Monomial algebras defined by Lyndon words.
type: Item
instance of: scholarly article
title: Monomial algebras defined by Lyndon words.
zbMATH Open document ID: 1310.16018
author: Tatiana Gateva-Ivanova
author: Gunnar Fløystad
published in: Journal of Algebra
publication date: 2014-07-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.6256
review text: \textit{D. J. Anick} studied a class of monomial algebras with finite global dimension [Trans. Am. Math. Soc. 291, 291-310 (1985; Zbl 0575.16002); ibid. 296, 641-659 (1986; Zbl 0598.16028)]. This paper is a further study of graded associative algebras for which the set of obstructions consists of Lyndon words. Let \(X=\{x_1,\ldots,x_g\}\) be a finite alphabet and \(K\) a field. The authors study monomial algebras \(A=K\langle X\rangle/(W)\), where \(W\) is an antichain of Lyndon words in \(X\) of arbitrary cardinality. The authors establish a Poincaré-Birkhoff-Witt type basis of \(A\) in terms of its Lyndon atoms \(N\); in general, \(N\) may be infinite. The authors prove that if \(A\) has polynomial growth of degree \(d\) then \(A\) has global dimension \(d\) and is standard finitely presented, with \(d-1\leq |W|\leq d(d-1)/2\). Furthermore, \(A\) has polynomial growth iff the set of Lyndon atoms \(N\) is finite. In this case \(A\) has a \(K\)-basis \(\mathcal N=\{l_1^{\alpha_1}l_2^{\alpha_2}\cdots l_d^{\alpha_d}\mid\alpha_i\geq 0,\;1\leq i\leq d\}\), where \(N=\{l_1,\ldots,l_d\}\). The authors also construct an extremal class of monomial algebras, called the Fibonacci-Lyndon algebras, \(F_n\), with global dimension \(n\) and of polynomial growth and uniquely determined up to isomorphism. The authors prove that the algebra \(F_6\) of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin-Schelter regular algebra.
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 68R15
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 16Z05
zbMATH DE Number: 6320560
zbMATH Keywords: Lyndon words
zbMATH Keywords: monomial algebras
zbMATH Keywords: algebras of polynomial growth
zbMATH Keywords: global dimension
zbMATH Keywords: Artin-Schelter regular algebras
zbMATH Keywords: graded algebras
zbMATH Keywords: Poincaré-Birkhoff-Witt bases
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
OpenAlex ID: W2074716363
arXiv ID: 1207.6256
cites work: On Monomial Algebras of Finite Global Dimension
cites work: On the Homology of Associative Algebras
cites work: Artin-Schelter regular algebras of dimension five
cites work: Q4728507
cites work: Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity
cites work: Standard Lyndon Bases of Lie Algebras and Enveloping Algebras
cites work: Q3659988
cites work: Q4529547
cites work: Q3134850
cites work: Lyndon words and Fibonacci numbers
cites work: Q3243334
cites work: Q4853962
cites work: Q3952128
cites work: Q3836502
DOI: 10.1016/J.JALGEBRA.2014.01.012
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description: scientific article