Paweł Matraś
supervisor: Michał Ziembowski
In ring theory algebras, which are rings with a structure of linear space, satisfying identities form an important class. For them, there are known different structure theorems, not true for rings in general. On the other hand, the problem of determining subalgebras satisfying a given identity in a fixed class of algebras is also interesting. One of the simplest examples of algebras with identity are commutative algebras.
The classical problem solved by Schur was to determine the maximal dimension of a commutative subalgebra of matrices over a field. Later, with some restrictions on a field, a characterization of commutative subalgebras of matrices with maximum dimensions up to conjugation was proven. In last years maximal dimensions of subalgebras of matrices with identities generalizing commutativity were found.
We develop study of these algebras, characterizing one of them up to conjugation. We also study the isomorphism problem and other structural problems.