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.
Let be a finite dimensional complex simple Lie algebra. (We also fix a Borel subalgebra and a Cartan subalgebra of Then the associated affine Kac-Moody Lie algebra is by definition the space together with the Lie bracket (for and where is the Killing form on normalized so that for the highest root of
The Lie algebra sits as a Lie subalgebra of as The Lie algebra admits a distinguished "parabolic" subalgebra We also define its "nil-radical" (which is an ideal of by and its "Levi component" (which is a Lie subalgebra of Clearly (as a vector space)
Also define and the Cartan subalgebra of Let (resp. be the Weyl group of (resp. Then acts on the dual space by linear automorphisms. Let be half the sum of the positive roots of and define by and where (as earlier) is the highest root of and is the associated coroot. Define the shifted action of on by for and Fix a positive integer Let be the set of dominant integral weights of and let be the fundamental alcove.
Define the loop algebra with Lie bracket for and Then can be viewed as a one-dimensional central extension of where the Lie algebra homomorphism is defined by and
Fix an irreducible (finite-dimensional) representation of and a number (to be called the level or central charge). Then we define the associated generalized Verma module for as where the has the same underlying vector space as on which acts trivially, the central element acts via the scalar and the action of is via the structure on
In the case when is a positive integer (in fact, it suffices to assume that where is the dual Coxeter number of has a unique irreducible quotient, denoted
It is easy to see that any vector is contained in a finite dimensional of In particular, the same property holds for any vector in
Consider the Lie subalgebra of spanned by where (resp. is a non-zero root vector of corresponding to the root (resp. and the coroot is to be thought of as an element of Then the Lie algebra is isomorphic to
A is said to be integrable if every vector is contained in a finite-dimensional of and also is contained in a finite-dimensional of
Then it follows easily from the that the irreducible module is integrable if and only if is an integer and where is the highest weight of