A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.Comment: 34 page

Algebras defined by Lyndon words and Artin-Schelter regularity

Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially we do not impose restrictions on $W$ and investigate the case when all algebras in $\mathfrak{C} (X, W)$ have polynomial growth and finite global dimension $d$. Next we consider classes $\mathfrak{C} (X, W)$ of algebras whose sets of obstructions $W$ are antichains of Lyndon words. The central question is "when a class $\mathfrak{C} (X, W)$ contains Artin-Schelter regular algebras?" Each class $\mathfrak{C} (X, W)$ defines a Lyndon pair $(N,W)$ which determines uniquely the global dimension, $gl\dim A$, and the Gelfand-Kirillov dimension, $GK\dim A$, for every $A \in \mathfrak{C}(X, W)$. We find a combinatorial condition in terms of $(N,W)$, so that the class $\mathfrak{C}(X, W)$ contains the enveloping algebra $U\mathfrak{g}$ of a Lie algebra $\mathfrak{g}$. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Groebner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimensions $6$ and $7$ occurring as enveloping $U = U\mathfrak{g}$ of standard monomial Lie algebras. The classification is made in terms of their Lyndon pairs $(N, W)$, each of which determines also the explicit relations of $U$

Veronese and Segre morphisms between non-commutative projective spaces

Number theory, Algebra and Geometr

Quadratic algebras of skew type and the underlying monoids

AbstractWe consider algebras over a field K defined by a presentation K〈x1,…,xn∣R〉, where R consists of n2 square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand–Kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Gateva-Ivanova and van den Bergh, and Jespers and Okniński considered special classes of such algebras in the contexts of noetherian algebras, Gröbner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang–Baxter equation

Semigroups of I-type

Assume that S is a semigroup generated by {x(1),..., x(n)}, and let U be the multiplicative free commutative semigroup generated by {u(1),..., u(n)}. We say that S is of I-type if there is a bijective upsilon : U --> S such that for all a is an element of U, {upsilon(u(1)a),..., upsilon(u(n)a)} = {x(1)upsilon(a),..., x(n)upsilon(a)}. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be of I-type is related to various other mathematical notions found in the literature. In particular we show that semigroups of I-type appear in the study of the set-theoretic solutions of the Yang-Baxter equation, in the theory of Bieberbach groups, and in the study of certain skew binomial polynomial rings which were introduced by the first author. (C) 1998 Academic Press