Modular representations of the Jordan superalgebras \(D(t)\) and \(K_3\)
type: Item
instance of: scholarly article
title: Modular representations of the Jordan superalgebras \(D(t)\) and \(K_3\)
zbMATH Open document ID: 1180.17012
author: Maria Trushina
published in: Journal of Algebra
publication date: 2008-10-17T00:00:00Z
review text: The irreducible Jordan representations of the simple superalgebra \(D(t)\) over an algebraically closed field of characteristic \(p\neq 2\) are classified. The right multiplication by the identity of the superalgebra \(D(t)\) on any irreducible superbimodule is either the zero map or \({1\over 2}\text{Id}\) or the identity \(\text{Id}\). In the first case the module is trivial, the second case for the superalgebra \(D(t)\) in zero characteristic was considered by \textit{C. Martínez} and \textit{E. Zelmanov} in Can. Math. Bull. 45, No. 4, 653--671 (2002; Zbl 1029.17024). The third case was studied independently by the same authors [Trans. Am. Math. Soc. 358, No. 8, 3637--3649 (2006; Zbl 1141.17024)] and by the author of the present paper [J. Algebra Appl. 4, No. 1, 1--14 (2004; Zbl 1062.17015)]. The author uses a Jordan superalgebra \(D(\lambda,\mu)\) such that \(D(1,t)=D(t)\). The irreducible representations of this more symmetric superalgebra \(D(\lambda,\mu)\) can be obtained from those of simple \(3\)-dimensional Lie algebra \(\text{sl}_2\). The \(F\)-superalgebra \(D(\lambda,\mu)\) can be described as the one whose even part is \(Fe_1\oplus F e_2\) where \(e_i^2=0\) (\(i=1,2\)), \(e_1e_2=0\), and odd part \(F x\oplus F y\) such that \(e_i x={1\over 2}x\), \(e_i y={1\over 2}y\) and \(x y=\lambda e_1+\mu e_2\). For any module over \(D(\lambda,\mu)\) the operators \({2\over{\lambda+\mu}}X\circ Y\), \({2\over{\lambda+\mu}}X^2\) and \({2\over{\lambda+\mu}}Y^2\) generate \(\text{sl}_2\) (here \(A\) stands for the right multiplication operator \(R_a\) of the module). Thus the irreducible representations of this algebra can be given in terms of those of \(\text{sl}_2\). The author finds six cases in the classification theorem. As a corollary she obtains a classification of the finite-dimensional irreducible representations of the Kaplansky superalgebra \(K_3\) in the case of characteristic \(p\neq 2\).
Mathematics Subject Classification ID: 17C70
Mathematics Subject Classification ID: 17C55
zbMATH DE Number: 5353577
zbMATH Keywords: Jordan superalgebra
zbMATH Keywords: Jordan representation
zbMATH Keywords: sl(2)
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.04.020
OpenAlex ID: W2019702107
cites work: Classification of simple z-graded lie superalgebras and simple jordan superalgebras
cites work: Specializations of Jordan Superalgebras
cites work: Unital bimodules over the simple Jordan superalgebra 𝐷(𝑡)
cites work: Irreducible representations of a simple three-dimensional lie algebra over a field of finite characteristic
cites work: IRREDUCIBLE REPRESENTATIONS OF A CERTAIN JORDAN SUPERALGEBRA
DOI: 10.1016/J.JALGEBRA.2008.04.020
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description: scientific article; zbMATH DE number 5353577