Monomial algebras defined by Lyndon words

Abstract

Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K <X> /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincar\'{e}-Birkhoff-Witt type basis of $A$ in terms of its \emph{Lyndon atoms} $N$, but, in general, $N$ may be infinite. We 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 $\mathfrak{N} = {l_1^{\alpha_{1}}l_2^{\alpha_{2}}... l_d^{\alpha_{d}} \mid \alpha_{i} \geq 0, 1 \leq i \leq d}$, where $N = {l_1, ...,l_d}$. We give an extremal class of monomial algebras, the Fibonacci-Lyndon algebras, $F_n$, with global dimension $n$ and polynomial growth, and show that the algebra $F_6$ of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin-Schelter regular algebra.