Monomial algebras defined by Lyndon words

Abstract

Assume that X={x1,…,xg} 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 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⩽|W|⩽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 N={l1α1l2α2⋯ldαd|αi⩾0,1⩽i⩽d}, where N={l1,…,ld}. We give an extremal class of monomial algebras, the Fibonacci–Lyndon algebras, Fn, with global dimension n and polynomial growth, and show that the algebra F6 of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin–Schelter regular algebra.