Quivers and Three-Dimensional Lie Algebras

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.Comment: 18 page

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On June 29, 2005, Valeo IP (www.valeoip.com), an online copyright licensing service, announced on its Website and to its employees that it is shutting down it operations effective September 2, 2005. The move comes after a volatile spring which saw the company engaged in a bitter lawsuit with former partner DataDepth (www.datadepth.com) and the abrupt departure of its President. The departure of Valeo IP leaves a gap in the growing, but confusing business of licensing digital content. Both the growth and the confusion come from the increasing awareness of the need to manage copyrights and use of digital content, with uncertainty as to how copyright law applies to digital content