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Quivers and Three-Dimensional Lie Algebras

We study a family of three-dimensional Lie algebras $L_μ$ that depend on a continuous parameter $μ$. 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 $L_μ$ are isomorphic to the path algebras of these quivers modulo certain ideals in the case that $μ$ 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 $L_μ$. In particular, though it is known that the Lie algebras $L_μ$ 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 $L_μ$ that are of finite or tame representation type.