Conjectures on Set-theoretic Solutions of the Yang-Baxter Equation
by
Tatiana Gateva-Ivanova
Instutute of Mathematics, Bulgarian Academy of Sciences and American University in Bulgaria

Let X be a finite set. An injective map r: X×X --> X×X is a set-theoretic solution of the Yang-Baxter equation (YBE) if r₁r₂r₁=r₂r₁r₂, where r_i are maps on X³ defined as r₁=r×Id_X, and r₂=Id_X×r. We shall refer to r simply as a solution (of YBE) and denote it by (r;X). With each involutive solution (r;X) of YBE we associate a semigroup S(r)=<X; R(r) > generated by X and with a set of defining relations R(r), where xy=zt is in R(r) iff r(x, y)=(z, t), and r(z, t)=(x, y). We call S(r) a Yang-Baxter semigroup.

It was shown in [4] that each binomial semigroup S₀ with relations of skew-polynomial type defines a nondegenerate involutive solution r₀ which acts trivially on diag(X×X). The bionomial semigroups S₀ were introduced and studied in [1] and [2] .

Conjecture 1, [3]. (r;X) is a nondegenerate involutive solution of the Yang-Baxter equation acting trivially on diag(X×X) if and only if the set X can be ordered so that the semigroup S(r) associated with r is a binomial semigroup of skew-polynomial type.

Conjecture 1 is equivalent to the following:

Conjecture 2. Let (r;X) be a nondegenerate involutive solution of YBE which acts trivially on diag(X×X). Then the set X can be ordered so that r(x, y)=(z, t), and x > y imply z < t, z < x, and y < t.

We show that Conjecture 2 is also equivalent to certain conjectures of Etingof and Schedler.

Theorem. Let X=<x₁, ... , x_n > be a finite set, let (r, X) be a solution of the Yang-Baxter equation, and let S(r)=<X; R(r) > be the Yang-Baxter semigroup associated with r. Suppose that at least one of the following conditions is satisfied:

(1) The number n is atmost 10.

(2) The number of generators n of S(r) is either a) atmost 23, or b) different from 2^k, (k is an arbitrary integer) and the set of relations R(r) satisfies the following:

(*) for every two non-commuting elements x and y of X, there exists a z in X such that xy=zx, or xy=yz is in R(r).

(3) The number n is arbitrary and the monomial W, generating the socle of the Koszul dual A^! of the semigroup algebra A=kS₀ over a field k can be presented as a product of all generators x₁, ... , x_n.

Then the set X can be ordered so that:

1) If xy=zt is in R(r) then the inequality x > y implies z < t, z < x, y < t; and

2) S(r) is a binomial semigroup with relations of skew-polynomial type. .

References

[1] T. Gateva-Ivanova, Noetherian properties of skew-polynomial rings with binomial relarions, Trans. Amer. Math. Soc. 343(1994), 203-219.

[2] T. Gateva-Ivanova, Skew-polynomial rings with binomial relations, J. Algebra, 185 (1996), 710-753.

[3] T. Gateva-Ivanova, M. Van den Bergh, Regularity of skew-polynomial rings with binomial relations, a talk to the Ring Theory Conference, 1996, Niskolc, Hungary.

[4] T. Gateva-Ivanova, M. Van den Bergh, Semigroups of I type, J. Algebra 206 (1998), 97-112.

Date received: July 14, 2000