On the freeness problem for truncated current algebras
type: Item
instance of: scholarly article
title: On the freeness problem for truncated current algebras
zbMATH Open document ID: 1479.17025
author: Germán Benitez
author: Henrique Rocha
published in: São Paulo Journal of Mathematical Sciences
publication date: 2021-07-28T00:00:00Z
review text: The Takiff algebras of a Lie algebra \(\mathfrak{g}\) are the Lie algebras \(\mathfrak{g} \otimes \mathbb{C}[x]/<x^m>\). This article concerns the structure of the universal enveloping algebras of certain Takiff algebras as modules over their centers. Let \(\mathfrak{g}\) be a finite dimensional complex semisimple Lie algebra. By a classical result of Kostant, \(\mathfrak{U(g)}\) is free over its center. It is known that this also holds if \(\mathfrak{g}\) is replaced by \(\mathfrak{g} \otimes \mathbb{C}[x]/<x^2>\). Here the authors prove that \(\mathfrak{U}(\mathfrak{gl}_3 \otimes \mathbb{C}[x]/<x^m>)\) is free over its center for all \(m\). In addition, they prove that \(\mathfrak{U}(\mathfrak{gl}_2 \otimes \mathbb{C}[x]/<x^m>)\) is free over its Gel'fand-Tsetlin algebra, the commutative subalgebra generated by the centers of the universal enveloping algebras of a nested collection of Levi subalgebras.
reviewed by: Charles H. Conley
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 7375692
zbMATH Keywords: current algebra
zbMATH Keywords: equidimensionality
zbMATH Keywords: Gelfand-Tsetlin
Wikidata QID: Q107169283
MaRDI profile type: Publication
full work available at URL: s40863-020-00204-1
OpenAlex ID: W3122988271
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DOI: 10.1007/S40863-020-00204-1
description: scientific article; zbMATH DE number 7375692