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The maximum dimension of a Lie nilpotent subalgebra of \(\mathbb{M}_n(F)\) of index \(m\). (English) Zbl 1456.16027

This paper under review is an attempt to answer a conjecture posed by J. Szigeti and L. Van Wyk in [Commun. Algebra 43, No. 11, 4783–4796 (2015; Zbl 1333.16003)]. The statement of this conjecture is rendered less cumbersome if expressed in terms of a function \(M(\ell, n)\) of positive integer arguments \(\ell\) and \(n\), defined as follows: \[ \begin{aligned} M(\ell, n) = & \max \Bigg\{\, \frac{1}{2} \left(n^2- \sum \limits _{i=1}^{\ell} k_i^2\right)+1 \, | \, k_1, k_2, \cdots, k_{\ell}\,\\ & \text{are nonnegative integers such that}\, \sum_{i=1}^{\ell} k_i=n \, \Bigg\} \end{aligned} \] Szigeti and van Wyk’s conjecture 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 \[ \dim_F R\leq M(m+1, n). \] This conjecture is eventually solved in the current joint work by the four authors. The case \(m=1\) reduces to a classical theorem of I. Schur [J. Reine Angew. Math. 130, 66–76 (1905; JFM 36.0140.01)], later generalized by N. Jacobson [Bull. Am. Math. Soc. 50, 431–436 (1944; Zbl 0063.03016)] 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 \(\dim_FR\leq \lfloor \frac{n^2}{4} \rfloor+1\). 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 simultaneously.
Reviewer: Wei Feng (Beijing)

MSC:

16S50 Endomorphism rings; matrix rings
16U80 Generalizations of commutativity (associative rings and algebras)
16R40 Identities other than those of matrices over commutative rings
17B99 Lie algebras and Lie superalgebras
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