Algebras defined by Lyndon words and Artin-Schelter regularity

Abstract:

Let $X= \{x_1, x_2, \cdots , x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak {C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially we do not impose restrictions on $W$ and investigate the case when the algebras in $\mathfrak {C} (X, W)$ have polynomial growth and finite global dimension $d$. Next we consider classes $\mathfrak {C} (X, W)$ of algebras whose sets of obstructions $W$ are antichains of Lyndon words. The central question is “when a class $\mathfrak {C} (X, W)$ contains Artin-Schelter regular algebras?” Each class $\mathfrak {C} (X, W)$ defines a Lyndon pair $(N,W)$, which, if $N$ is finite, determines uniquely the global dimension, $gl\,dimA$, and the Gelfand-Kirillov dimension, $GK dimA$, for every $A \in \mathfrak {C}(X, W)$. We find a combinatorial condition in terms of $(N,W)$, so that the class $\mathfrak {C}(X, W)$ contains the enveloping algebra $U\mathfrak {g}$, of a Lie algebra $\mathfrak {g}$. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Gröbner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimension $6$ and $7$ occurring as enveloping $U = U\mathfrak {g}$ of standard monomial Lie algebras. The classification is made in terms of their Lyndon pairs $(N, W)$, each of which determines also the explicit relations of $U$.