TY - JOUR

T1 - Involutive yang-baxter groups

AU - Cedo, Ferran

AU - Jespers, Eric

AU - Rio, Angel Del

PY - 2010/5/1

Y1 - 2010/5/1

N2 - In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one- to-one correspondence between involutive non-degenerate solutions on finite sets and groups of I-type. A group G of I-type is a group isomorphic to a subgroup of Fan ⋊ Symn so that the projection onto the first component is a bijective map, where Fan is the free abelian group of rank n and Symn is the symmetric group of degree n. The projection of G onto the second component Symn we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld's problem for involutive non- degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G, classify the groups of I-type with G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of I-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang- Baxter equation) with a prescribed associated IYB group. © 2009 American Mathematical Society.

AB - In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one- to-one correspondence between involutive non-degenerate solutions on finite sets and groups of I-type. A group G of I-type is a group isomorphic to a subgroup of Fan ⋊ Symn so that the projection onto the first component is a bijective map, where Fan is the free abelian group of rank n and Symn is the symmetric group of degree n. The projection of G onto the second component Symn we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld's problem for involutive non- degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G, classify the groups of I-type with G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of I-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang- Baxter equation) with a prescribed associated IYB group. © 2009 American Mathematical Society.

U2 - 10.1090/S0002-9947-09-04927-7

DO - 10.1090/S0002-9947-09-04927-7

M3 - Article

SN - 0002-9947

VL - 362

SP - 2541

EP - 2558

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

ER -