Quivers and Three-Dimensional Lie Algebras

TL;DR

This paper explores the connection between three-dimensional Lie algebras parameterized by  and certain quivers, revealing how their representation theories relate and identifying conditions under which their categories are finite or tame.

Contribution

It introduces quivers associated with the Lie algebras and establishes isomorphisms between their enveloping algebras and quiver path algebras, linking Lie algebra representations to quiver theory.

Findings

Isomorphism between enveloping algebras and quiver path algebras for rational and non-rational 

Representation categories of these Lie algebras relate to affine type A quivers

Restrictions on weight decompositions yield finite or tame representation subcategories

Abstract

We study a family of three-dimensional Lie algebras $L_\mu$ that depend on a continuous parameter $\mu$. We introduce certain quivers, which we denote by $Q_{m,n}$ $(m,n \in \mathbb{Z})$ and $Q_{\infty \times \infty}$, and prove that idempotented versions of the enveloping algebras of the Lie algebras $L_{\mu}$ are isomorphic to the path algebras of these quivers modulo certain ideals in the case that $\mu$ is rational and non-rational, respectively. We then show how the representation theory of the quivers $Q_{m,n}$ and $Q_{\infty\times\infty}$ can be related to the representation theory of quivers of affine type $A$, and use this relationship to study representations of the Lie algebras $L_\mu$. In particular, though it is known that the Lie algebras $L_\mu$ are of wild representation type, we show that if we impose certain restrictions on weight decompositions, we obtain full subcategories of the category of representations of $L_\mu$ that are of finite or tame representation type.