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.
Fix a positive integer We first recall the definition of fusion product for positive level representations.
For any two there is a fusion product which is again an integrable representation of with the same central charge It is given as: where is the dimension of the space of vacua for the Riemann sphere with three punctures 0, 1, and the representations and attached to them respectively (cf. [TUY1989]).
Let be the affine Kac-Moody group associated to the Lie algebra and its parabolic subgroup (corresponding to the Lie subalgebra (cf. [Kum1987, §1]). Then is a projective ind-variety. Now, given a finite-dimensional algebraic representation of we can consider the associated homogeneous vector bundle on and the corresponding Euler-Poincaré characteristic (which is a virtual
Recall that is determined in [Kum1987, Corollary 3.11] (and also in [Mat1988]).
We give a new definition of a fusion product in the following.
For any positive integer and define as virtual where the (cf. §1.1 for the notation
Let be the free Abelian group generated by Then gives rise to a product in
Let be the (simple) simply-connected complex algebraic group with Lie algebra and let be its representation ring, i.e., is the free Abelian group generated by the which is a ring under the usual tensor product of Define the map by where is the homogeneous vector bundle on associated to the and is the dual vector bundle on
We have the following lemma.
The kernel of is an ideal of Moreover, is a homomorphism with respect to the product in In particular, is an associative (and commutative) algebra under