Quivers and Three Dimensional Solvable Lie Algebras

Jeffrey Pike¹

¹Dept. of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa, Ontario K1N 6N5, Canada

Journal of Lie Theory, Tome 27 (2017) no. 3, pp. 707-726

Résumé

We study a family of three-dimensional solvable 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 natural full subcategories of the category of representations of $L_\mu$ that are of finite or tame representation type.

Détail

Classification : 17B10, 16G20, 22E47
Mots-clés : Lie algebra, quiver, path algebra, preprojective algebra, representation