The structure of subalgebras of full matrix algebras over a field satisfying the identity \([x_1, y_1][x_2, y_2] \cdots [x_q, y_q]=0\) (Q6588191)

The authors call an associative algebra over a field \(K\) a \emph{\(\mathrm{D}_q\) algebra} if it satisfies the polynomial identity \([x_1,y_1]\cdots [x_q,y_q]=0\), where \([x,y]:=xy-yx\). For example, \(\mathrm{D}_1\) algebras are the commutative algebras, and the algebra of \(q\times q\) upper triangular matrices is a \(\mathrm{D}_q\) algebra. The main result of the paper is a classification of maximal (with respect to inclusion) \(\mathrm{D}_q\) subalgebras of \(M_n(K)\) (the algebra of \(n\times n\) matrices over \(K\)) up to conjugacy (with respect to the group of invertible \(n\times n\) matrices over \(K\)), and also up to isomorphism. For a \(q\)-tuple \((n_1,\dots,n_q)\) of positive integers with \(n_1+\cdots+n_q=n\) (where \(q\ge 2\)) and given maximal commutative subalgebras \(A_i\) of \(M_{n_i}(K)\), consider the set of block upper triangular \(n\times n\) matrices, whose \(i\)th diagonal block belongs to \(A_i\) for \(i=1,\dots,q\), and whose blocks above the diagonal are arbitrary. Then this is a maximal \(\mathrm{D}_q\) subalgebra of \(M_n(K)\), and any maximal \(\mathrm{D}_q\) subalgebra in \(M_n(K)\) (which is not a \(\mathrm{D}_{ q-1}\) subalgebra) is conjugate to a subalgebra of this form for a unique \(q\)-tuple \((n_1,\dots,n_q)\) and unique (up to conjugacy) subalgebras \(A_i\) of \(M_{n_i}(K)\). In fact the \(q\)-tuple \((n_1,\dots,n_q)\) and the isomorphism class of \(A_i\) for \(i=1,\dots,q\) are uniquely determined already by the isomorphism class of the given \(\mathrm{D}_q\) subalgebra. When \(K\) is algebraically closed, then two maximal \(\mathrm{D}_q\) subalgebras are isomorphic if and only if they are conjugate.