Algebras defined by Lyndon words and Artin-Schelter regularity

Let X = { x 1 , x 2 , ⋯ , x n } X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K K be a field. We study classes C ( X , W ) \mathfrak {C}(X, W) of graded K K -algebras A = K ⟨ X ⟩ / I A = K\langle X\rangle / I , generated by X X and with a fixed set of obstructions W W . Initially we do not impose restrictions on W W and investigate the case when the algebras in C ( X , W ) \mathfrak {C} (X, W) have polynomial growth and finite global dimension d d . Next we consider classes C ( X , W ) \mathfrak {C} (X, W) of algebras whose sets of obstructions W W are antichains of Lyndon words. The central question is “when a class C ( X , W ) \mathfrak {C} (X, W) contains Artin-Schelter regular algebras?” Each class C ( X , W ) \mathfrak {C} (X, W) defines a Lyndon pair ( N , W ) (N,W) , which, if N N is finite, determines uniquely the global dimension, g l d i m A gl\,dimA , and the Gelfand-Kirillov dimension, G K d i m A GK dimA , for every A ∈ C ( X , W ) A \in \mathfrak {C}(X, W) . We find a combinatorial condition in terms of