Lie solvability in matrix algebras

ABSTRACT

If an algebra $\mathcal{A}$ satisfies the polynomial identity $$[x_1,y_1][x_2,y_2]\cdots [x_{2^m},y_{2^m}]=0$$(for short, $\mathcal{A}$ is ${\text{D}}_{2^m}$), then $\mathcal{A}$ is trivially Lie solvable of index $m+1$ (for short, $\mathcal{A}$ is ${\text{Ls}}_{m+1}$). We prove that the converse holds for subalgebras of the upper triangular matrix algebra $U_n(R),R$ any commutative ring, and $n\ge 1$. We also prove that if a ring S is ${\text{D}}_2$ (respectively, ${\text{Ls}}_2$), then the subring $U_m^{\star }(S)$ of $U_m(S)$ comprising the upper triangular $m\times m$ matrices with constant main diagonal, is ${\text{D}}_{2^{⌈{log}_{2}m⌉}}$ (respectively, ${\text{Ls}}_{⌈{log}_{2}m⌉+1}$) for all $m\ge 2$. We also study two related questions, namely whether, for a field F, an ${\text{Ls}}_2$ subalgebra of $M_n(F),$ for some n, with (F-)dimension larger than the maximum dimension $2+⌊\frac{3n^2}{8}⌋$ of a ${\text{D}}_2$ subalgebra of $M_n(F)$, exists, and whether a ${\text{D}}_2$ subalgebra of $U_n(F)$ with (the mentioned) maximum dimension, other than the typical ${\text{D}}_2$ subalgebras of $U_n(F)$ with maximum dimension, which were described by Domokos and refined by van Wyk and Ziembowski, exists. Partial results with regard to these two questions are obtained.

KEYWORDS:

• Lie solvable
• Lie nilpotent
• commutator
• polynomial identity
• matrix algebra
• upper triangular matrix algebra
• subalgebra
• dimension

AMS SUBJECT CLASSIFICATIONS:

• 16S50
• 16U80
• 16R10
• 16R40

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the National Research Foundation of South Africa [grant number UID 72375]. All opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and therefore the National Research Foundation does not accept any liability in regard thereto. The second author was supported by the Polish National Science Centre [grant number UMO-2013/09/D/ST1/03669].