Graded subalgebras associated with a subalgebra of a graded free Lie algebra
type: Item
instance of: scholarly article
title: Graded subalgebras associated with a subalgebra of a graded free Lie algebra
zbMATH Open document ID: 0564.17001
DOI: 10.1016/0022-4049(85)90090-8
author: Yusuf Ünlü
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: If \(L=\oplus_{n\geq 1}L^ n\) is a graded free Lie algebra over a field K and A is a subalgebra of L, let gr A (resp. Gr A) denote the subalgebra of L generated by the homogeneous components of minimum (resp. maximum) degree of elements of A. It is shown that for any positive integer r, gr \(A_ r=(gr A)_ r\) and Gr \(A_ r=(Gr A)_ r\) where \(B_ r\), for a Lie algebra B denotes the r-th term of the lower central series of B. As an application of this result it is shown that if B is a free subalgebra of a free nilpotent Lie algebra N and Y is a free generating set of N, then \(gr_ Y B\) is also free where the gradation of N is the one induced by Y. It is known [\textit{M. Boral}, Proc. Am. Math. Soc. 94, 23-28 (1985; Zbl 0532.17004)] that the only free non-abelian ideal in a free nilpotent Lie algebra is the algebra itself. A simple proof of this result is given as another application.
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 3899119
zbMATH Keywords: graded subalgebras
zbMATH Keywords: graded free Lie algebra
zbMATH Keywords: homogeneous components
zbMATH Keywords: lower central series
zbMATH Keywords: free subalgebra
zbMATH Keywords: free nilpotent Lie algebra
Wikidata QID: Q115364452
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90090-8
OpenAlex ID: W2043599936
cites work: Free nilpotent and nilpotent quadratic Lie algebras
cites work: Q4095018
cites work: Primitive elements of free Lie algebras
description: scientific article; zbMATH DE number 3899119