On the functor of Arakawa, Suzuki and Tsuchiya
We study commutative vertex operator algebras. These algebras are isomorphic to certain subalgebras in Kac-Moody vertex operator algebras. We describe systems of relations and degenerations to quadratic algebras. Our approach leads to the fermionic formulas for characters.
We give an alternative proof of the main result of ; the proof relies on Brion’s theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra widehatsln(C). Our approach is to assign integer points of a certain polytope to vectors comprising a monomial basis of the subspace and then compute the character by using (a variation of) Brion’s theorem.