On Lie \(k\)-algebras
type: Item
instance of: scholarly article
title: On Lie \(k\)-algebras
zbMATH Open document ID: 0844.17001
author: Michelle L. Wachs
author: Philip J. Hanlon
published in: Advances in Mathematics
publication date: 1996-09-08T00:00:00Z
review text: A Lie \(k\)-algebra is a bigraded vector space with a \((k + 1)\)-ary skew-symmetric operation which preserves the grading, which is alternating and which satisfies an identity extending the Jacobi identity of Lie superalgebras (the case \(k = 1)\). The symmetric group \(S_n\) acts on the multilinear part of degree \(n\) of the free Lie \(k\)-algebra. On the other hand, \(S_n\) acts on the top homology of the poset of partitions of \(\{1, \dots, n\}\) whose blocks have cardinality \(\equiv 1 \bmod k\). The main result is that when \(n \equiv 1 \bmod k\), these two characters coincide. A basis of the multilinear part is also constructed, extending the basis of left to right bracketings in the free Lie algebra. Furthermore, the authors extend the Koszul complex of Lie algebras, and compute the homology of the evenly generated free Lie \(k\)-algebra.
reviewed by: Christophe Reutenauer
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 812543
zbMATH Keywords: free Lie \(k\)-algebra
zbMATH Keywords: symmetric group
zbMATH Keywords: Koszul complex
zbMATH Keywords: Lie algebras
zbMATH Keywords: homology
MaRDI profile type: Publication
full work available at URL: aima.1995.1038
OpenAlex ID: W2073797163
DOI: 10.1006/AIMA.1995.1038
description: scientific article; zbMATH DE number 812543