ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

Mansuroglu, Nil and Stöhr, Ralph (2013) ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS. Internat. J. Algebra Comput. 23 (2013), no. 1, 205-213, 23 (1). pp. 205-213.

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Abstract

Let $L$ be a free Lie algebra of finite rank over a field $K$ and let $L_{n}$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_m,L_n]$ for all $m$ and $n$ were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form $[L_m,L_n,L_k]=[[L_m,L_n],L_k]$. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field $K$. For example, the dimension of $[L_2,L_2,L_1]$ over fields of characteristic $2$ is different from the dimension over fields of characteristic other than $2$. Our main result are formulae for the dimension of $[L_m,L_n,L_k]$. Under certain conditions on $m$, $n$ and $k$ they lead to explicit formulae that do not depend on the characteristic of $K$, and express the dimension of $[L_m,L_n,L_k]$ in terms of Witt's dimension function.

Item Type:	Article
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 17 Nonassociative rings and algebras
Depositing User:	Prof Ralph Stöhr
Date Deposited:	13 Mar 2013
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1955

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