We study associative multiplications in semisimple 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-types.

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