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.
We denote equipped with the fusion product (resp. by (resp. Recall that the associativity of follows from the factorization rule (cf. [TUY1989]) for with punctures.
For any let be the fundamental weight corresponding to We have the following lemma.
The products and in coincide if and only if for all In fact, the products and in coincide if and only if for all
As a consequence of the above lemma, together with some results of [Agr1995, Corollary 4.3], [BMi1995] and some partial determination of for those such that we obtain the following result.
For any simple (simply-connected) group of type or the products and in coincide. In particular, for these groups, the map (cf. Definition 3.2) is an algebra homomorphism with respect to the product in
(a) Of course for all if Conjecture 2.3 is true. In particular, the validity of Conjecture 2.3 will imply that and coincide in (b) The "in particular" statement of the above theorem is due to Faltings [Fal1994, Appendix], proved via case by case computation. In fact, this result of Faltings was the main motivation behind our work. (c) It is likely that we can prove Theorem 4.2 for all and all positive integer by combining our proof of the above Theorem and some results of Finkelberg [Fin1993]. (Some details in [Fin1993] are not clear to me as yet.)
Finally, we can prove the validity of Conjecture (2.3) for
Conjecture 2.3 is true for the group In particular. Theorem 2.4 is true in this case. In fact, in this case, one has a rather precise description of and hence of for all