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.

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