Quivers and Three-Dimensional Lie Algebras

2014/09/23 by Jeffrey Pike, Pike, Jeffrey
Mathematics · #16G20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary: 17B10 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary: 22E47 #math.RA #math.RT #msc:16G20 #msc:17B10 #msc:22E47

paper · pdf · doi:10.48550/arxiv.1409.6376

18 pages

arxiv created 2014/09/23 · openalex publication_date 2014/09/23 · arxiv updated 2014/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ∈ ℤ) and Q_{∞ × ∞}, 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_∞×∞ 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.

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