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

Abstract

Let $\lie g$ be a simple Lie algebra and let $\bs^{\lie g}$ be the locally finite part of the algebra of invariants $(_\bc\bv\otimes S(\lie g))^{\lie g}$ where $\bv$ is the direct sum of all simple finite-dimensional modules for $\lie g$ and $S(\lie g)$ is the symmetric algebra of $\lie g$. 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 $\bs^{\lie g}_Ψ(\le_Ψλ)$ of $\bs^{\lie g}$ and prove that the algebra is Koszul of global dimension at most the cardinality of $Ψ$. Using this we then construct naturally an infinite-dimensional 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.

25 pages