A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Abstract

Let g be a finite-dimensional simple Lie algebra and let S g be the locally finite part of the algebra of invariants ( End C V ⊗ S ( g ) ) g where V is the direct sum of all simple finite-dimensional modules for g and S ( g ) is the symmetric algebra of g . Given an integral weight ξ, let Ψ = Ψ ( ξ ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra S Ψ g ( ⩽ Ψ λ ) of S g and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.