Algebras defined by Lyndon words and Artin-Schelter regularity

The classification of Artin-Schelter regular algebras and finding new classes of regular algebras are fundamental problems
in noncommutative algebraic geometry. In this talk we consider classes C(X, W) of associative graded algebras A over a
field K, generated by a finite set X and with a fixed obstructions set W, where W is a finite antichain of Lyndon words
in X. The main question is: when such a class contains an AS regular algebra? Given the global dimension d of A, we
that the order of W satisfies the inequality d-1 ≤ |W| ≤ d(d-1)/2. We prove that W is unique, and  C(X, W) contains regular
algebras,  whenever |W| attains the lower, or the upper bounds.  If the order of W is d(d-1)/2,  C(X, W) contains numerous
non-isomorphic PBW regular algebras, each of which defines a set-theoretic solution of the Yang-Baxter equation.