Involutive Yang-Baxter groups

Authors:
Ferran Cedó, Eric Jespers and Ángel del Río

Journal:
Trans. Amer. Math. Soc. **362** (2010), 2541-2558

MSC (2010):
Primary 81R50, 20F29, 20B35, 20F16

Published electronically:
December 3, 2009

MathSciNet review:
2584610

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Abstract | References | Similar Articles | Additional Information

Abstract: 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 -type. A group of -type is a group isomorphic to a subgroup of so that the projection onto the first component is a bijective map, where is the free abelian group of rank and is the symmetric group of degree . The projection of onto the second component 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 , classify the groups of -type with 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 -type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.

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Additional Information

**Ferran Cedó**

Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Email:
cedo@mat.uab.cat

**Eric Jespers**

Affiliation:
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

Email:
efjesper@vub.ac.be

**Ángel del Río**

Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain

Email:
adelrio@um.es

DOI:
http://dx.doi.org/10.1090/S0002-9947-09-04927-7

Keywords:
Yang-Baxter equation,
involutive non-degenerate solutions,
group of $I$-type,
finite solvable group.

Received by editor(s):
March 26, 2008

Published electronically:
December 3, 2009

Additional Notes:
The first author was partially supported by grants of DGI MEC-FEDER (Spain) MTM2005-00934 and Generalitat de Catalunya 2005SGR00206

The second author was partially supported by grants of Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium) and Flemish-Polish bilateral agreement BIL2005/VUB/06.

The third author was partially supported by grants of DGI MEC-FEDER (Spain) MTM2006-06865 and Fundación Séneca of Murcia 04555/GERM/06.

Article copyright:
© Copyright 2009
American Mathematical Society