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 Poincaré-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 $d-1\le |W|\le d(d-1)/2$. Furthermore, $A$ has polynomial growth iff the set of Lyndon atoms $N$ is finite. In this case $A$ has a $K$-basis $\mathfrak{N}=l_1^{{\alpha }_1}{l}_2^{{\alpha }_2}...l_d^{{\alpha }_d}\mid {\alpha }_i\ge 0,1\le i\le d$, 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.

Cited by in corpus (2)

• Algebras defined by Lyndon words and Artin-Schelter regularity
• The structure of connected (graded) Hopf algebras