On the maximum dimensions of subalgebras of Mn(K) satisfying two related identities

Abstract

For an arbitrary $q\ge 2$, we find an upper bound for the dimension of a subalgebra of the full matrix algebra M${}_n\left(K\right)$ over an arbitrary field K satisfying the identity$\left[\right[x_1,y_1],z_1]\cdot \left[\right[x_2,y_2],z_2]\cdot \cdots \cdot \left[\right[x_q,y_q],z_q]=0,$ and we show that this upper bound is sharp by presenting an example in block triangular form of a subalgebra of M${}_n\left(K\right)$ with dimension equal to the obtained upper bound. We apply this result to Lie solvable algebras of index 2, i.e., algebras satisfying the identity $\left[\right[x_1,y_1],[x_2,y_2\left]\right]=0$. To be precise, for $n\le 4$, we find the sharp upper bound for the dimension of a Lie solvable subalgebra of M${}_n\left(K\right)$ of index 2, and for $n>4$, we obtain the relatively tight (at least for small values of $n>4$) interval$[2+\lfloor \frac{3n^2}{8}\rfloor ,2+\lfloor \frac{5n^2}{12}\rfloor ]$ for the maximum dimension of a Lie solvable subalgebra of M${}_n\left(K\right)$ of index 2, the exact value of which is not known.