Exponential Schreier's formula for free Lie algebras and its applications
type: Item
instance of: scholarly article
title: Exponential Schreier's formula for free Lie algebras and its applications
zbMATH Open document ID: 0983.17007
DOI: 10.1007/BF02366349
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-08-16T00:00:00Z
review text: The author studies the growth function \(c_n({\mathbb V})\) for a variety of Lie algebras \({\mathbb V}\), where \(c_n({\mathbb V})\) is the dimension of the linear hull of the set of multilinear words with \(n\) different letters in the free algebra of the variety \({\mathbb V}\). For every non-trivial variety of linear algebras \({\mathbb V}\), some complexity function \({\mathcal C}({\mathbb V},z)\) is constructed which is an entire function of a complex variable. Namely, \({\mathcal C}({\mathbb V},z)=\sum_{n=1}^{\infty} c_n({\mathbb V})z^n/n!\), \(z\) is a complex variable. The author obtains the exponential analogue of Schreier's formula for free groups (Theorem 2.2). Further he applies this result to the determination of explicit formulas for the complexity function of the product of Lie varieties (Theorem 3.1) and that of the polynomial variety (Theorem 3.2). Finally, the author finds asymptotics for some concrete varieties (Theorems 4.3 and 4.4).
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 1315228
zbMATH Keywords: Lie algebra
zbMATH Keywords: free algebra
zbMATH Keywords: variety of algebras
zbMATH Keywords: exponential generating function
zbMATH Keywords: growth function
author: Victor Petrogradsky
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: Q4146776
cites work: Q3243472
cites work: Q5732796
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cites work: Intermediate growth in Lie algebras and their enveloping algebras
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full work available at URL: https://doi.org/10.1007/bf02366349
OpenAlex ID: W2019325464
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description: scientific article; zbMATH DE number 1315228