The structure of subalgebras of full matrix algebras over a field satisfying the identity [x_1, y_1][x_2, y_2] ... [x_q, y_q] = 0 (2403.19557v1)

Abstract: A subalgebra of the full matrix algebra Mn(K), K a field, satisfying the identity [x1, y1][x2, y2]...[xq, yq] = 0 is called a Dq subalgebra of Mn(K). In the paper we deal with the structure, conjugation and isomorphism problems of maximal Dq subalgebras of Mn(K). We show that a maximal Dq subalgebra A of Mn(K) is conjugated with a block triangular subalgebra of Mn(K) with maximal commutative diagonal blocks. By analysis of conjugations, the sizes of the obtained diagonal blocks are uniquely determined. It reduces the problem of conjugation of maximal Dq subalgebras of Mn(K) to the analogous problem in the class of commutative subalgebras of Mn(K). Further examining conjugations, in case A is contained in the upper triangular matrix algebra Un(K), we prove that A is already in a block triangular form. We consider the isomorphism problem in a certain class of maximal Dq subalgebras of Mn(K) which contain all Dq subalgebras of Mn(K) with maximum dimension. In case K is algebraically closed, we invoke Jacobson's characterization of maximal commutative subalgebras of Mn(K) with maximum (K-)dimension to show that isomorphic subalgebras in this class are already conjugated. To illustrate it, we invoke results from [17] and find all isomorphism (equivalently conjugation) classes of Dq subalgebras of Mn(K) with maximum possible dimension, in case K is algebraically closed.