Algebras defined by Lyndon words and Artin-Schelter regularity

Abstract

LetX={x1,x2,⋯,xn}X= \{x_1, x_2, \cdots , x_n\}be a finite alphabet, and letKKbe a field. We study classesC(X,W)\mathfrak {C}(X, W)of gradedKK-algebrasA=K⟨X⟩/IA = K\langle X\rangle / I, generated byXXand witha fixed set of obstructionsWW. Initially we do not impose restrictions onWWand investigate the case when the algebras inC(X,W)\mathfrak {C} (X, W)have polynomial growth and finite global dimensiondd. Next we consider classesC(X,W)\mathfrak {C} (X, W)of algebras whose sets of obstructionsWWare antichains of Lyndon words. The central question is “when a classC(X,W)\mathfrak {C} (X, W)contains Artin-Schelter regular algebras?” Each classC(X,W)\mathfrak {C} (X, W)defines a Lyndon pair(N,W)(N,W), which, ifNNis finite, determines uniquely the global dimension,gldimAgl\,dimA, and the Gelfand-Kirillov dimension,GKdimAGK dimA, for everyA∈C(X,W)A \in \mathfrak {C}(X, W). We find a combinatorial condition in terms of(N,W)(N,W), so that the classC(X,W)\mathfrak {C}(X, W)contains the enveloping algebraUgU\mathfrak {g}, of a Lie algebrag\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 dimension66and77occurring as envelopingU=UgU = U\mathfrak {g}ofstandard monomial Lie algebras. The classification is made in terms of their Lyndon pairs(N,W)(N, W), each of which determines also the explicit relations ofUU.