An analogue of Wagner's theorem for decompositions of matrix algebras

Wagner's celebrated theorem states that a finite affine plane whose collineation group is transitive on lines is a translation plane. The notion of an orthogonal decomposition (OD) of a classically semisimple associative algebra introduced by the author allows one to draw an analogy between finite affine planes of order $n$ and ODs of the matrix algebra $M_n(\mathbb C)$ into a sum of subalgebras conjugate to the diagonal subalgebra. These ODs are called WP-decompositions and are equivalent to the well-known ODs of simple Lie algebras of type $A_{n-1}$ into a sum of Cartan subalgebras. In this paper we give a detailed and improved proof of the analogue of Wagner's theorem for WP-decompositions of the matrix algebra of odd non-square order an outline of which was earlier published in a short note in “Russian Math. Surveys” in 1994. In addition, in the framework of the theory of ODs of associative algebras, based on the method of idempotent bases, we obtain an elementary proof of the well-known Kostrikin–Tiep theorem on irreducible ODs of Lie algebras of type $A_{n-1}$ in the case where $n$ is a prime-power.