Leibniz representations of Lie algebras
type: Item
instance of: scholarly article
title: Leibniz representations of Lie algebras
zbMATH Open document ID: 0855.17018
author: J.-L. Loday
author: Teimuraz Pirashvili
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: The authors examine the category \(L({\mathfrak g})\) of finite-dimensional Leibniz representation [the authors, Math. Ann. 296, 139-158 (1993; Zbl 0821.17022)] of the finite-dimensional semisimple Lie algebra \({\mathfrak g}\). First, they notice that \(L({\mathfrak g})\) is not semisimple even when the characteristic of the field \(k\) is 0. One of the aims of the paper is to prove that this category has global dimension 2. Next, the authors obtain the description of the quiver of the category \(L({\mathfrak g})\) in the sense of \textit{P. Gabriel} [Manuscr. Math. 6, 71-103 (1962; Zbl 0232.08001)] and, in the end, they prove that this category is tame only for \(g= sl_2\), and give the complete list of indecomposable objects in that case.
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 16G99
Mathematics Subject Classification ID: 17A32
zbMATH DE Number: 883767
zbMATH Keywords: category of finite-dimensional Leibniz representations
zbMATH Keywords: tameness
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: global dimension 2
zbMATH Keywords: quiver
zbMATH Keywords: indecomposable objects
Wikidata QID: Q115396102
reviewed by: Jan Kubarski
MaRDI profile type: Publication
OpenAlex ID: W2029356977
DOI: 10.1006/JABR.1996.0127
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description: scientific article; zbMATH DE number 883767