Algebraic structures connected with pairs of compatible associative algebras
Abstract
We study associative multiplications in semisimple associative algebras over C compatible with the usual one or, in other words, linear deformations of semisimple associative algebras over C. It turns out that these deformations are in onetoone correspondence with representations of certain algebraic structures, which we call Mstructures in the matrix case and PMstructures in the case of direct sums of several matrix algebras. We also investigate various properties of PMstructures, provide numerous examples and describe an important class of PMstructures. The classification of these PMstructures naturally leads to affine Dynkin diagrams of A, D, Etype.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2005
 DOI:
 10.48550/arXiv.math/0512499
 arXiv:
 arXiv:math/0512499
 Bibcode:
 2005math.....12499O
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory;
 High Energy Physics  Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 17B80;
 17B63;
 32L81;
 14H70
 EPrint:
 29 pages, Latex. The case of semisimple algebras A and B is completed (Chapter 4). A construction of compatible products is added (Chapter 1)