Quivers with relations arising from Koszul algebras of $\mathfrak g$-invariants

TL;DR

This paper explicitly describes the structure of certain infinite-dimensional Koszul algebras associated with complex simple Lie algebras using quivers with relations, focusing on types A and C.

Contribution

It provides an explicit description of the algebra structure via quivers with relations for specific Lie algebra types, expanding understanding of these Koszul algebras.

Findings

Explicit quiver with relations for type A and C algebras

Descriptions of infinite families of quivers and finite dimensional algebras

Structural insights into Koszul algebras from Lie algebra invariants

Abstract

Let $\mathfrak g$ be a complex simple Lie algebra and let $\Psi$ be an extremal set of positive roots. One associates with $\Psi$ an infinite dimensional Koszul algebra $\bold S_\Psi^{\lie g}$ which is a graded subalgebra of the locally finite part of $((\bold V)^{op}\tensor S(\lie g))^{\lie g}$, where $\bold V$ is the direct sum of all simple finite dimensional $\lie g$-modules. We describe the structure of the algebra $\bold S_\Psi^{\lie g}$ explicitly in terms of an infinite quiver with relations for $\lie g$ of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensional algebras arising from this construction.