Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity

We study quadratic algebras over a field $\textbf{k}$. We prove
that the class of  $n$-generated PBW algebras with finite global
dimension and polynomial growth contains a unique (up to
isomorphism) monomial algebra, $A= \textbf{k} \langle x_1,\cdots,x_n
\rangle /(x_jx_i \mid 1 \leq i < j \leq n)$.

The main result shows that for an $n$-generated quantum binomial algebra
$A$ the following conditions are equivalent:
(i) $A$ is a PBW algebra with  finite global dimension.
(ii) $A$ is a PBW algebra with polynomial growth.
(iii) $A$ is  an Artin-Schelter regular PBW algebra.
(iv)  $A$ is a Yang-Baxter algebra, that is the set of relations
$\Re$ defines canonically  a solution of the
Yang-Baxter equation.
(v) $A$ is a binomial skew polynomial ring, with respect to some
enumeration of $X$.
(vi) $H_A(z)= \frac{1}{(1-z)^n}$.
(vii) The Koszul dual $A^{!}$ is a quantum Grassman algebra.

It follows then that the problem of classification of Artin-Schelter
regular PBW algebras with quantum binomial relations and global dimension
$n$ is equivalent to the classification of square-free set-theoretic
solutions of YBE, $(X,r),$ on sets $X$ of order  $n$. Even  under these
strong restrictions on the shape of the relations, the problem remains
highly nontrivial. However for reasonably small $n$ (say $n\leq 10$)
the square-free solutions of YBE $(X,r)$ are known. A possible
classification for general $n$ can be based on the so called
multipermutation level of  the solutions.