## A combinatorial approach to the set-theoretic solutions of the YangBaxter equation

Venue: | J.Math.Phys |

Citations: | 14 - 6 self |

### BibTeX

@ARTICLE{Gateva-ivanova_acombinatorial,

author = {Tatiana Gateva-ivanova},

title = {A combinatorial approach to the set-theoretic solutions of the YangBaxter equation},

journal = {J.Math.Phys},

year = {},

pages = {3828--3858}

}

### OpenURL

### Abstract

Abstract. A bijective map r: X 2 − → X 2, where X = {x1, · · · , xn} is a finite set, is called a set-theoretic solution of the Yang-Baxter equation (YBE) if the braid relation r12r23r12 = r23r12r23 holds in X 3. A non-degenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called squarefree 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 squarefree 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 Poincaré-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. 1.

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Citation Context ...on V ⊗ V for the vector space V spanned by X, which is, clearly, a solution of 1.1. Various works dealing with set-theoretic solutions appeared during the last decade, cf. [32], [17], [14], [6], [7], =-=[30]-=-, [21], [24], [27]. The purpose of this paper is first to present some recent conjectures on the set-theoretic solutions of the Yang-Baxter equation, and to give an account of the research in this are... |

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Citation Context ... properties of the solutions inspired the following Conjecture, which we reported first in a talk at the International Conference in Ring Theory, Miskolc 1996, see also [11], [12]. Conjecture 1 2.18. =-=[13]-=- Let (X, r) be a square-free (non-degenerate, involutive) solution of the Yang-Baxter equation. Then the set X can be ordered so, that the associated semigroup S = S(X, r) is of skew-polynomial type. ... |

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Citation Context ...he semigroup algebra k[S] is right Noetherian and has finite Gelfand-Kirillov dimension, then k[S] satisfies a polynomial identity, and S satisfies a semigroup identity. Condition 6.1.12 follows from =-=[28]-=-. The Koszul dual algebra A ! is defined in [23]. Condition 6.1.10 follows from the fact that a Koszul algebra A of finite global dimension is Gorenstein if and only if A ! is Frobenius, cf [29], Prop... |

1 |
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Citation Context ...l these results in Theorem 6.1. My student, M.S. Garcia Roman has shown that for an explicitly given solution (X, r), condition 2.26.3 is equivallent to a standard problem from Linear Programming. In =-=[15]-=- is presented a matched pairs approach to the set-theoretic solutions of the Yang-Baxter equation. One of the main results in [15], given here as Theorem 5.6 covers all known constructions of solution... |