Exponential generating functions and complexity of Lie varieties
type: Item
instance of: scholarly article
title: Exponential generating functions and complexity of Lie varieties
zbMATH Open document ID: 1014.17005
DOI: 10.1007/BF02780183
published in: Israel Journal of Mathematics
publication date: 2000-02-15T00:00:00Z
review text: Let \({\mathfrak V}\) be a variety of Lie algebras, \(c_n({\mathfrak V})\) be the dimension of the linear span of all multilinear words with \(n\) distinct letters in the free algebra of the variety \({\mathfrak V}\). For every non-trivial variety of linear algebras \({\mathfrak V}\), some complexity function \({\mathcal C}({\mathfrak V},z)\) is constructed which is an entire function of a complex variable. Namely, \({\mathcal C}({\mathfrak V},z)=\sum_{n=1}^{\infty} c_n({\mathfrak V}) z^n/n!\), \(z\) is a complex variable. In the article under review, the author introduces the notion of complexity for varieties of Lie algebras. The main goal of the author is to specify the complexity of a product of two varieties in terms of the complexities of multiplicands. The main observation is that \({\mathcal C}({\mathfrak {MV}},z)\) behaves like a composition of functions \({\mathcal C}({\mathfrak M},z)\), \(\exp(z)\), and \({\mathcal C}({\mathfrak V},z)\).
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 1389714
zbMATH Keywords: Lie algebra
zbMATH Keywords: free algebra
zbMATH Keywords: variety of algebras
zbMATH Keywords: exponential generating function
zbMATH Keywords: product of varieties
author: Victor Petrogradsky
reviewed by: Boris Vernikov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02780183
OpenAlex ID: W1984777518
description: scientific article; zbMATH DE number 1389714