Algebraic structures connected with pairs of compatible associative algebras
Abstract
We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A, D, E-type.
Cite
@article{arxiv.math/0512499,
title = {Algebraic structures connected with pairs of compatible associative algebras},
author = {Alexander Odesskii and Vladimir Sokolov},
journal= {arXiv preprint arXiv:math/0512499},
year = {2007}
}
Comments
29 pages, Latex. The case of semi-simple algebras A and B is completed (Chapter 4). A construction of compatible products is added (Chapter 1)