Jordan superalgebras of vector type and projective modules
type: Item
instance of: scholarly article
title: Jordan superalgebras of vector type and projective modules
zbMATH Open document ID: 1284.17018
DOI: 10.1134/S0037446612020267
published in: Siberian Mathematical Journal
publication date: 2012-11-14T00:00:00Z
review text: The aim of the paper is to study connections between Jordan superalgebras of vector type and finitely generated projective modules of rank \(1\) over integral domains. Given an associative commutative \(F\)-algebra \(\Gamma\) with a nonzero derivation \(D\) we denote by \(\Gamma x\) a copy of \(\Gamma\) and define in \(\Gamma\oplus\Gamma x\) the product: \[ a\cdot b:=ab,\quad a\cdot bx:=(ab)x,\quad ax\cdot b:=(ab)x,\quad (ax)\cdot(bx)=D(a)b-aD(b), \] for \(a,b\in\Gamma\) and denoting by \(ab\) the product in \(\Gamma\). This new algebra \(\Gamma\oplus\Gamma x\) is denoted by \(J(\Gamma,D)\) and it turns out to be a Jordan superalgebra (of vector type) whose even part is \(A:=\Gamma\) and odd part \(M:=\Gamma x\). The superalgebra \(J(\Gamma,D)\) is simple if and only if \(\Gamma\) is \(D\)-simple. It is proved in the paper that if \(J=A\oplus M\) is a unital simple Jordan superalgebra such that the even part \(A\) is an associative algebra and the odd part \(M\) an associative \(A\)-module (and \(J\) is not a superalgebra of a nondegenerate bilinear superform) then \(M=Ax_1+\cdots Ax_n\) for some \(x_i\in M\) and the product in \(M\) is given by \[ ax_i\cdot bx_j=\gamma_{ij} ab+D_{ij}(a)b-aD_{ji}(b),\quad i,j=1,\dots,n, \] where \(\gamma_{ij}\in A\) and \(D_{ij}\) is a derivation of \(A\). The algebra \(A\) is differentiably simple with respect to the set of derivations \(\Delta=\{D_{ij}\}\) and \(M\) is a projective \(A\)-module if rank \(1\). Furthermore \(J\) is a subalgebra of \(J(\Gamma,D)\). The following is also proved in the work under review. Take \(A\) to be an integral domain and \(M=Ax_1+\cdots Ax_n\) a finitely generated projective \(A\)-module of rank \(1\). Consider a derivation \(A\to (M\otimes_A M)^*\) such that \(a\mapsto\bar a\). Take then a set of derivations \(\Delta=\{D_{ij}: i,j=1,\dots,n\}\) such that \(D_{ij}(a):=\bar a(x_i\otimes x_j)\). Fix elements \(\gamma_{ij}\in A\) such that \(\gamma_{ij}=-\gamma_{ji}\) and define in the vector space …
Mathematics Subject Classification ID: 17C70
zbMATH DE Number: 6105695
zbMATH Keywords: Jordan superalgebra of vector type
zbMATH Keywords: differentiably simple algebra
zbMATH Keywords: integral domain
zbMATH Keywords: field of quotients
zbMATH Keywords: polynomial algebra
zbMATH Keywords: projective module
zbMATH Keywords: Picard group
author: V. N. Zhelyabin
reviewed by: Candido Martín González
MaRDI profile type: Publication
cites work: The component groups of nilpotents in exceptional simple real lie algebras
cites work: PRIME ALGEBRAS AND ABSOLUTE ZERO DIVISORS
cites work: Q4006530
cites work: Speciality and non-speciality of two Jordan superalgebras
cites work: Specializations of Jordan Superalgebras
cites work: Q4677699
cites work: Q4675476
cites work: Differentiably simple rings of prime characteristic
cites work: Vector Bundles and Projective Modules
cites work: Differential algebras and simple Jordan superalgebras
cites work: New examples of simple Jordan superalgebras over an arbitrary field of characteristic 0
cites work: Classification of linearly compact simple Jordan and generalized Poisson superalgebras
cites work: Classification of simple z-graded lie superalgebras and simple jordan superalgebras
cites work: Q4220222
cites work: Q4471580
cites work: Simple finite-dimensional Jordan superalgebras of prime characteristic
cites work: Graded simple Jordan superalgebras of growth one
cites work: Simple Jordan superalgebras with semisimple even part.
cites work: Q4224184
description: scientific article; zbMATH DE number 6105695