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arXiv · 1608.04562

The maximum dimension of a Lie nilpotent subalgebra of M_{n}(F) of index m

J. Szigeti·J. van den Berg·L. van Wyk·M. Ziembowski
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Abstract

The main result of this paper is the following: if F is any field and R is any F-subalgebra of the algebra of nxn matrices over F with Lie nilpotence index m, then the F-dimension of R is less or equal than M(m+1,n), where M(m+1,n) is the maximum of a certain expression of m+1 nonnegative integers and n. 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 the full nxn matrix algebra over F, then the F-dimension of R is less or equal than ((n^2)/4)+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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J. Szigeti, J. van den Berg, L. van Wyk, M. Ziembowski. 2016-08-16. The maximum dimension of a Lie nilpotent subalgebra of M_{n}(F) of index m. https://arxiv.org/abs/1608.04562

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