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May 2009 Algebras with ternary law of composition and their realization by cubic matrices
V. Abramov, R. Kerner, O. Liivapuu, S. Shitov
J. Gen. Lie Theory Appl. 3(2): 77-94 (May 2009). DOI: 10.4303/jglta/S090201

Abstract

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a~triple product to be continuous mapping, we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions, we construct a~ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.

Citation

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V. Abramov. R. Kerner. O. Liivapuu. S. Shitov. "Algebras with ternary law of composition and their realization by cubic matrices." J. Gen. Lie Theory Appl. 3 (2) 77 - 94, May 2009. https://doi.org/10.4303/jglta/S090201

Information

Published: May 2009
First available in Project Euclid: 7 October 2011

zbMATH: 1275.17004
MathSciNet: MR2504871
Digital Object Identifier: 10.4303/jglta/S090201

Subjects:
Primary: 17A40, 20N10

Rights: Copyright © 2009 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

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Vol.3 • No. 2 • May 2009
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