2016/08/16 by Jenö Szigeti, J. van den Berg, Szigeti, J. +9 · 1 citation
Engineering · Mathematics · #16U80 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Primary 16S50 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #graph theory and CDMA systems #secondary 16R40
paper · pdf · doi:10.48550/arxiv.1608.04562
openalex publication_date 2016/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main result of this paper is the following: if F is any field and R is\nany F-subalgebra of the algebra of nxn matrices over F with Lie nilpotence\nindex m, then the F-dimension of R is less or equal than M(m+1,n), where\nM(m+1,n) is the maximum of a certain expression of m+1 nonnegative integers and\nn. The case m=1 reduces to a classical theorem of Schur (1905), later\ngeneralized by Jacobson (1944) to all fields, which asserts that if F is an\nalgebraically closed field of characteristic zero, and R is any commutative\nF-subalgebra of the full nxn matrix algebra over F, then the F-dimension of R\nis less or equal than ((n2)/4)+1. Examples constructed from block upper\ntriangular matrices show that the upper bound of M(m+1,n) cannot be lowered for\nany choice of m and n. An explicit formula for M(m+1,n) is also derived.\n