Monomial algebras defined by Lyndon words.
\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.
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