Fusion Product of Positive Level Representations and Lie Algebra Homology

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 26 March 2015

This is an excerpt of the paper Fusion Product of Positive Level Representations and Lie Algebra Homology by Shrawan Kumar, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA.

Preliminaries and Notation

Let 𝔤 be a finite dimensional complex simple Lie algebra. (We also fix a Borel subalgebra 𝔟 and a Cartan subalgebra 𝔥𝔟 of 𝔤.) Then the associated affine Kac-Moody Lie algebra is by definition the space 𝔤𝔤 [t±1]K together with the Lie bracket (for X,Y𝔤 and P,Q[t±1]) [XP,YQ]= [X,Y]PQ+ (X,Y)Rest=0(dPdtQ)) K,and [𝔤,K]=0, where ·,·) is the Killing form on 𝔤, normalized so that θ,θ=2 for the highest root θ of 𝔤.

The Lie algebra 𝔤 sits as a Lie subalgebra of 𝔤 as 𝔤t0. The Lie algebra 𝔤 admits a distinguished "parabolic" subalgebra 𝔭𝔤[t] K. We also define its "nil-radical" 𝔲 (which is an ideal of 𝔭) by 𝔲𝔤t[t], and its "Levi component" (which is a Lie subalgebra of 𝔭) 𝔭0𝔤 t0K. Clearly (as a vector space) 𝔭=𝔲𝔭0.

Also define 𝔲-=𝔤t-1[t-1]𝔤, and the Cartan subalgebra 𝔥=𝔥t0K of 𝔤. Let W (resp. W) be the Weyl group of 𝔤 (resp. 𝔤). Then W acts on the dual space 𝔤* by linear automorphisms. Let ρ𝔥* be half the sum of the positive roots of 𝔤 and define ρ𝔥* by ρ|𝔥=ρ and ρ(K)=1+ ρ,θ= dual Coxeter number of𝔤, where θ (as earlier) is the highest root of 𝔤 and θ is the associated coroot. Define the shifted action of W on 𝔥* by w*β=w(β+ρ)-ρ, for β𝔥* and wW. Fix a positive integer . Let P+𝔥* be the set of dominant integral weights of 𝔤 and let P+{λP+:λ,θ} be the fundamental alcove.

Define the loop algebra Ω(𝔤)𝔤[t±1] with Lie bracket [XP,YQ]=[X,Y]PQ, for X,Y𝔤 and P,Q[t±1]. Then 𝔤 can be viewed as a one-dimensional central extension of Ω(𝔤): 0K𝔤π Ω(𝔤)0, (1.1) where the Lie algebra homomorphism π is defined by π(XP)=XP and π(K)=0.

Irreducible Representations of 𝔤

Fix an irreducible (finite-dimensional) representation V of 𝔤 and a number (to be called the level or central charge). Then we define the associated generalized Verma module for 𝔤 as M(V,)=U(𝔤) U(𝔭) I(V), where the 𝔭-module I(V) has the same underlying vector space as V on which 𝔲 acts trivially, the central element K acts via the scalar and the action of 𝔤=𝔤t0 is via the 𝔤-module structure on V.

In the case when is a positive integer (in fact, it suffices to assume that -h, where h is the dual Coxeter number of 𝔤), M(V,) has a unique irreducible quotient, denoted L(V,).

It is easy to see that any vector vM(V,) is contained in a finite dimensional 𝔤-submodule of M(V,). In particular, the same property holds for any vector in L(V,).

Consider the Lie subalgebra 𝔯0 of 𝔤 spanned by {Yθt,θ1,Xθt-1}, where Yθ (resp. Xθ) is a non-zero root vector of 𝔤 corresponding to the root -θ (resp. θ) and the coroot θ is to be thought of as an element of 𝔥. Then the Lie algebra 𝔯0 is isomorphic to sl(2).

A 𝔤-module W is said to be integrable if every vector vW is contained in a finite-dimensional 𝔤-submodule of W and also v is contained in a finite-dimensional 𝔯0-submodule of W.

Then it follows easily from the sl(2)-theory that the irreducible module L(V,) is integrable if and only if is an integer and (λ,θ), where λ is the highest weight of V.

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