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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: J. Szigeti, J. van den Berg, L. van Wyk and M. Ziembowski
Journal: Trans. Amer. Math. Soc. 372 (2019), 4553-4583
MSC (2010): Primary 16S50, 16U80; Secondary 16R40, 17B99
Published electronically: June 21, 2019
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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

$\displaystyle {\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.

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Additional Information

J. Szigeti
Affiliation: Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary

J. van den Berg
Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa

L. van Wyk
Affiliation: Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, Stellenbosch, South Africa

M. Ziembowski
Affiliation: Faculty of Mathematics and Information Science, Technical University of Warsaw, 00-661 Warsaw, Poland

Keywords: Lie nilpotent, matrix algebra, Lie algebra, commutative subalgebra, dimension
Received by editor(s): July 11, 2017
Published electronically: June 21, 2019
Additional Notes: The first author was partially supported by the National Research, Development and Innovation Office of Hungary (NKFIH) K119934.
The second and third authors were supported by the National Research Foundation of South Africa under grant numbers UID 85784 and UID 72375, respectively. 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 fourth author was supported by the Polish National Science Centre grant UMO-2017/25/B/ST1/00384.
The second author is the corresponding author.
Article copyright: © Copyright 2019 American Mathematical Society