The maximum dimension of a Lie nilpotent subalgebra of $\boldsymbol {\mathbb {M}_n(F)}$ of index $\boldsymbol {m}$

Abstract:

The main result of this paper is the following: if $F$ is any field and $R$ any $F$-subalgebra of the algebra $\mathbb {M}_n(F)$ of $n\times n$ matrices over $F$ with Lie nilpotence index $m$, then \[ {\dim }_{F}R \leqslant M(m+1,n), \] where $M(m+1,n)$ is the maximum of $\frac {1}{2}\!\left (n^{2}-\sum _{i=1}^{m+1}k_{i}^{2}\right )+1$ subject to the constraint $\sum _{i=1}^{m+1}k_{i}=n$ and $k_{1},k_{2},\ldots ,k_{m+1}$ nonnegative integers. This answers in the affirmative a conjecture by the first and third authors. The case $m=1$ reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if $F$ is an algebraically closed field of characteristic zero and $R$ is any commutative $F$-subalgebra of $\mathbb {M}_{n}(F)$, then ${\dim }_{F}R \leqslant \left \lfloor \frac {n^{2}}{4}\right \rfloor +1$. Examples constructed from block upper triangular matrices show that the upper bound of $M(m+1,n)$ cannot be lowered for any choice of $m$ and $n$. An explicit formula for $M(m+1,n)$ is also derived.