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

TL;DR

This paper constructs a family of Koszul algebras from finite-dimensional representations of simple Lie algebras, revealing new algebraic structures with controlled global dimensions related to root subsets.

Contribution

It introduces a novel construction of Koszul algebras associated with Lie algebra representations, extending the understanding of their algebraic and homological properties.

Findings

Finite-dimensional subalgebras are Koszul with bounded global dimension.

An infinite-dimensional Koszul algebra with maximal global dimension is constructed.

Results connect Lie algebra representations with algebraic and homological structures.

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 $\xi$, let $\Psi=\Psi(\xi)$ be the subset of roots which have maximal scalar product with $\xi$. Given a dominant integral weight $\lambda$ and $\xi$ such that $\Psi$ is a subset of the positive roots we construct a finite-dimensional subalgebra $\bs^{\lie g}_\Psi(\le_\Psi\lambda)$ of $\bs^{\lie g}$ and prove that the algebra is Koszul of global dimension at most the cardinality of $\Psi$. Using this we then construct naturally an infinite-dimensional Koszul algebra of global dimension equal to the cardinality of $\Psi$. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.