Algebras defined by Lyndon words and Artin-Schelter regularity

Let X = { x 1 , 2 ⋯ n stretchy="false">} X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K"> K encoding="application/x-tex">K field. We study classes alttext="German upper C left-parenthesis W right-parenthesis"> C stretchy="false">( W stretchy="false">) encoding="application/x-tex">\mathfrak {C}(X, W) of graded -algebras A K mathematical left-angle right-angle slash I"> A stretchy="false">⟨<!-- ⟨ stretchy="false">⟩<!-- ⟩ <mml:mo>/ I encoding="application/x-tex">A = K\langle X\rangle / I , generated by X"> encoding="application/x-tex">X with a fixed set obstructions W"> encoding="application/x-tex">W . Initially we do not impose restrictions on investigate the case when algebras in {C} (X, have polynomial growth global dimension alttext="d"> d encoding="application/x-tex">d Next consider whose sets obstructions are antichains Lyndon words. The central question is “when class contains Artin-Schelter regular algebras?” Each defines pair alttext="left-parenthesis N N encoding="application/x-tex">(N,W) which, if N"> encoding="application/x-tex">N finite, determines uniquely dimension, alttext="g l d i m A"> g l i m encoding="application/x-tex">gl\,dimA Gelfand-Kirillov G G encoding="application/x-tex">GK dimA for every element-of German ∈<!-- ∈ \in \mathfrak find combinatorial condition terms so that enveloping algebra U g"> U mathvariant="fraktur">g encoding="application/x-tex">U\mathfrak {g} Lie introduce monomial defined words, prove results Gröbner-Shirshov bases ideals Lyndon-Lie monomials. Finally classify all two-generated alttext="6"> 6 encoding="application/x-tex">6 alttext="7"> 7 encoding="application/x-tex">7 occurring as encoding="application/x-tex">U U\mathfrak standard algebras. classification made their pairs encoding="application/x-tex">(N, each which also explicit relations U"> encoding="application/x-tex">U