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Journal of Lie Theory 27 (2017), No. 3, 707--726
Copyright Heldermann Verlag 2017

Quivers and Three Dimensional Solvable Lie Algebras

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


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.

Keywords: Lie algebra, quiver, path algebra, preprojective algebra, representation.

MSC: 17B10, 16G20; 22E47

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