Linear associative deformations of matrix multiplication
V.Sokolov (Landau Inst, Moscow)

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. 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 describe an important class of
$PM$-structures. The classification of these $PM$-structures leads to the
affine Dynkin diagrams of $\tilde A_{2 k-1},$ $\tilde D_{k},$ $\tilde
E_{6},$ $\tilde E_{7},$ and $\tilde E_{8}$-type.  We investigate in details
the multiplications of the $\tilde A_{2 k-1}$-type and integrable matrix
ODEs and PDEs generated by them.