On blocks in restricted representations of Lie superalgebras of Cartan type
type: Item
instance of: scholarly article
title: On blocks in restricted representations of Lie superalgebras of Cartan type
zbMATH Open document ID: 1486.17013
author: Feifei Duan
author: Bin Shu
author: Yu-Feng Yao
publication date: 2021-05-04T00:00:00Z
review text: Let \(\mathbf{k}\) be an algebraically closed field of finite characteristic \(p>3\). And let \(\mathfrak{g}\) be a either one of the simple Lie superalgebras \(W(n)\), \(S(n)\), \(H(n)\) of Cartan type or toral extensions \(\bar S(n)\), \(\bar H(n)\) of the last two cases. In all these cases the Lie superalgebra \(\mathfrak{g}\) is naturally equipped with the structure of a restricted Lie superalgebra. The authors consider the category \(U_0(\mathfrak{g})\mathbf{-mod}\) of restricted representations of \(\mathfrak{g}\) and prove that is it of one block for all five cases of \(\mathfrak{g}\) above. The case of \(\mathfrak{g}=W(n)\) was already treated in [\textit{Y. Yao} and \textit{B. Shu}, Chin. Ann. Math., Ser. B 34, No. 6, 921--926 (2013; Zbl 1343.17013)]. In the given paper the authors first prove that \(U_0(\mathfrak{g})\mathbf{-mod}\) is of one block modulo parities for \(\mathfrak{g}\) being \(\bar S(n)\) (Theorem 3.4) or \(\bar H(n)\), \(n\ge 5\) (Theorem 4.5). Then they prove that the same statement is true for \(S(n)\) and \(H(n)\). Finally, they complete the proof by using the Degenerate Parity Lemma (Lemma 2.6), which states that in the category \(U_0(\mathfrak{g})\mathbf{-mod}\) the irreducible modules \(L(\lambda)\) and \(\Pi L(\lambda)\) are of the same block. Here \(\Pi\) denotes the parity change functor in the category of \(\mathbf{k}\) superspaces.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 7343748
zbMATH Keywords: Lie superalgebra of Cartan type
zbMATH Keywords: restricted representation
zbMATH Keywords: block
reviewed by: Boris M. Dubrov
MaRDI profile type: Publication
full work available at URL: s10114-020-9469-z
OpenAlex ID: W3083578934
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DOI: 10.1007/S10114-020-9469-Z
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 7343748