2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106341We 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.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)Quantum AlgebraHigh Energy Physics - TheoryRings and AlgebrasRepresentation TheoryExactly Solvable and Integrable Systems17B80, 17B63, 32L81, 14H70text