\(U(\mathfrak{h})\)-free modules over the block algebra \(\mathcal{B}(q)\)
type: Item
instance of: scholarly article
title: \(U(\mathfrak{h})\)-free modules over the block algebra \(\mathcal{B}(q)\)
zbMATH Open document ID: 1476.17005
author: Xiangqian Guo
author: Mengjiao Wang
author: Xuewen Liu
published in: Journal of Geometry and Physics
publication date: 2021-09-16T00:00:00Z
review text: The Block Lie algebra \(\mathcal{B}(q)\) for \(q \in \mathbb{C}\) is the Lie algebra over \(\mathbb{C}\) with basis \(\{L_n \mid n=(n_1,n_2) \in \mathbb{Z}^2\}\) subject to the relation \([L_m, L_n] = (n_1(m_2 + q)-m_1(n_2 + q))L_{m+n}\). It has an extension \(\mathcal{B}'(q)\) by the derivations \(\partial_1\) and \(\partial_2\) with \(\partial_i(L_n)=n_i L_{n_i}\). The extended Lie algebra has the Cartan subalgebra \(\mathfrak{h}=\mathbb{C} \partial_1 + \mathbb{C} \partial_2\), i.e., a maximal abelian subalgebra with semisimple adjoint action on the whole algebra. The paper under review introduces a new class of modules of \(\mathcal{B}(q)\) and \(\mathcal{B}'(q)\), and describes all the simple objects among them. It is shown that these modules exhaust all \(U(\mathfrak{h})\)-free modules of rank one.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 7396040
zbMATH Keywords: block algebra
zbMATH Keywords: irreducible modules
zbMATH Keywords: U(h)-free modules
reviewed by: Shintarou Yanagida
MaRDI profile type: Publication
full work available at URL: j.geomphys.2021.104333
OpenAlex ID: W3183820539
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DOI: 10.1016/J.GEOMPHYS.2021.104333
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