Associative Superalgebras with Homogeneous Symmetric Structures

A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e., even or odd), and associative bilinear form on A. In this article, we show that any associative superalgebra with non-null product cannot admit simultaneously even-symmetric and...

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Bibliographic Details
Published in:Communications in algebra 2012-04, Vol.40 (4), p.1234-1259
Main Authors: Ayadi, Imen, Benayadi, Saïd
Format: Article
Language:English
Subjects:
Associative superalgebras
Generalized double extension
Homogeneous symmetric structures
Inductive description
Mathematics
Rings and Algebras
Simple associative superalgebras
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Summary:A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e., even or odd), and associative bilinear form on A. In this article, we show that any associative superalgebra with non-null product cannot admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure, and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descriptions of even-symmetric associative superalgebras and odd-symmetric associative superalgebras. We obtain also an other interesting description of odd-symmetric associative superalgebras whose even parts are semi-simple bimodules without using the notions of double extensions.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2010.549160