  Given an associative multiplication in matrix algebra compatible with the
usual one or, in other words, linear deformation of matrix algebra, we
construct a solution to the classical Yang-Baxter equation. We also develop a
theory of such deformations and construct numerous examples. It turns out that
these deformations are in one-to-one correspondence with representations of
certain algebraic structures, which we call M-structures. We also describe an
important class of M-structures related to the affine Dynkin diagrams of A, D,
E-type. These M-structures and their representations are described in terms of
quiver representations.
