z)δ(w − z) − δr,s+1 n∑ j=r+2 arj(z)a ∗ r−1,j(w)a ∗ r,r+1(z)δ(w − z) + a∗r,r+1(z)a ∗ r+1,r+2(z)δs,r+1(γbs(w)δ(w − z) − γ 2 (b+s−1(w) + b+s+1(w))δ(w − z) + γ2 2 δ̇(w − z))) + ((−2δs+1,ras,s+1(z)a ∗ s,s+1(z)a ∗ s,s+2(w) + 2δr+1,sas,s+1(z)a ∗ s,s+1(z)a ∗ s−1,s+1(w) − δr,s+1( n∑ j=s+2 (asj(z)a ∗ sj(z)− as+1,j(z)a ∗ s+1,j(z))a ∗ s,s+2(w) + as,s+1(w)a ∗ s,s+1(z)a ∗ s,s+2(z))− n∑ j=s+3 as+2,k(w)a ∗ s+1,j(z)a ∗ s,s+1(z) R. A. Martins 49 + δs,r+1 n∑ j=s+2 asj(z)a ∗ s−1,j(w)a ∗ s,s+1(z) + δs,r+1 n∑ j=s+2 (asj(z)a ∗ sj(z)− as+1,j(z)a ∗ s+1,j(z))a ∗ s−1,s+1(w) + δs+1,r(− r−2∑ j=1 aj,r−1(w)a ∗ j,r+1(z) + n∑ j=r+2 ar+1,j(z)a ∗ r−1,j(w)) + δs,r+1( r−1∑ j=1 ajr(z)a ∗ j,r+2(w)− n∑ j=r+3 ar+2,j(w)a ∗ rj(z)))δ(w − z) + (δr+1,sa ∗ s−1,s+1(z)− δr,s+1a ∗ r−1,r+1(z))((γbs(z) − γ 2 (b+s−1(z) + b+s+1(z)))δ(w − z) + γ 2 δ̇(w − z))− γ2 2 δr,s+1ȧr−1,r+1(z)δ(w − z)) + (δsrγa ∗ s,s+1(z) 2(bs(w) − 1 2 (b+s−1(w) + b+s+1(w)))δ(w − z) + δsr γ2 2 a∗s,s+1(z) 2δ̇(w − z) − a∗s,s+1(w)a ∗ s+1,s+2(w)δr,s+1(γbr(z)δ(z − w) − γ 2 (b+r−1(z) + b+r+1(z))δ(z − w) + γ2 2 δ̇(z − w)) + (δs,r+1a ∗ s−1,s+1(z)− δs+1,ra ∗ r−1,r+1(z))((γbr(z)− γ 2 (b+r−1(z) + b+r+1(z)))δ(w − z) + γ2 2 δ̇(w − z))− γ2 2 δr,s+1ȧr−1,r+1(z)δ(w − z) + γ2a∗r,r+1(z)a ∗ s,s+1(w)(δrs − 1 2 (δr,s+1 + δs,r+1))δ̇(w − z)). After doing the calculations above, the term in δrs become equal to zero. So we have only terms in δr,s+1 and δr,s−1. Then: [ρJ(Er)(z), ρJ(Es)(w)] = 0 if \|r − s\|6= 1, and (R5) is verified. To finish we have: [ρJ(Fr)(z1), ρJ(Fr)(z2), ρJ(Fs)(w)] := [ρJ(Fr)(z1), [ρJ(Fr)(z2), ρJ(Fs)(w)]] = [−ar,r+1(z1), (ar−1,r+1(w)− r−2∑ j=1 aj,r+1(w)a ∗ j,r−1(w))δ(w − z2)] + [ r−1∑ j=1 aj,r+1(z1)a ∗ jr(z1), ar−1,r+1(w) 50 Free field realizations of certain modules − r−2∑ l=1 al,r+1(w)a ∗ l,r−1(w))δ(w − z2)] = 0 + 0 = 0. Similarly for s = r + 1. If s = r − 1 we have: (a) [ar,r+1(z1)a ∗ r,r+1(z1) 2, [ρJ(Er)(z2), ρJ(Er−1)(w)]] = −ar,r+1(w)a ∗ r,r+1(z1) 2a∗r−1,r+1(w)δ̇(z1 − z2)δ(w − z2) + n∑ j=r+2 (arj(z2)a ∗ rj(z2)− ar+1,j(z2)a ∗ r+1,j(z2)) × a∗r,r+1(z1) 2a∗r−1,r(w)δ(z1 − z2)δ(w − z2) + n∑ j=r+2 arj(z2)a ∗ r−1,j(w)a ∗ r,r+1(z1) 2δ(z1 − z2)δ(w − z2) + a∗r,r+1(z1) 2a∗r−1,r(w)δ(z1 − z2)δ(w − z2)(γbr(z2) − γ 2 (b∗r−1(z2) + b+r+1(z2))) − γ2 2 a∗r,r+1(z1) 2a∗r−1,r(w)δ̇(w − z1)δ(z2 − w). (b) [ n∑ j=r+2 (arj(z1)a ∗ rj(z1)− ar+1,j(z1)a ∗ r+1,j(z1))a ∗ r,r+1(z1), [ρJ(Er)(z2), ρJ(Er−1)(w)]] = − n∑ j=r+2 (arj(z1)a ∗ rj(z1)− ar+1,j(z1)a ∗ r+1,j(z1)) × a∗r,r+1(z1) 2a∗r−1,r(w)δ(z1 − z2)δ(w − z2) − n∑ j=r+2 (arj(z1)a ∗ rj(z1)− ar+1,j(z1)a ∗ r+1,j(z1)) × a∗r,r+1(z1)a ∗ r−1,r+1(w)δ(z1 − z2)δ(w − z2) − 2 n∑ j=r+2 ar+1,j(z1)a ∗ rj(w)a ∗ r,r+1(z1)a ∗ r−1,r(w)δ(z1 − z2)δ(w − z2) − n∑ j=r+2 arj(z2)a ∗ r−1,j(w)a ∗ r,r+1(z1) 2δ(z1 − z2)δ(w − z2) − n∑ j=r+2 ar+1,j(z1)a ∗ r−1,j(w)a ∗ r,r+1(z1)δ(w − z2)δ(z1 − z2). R. A. Martins 51 (c) [ r−1∑ j=1 ajr(z1)a ∗ j,r+1(z1) − n∑ j=r+2 ar+1,j(z1)a ∗ rj(z1), [ρJ(Er)(z2), ρJ(Er−1)(w)]] = (a∗r−1,r+1(z1)a ∗ r,r+1(w) n∑ j=r+2 (a∗rj(z2)arj(z2)− a∗r+1,j(z2)ar+1,j(z2)) − a∗r−1,r+1(z1)a ∗ r,r+1(w)(γbr(z1)− γ 2 (b+r−1(z1) + b+r+1(z1))) + γ2 2 a∗r−1,r+1(z1)ȧ ∗ r,r+1(w) + 2 n∑ j=r+2 ar+1,j(z1)a ∗ rj(z1)a ∗ r−1,r(w)a ∗ r,r+1(w) + n∑ j=r+2 ar+1,j(z1)a ∗ r−1,j(w)a ∗ r,r+1(z2))δ(z1 − z2)δ(w − z2). (d) [−γa∗r,r+1(z1)(br(z1)− 1 2 a∗r,r+1(z1)(b + r−1(z1) + b+r+1(z1))) − γ2 2 ȧ∗r,r+1(z1), [ρJ(Er)(z2), ρJ(Er−1)(w)]] = −a∗r,r+1(z2) 2a∗r−1,r(w)δ(z1 − z2)δ(w − z2)(γbr(z1) − γ 2 (b+r−1(z1) + b+r+1(z1))) − γ2 2 a∗r,r+1(z2) 2a∗r−1,r(w)δ̇(z1 − w)δ(w − z2) + a∗r,r+1(z2)a ∗ r−1,r+1(w)δ(z1 − z2)δ(w − z2)(γbr(z1) − γ 2 (b+r−1(z1) + b+r+1(z1))) − γ2 2 a∗r,r+1(z2)a ∗ r−1,r+1(w)δ̇(z1 − z2)δ(w − z1) − γ2a∗r,r+1(z1)a ∗ r−1,r(w)a ∗ r,r+1(w)δ(z2 − w)δ̇(z2 − z1) + γ2 2 a∗r,r+1(z1)a ∗ r−1,r+1(z2)δ̇(z1 − z2)δ(w − z2). Adding the four summations up we have: [ρJ(Er)(z1), ρJ(Er)(z2), ρJ(Er−1)(w)] = 0 Similarly: [ρJ(Er)(z1), ρJ(Er)(z2), ρJ(Er+1)(w)] = 0 and then (R6) is proved. 52 Free field realizations of certain modules 6. Acknowledgment This work is part of the Ph.D. Thesis of the author, who was supported by FAPESP (Process number: 2008/06860-3). The author is grateful to his supervisor V. Futorny and to B. Cox for stimulating discussions. References [BBFK] Bekkert, V., Benkart, G., Futorny, V., Kashuba, I., New Irreducible modules for Heisenberg and Affine Lie Algebras, preprint 2010. [C1] COX, B., Fock Space Realizations of Imaginary Verma Modules, Algebras and Representation Theory (2005) 8: 173-206 [C2] B. Cox, Verma modules induced from nonstandard Borel subalgebras, Pacific J. Math. 165 (1994), 269–294. [CF] Cox, B., Futorny V. Intermediate Wakimoto modules for affine sl(n+1). Journal of Physics. A, Mathematical and General, v. 37, n. 21, p. 5589-5603, 2004. [DFG] I. Dimitrov, V. Futorny, and D. Grantcharov, Parabolic sets of roots, Groups, Rings and Group Rings, 61–73, Contemp. Math., 499 Amer. Math. Soc., Providence, RI, 2009. [F1] FUTORNY, V.M., Imaginary Verma modules for affine Lie algebra, Canad. Math. Bull. 37(2)(1994), 213-218 [F2] V. Futorny, The parabolic subsets of root systems and corresponding repre- sentations of affine Lie algebras, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989), 45–52, Contemp. Math., 131, Part 2, Amer. Math. Soc., Providence, RI, 1992. [F3] V. Futorny, Irreducible non-dense A (1) 1 -modules, Pacific J. Math. 172 (1996), 83-99. [F4] V. Futorny, Representations of Affine Lie Algebras, Queen’s Papers in Pure and Applied Math., 106 Queen’s University, Kingston, ON, 1997. [FS] V. Futorny and H. Saifi, Modules of Verma type and new irreducible rep- resentations for affine Lie algebras, Representations of algebras (Ottawa, ON, 1992), 185–191, CMS Conf. Proc. 14 Amer. Math. Soc., Providence, RI, 1993. [FK] V. Futorny and I.Kashuba, Induced modules for Kac-Moody Lie algebras, SIGMA - Symmetry, Integrability and Geometry: Methods and Applications 5 (2009), 026. [JK] H.P. Jakobsen and V. Kac, A new class of unitarizable highest weight rep- resentations of infinite dimensional Lie algebras, Nonlinear equations in classical and quantum field theory (Meudon/Paris, 1983/1984), 1–20, Lecture Notes in Phys. 226 Springer, Berlin, 1985. Contact information R. A. Martins Instituto de Matemática e Estat́istica, Universi- dade de São Paulo, Rua do Matão, 1010- Cidade Universitária, São Paulo SP, Brasil E-Mail: renatoam@ime.usp.br Received by the editors: 15.07.2011 and in final form 15.07.2011.

Free field realizations of certain modules for affine Lie algebra slˆ(n,C)

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