Involutive YangBaxter groups
Authors:
Ferran Cedó, Eric Jespers and Ángel del Río
Journal:
Trans. Amer. Math. Soc. 362 (2010), 25412558
MSC (2010):
Primary 81R50, 20F29, 20B35, 20F16
Published electronically:
December 3, 2009
MathSciNet review:
2584610
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Additional Information
Abstract: In 1992 Drinfeld posed the question of finding the settheoretic solutions of the YangBaxter equation. Recently, GatevaIvanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a grouptheoretical interpretation of involutive nondegenerate solutions. Namely, there is a onetoone correspondence between involutive nondegenerate 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 YangBaxter group (IYB group). This suggests the following strategy to attack Drinfeld's problem for involutive nondegenerate settheoretic 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 nonobvious method to construct infinitely many groups of type (and hence infinitely many involutive nondegenerate settheoretic solutions of the YangBaxter 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/S0002994709049277
Keywords:
YangBaxter equation,
involutive nondegenerate 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 MECFEDER (Spain) MTM200500934 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 FlemishPolish bilateral agreement BIL2005/VUB/06.
The third author was partially supported by grants of DGI MECFEDER (Spain) MTM200606865 and Fundación Séneca of Murcia 04555/GERM/06.
Article copyright:
© Copyright 2009
American Mathematical Society
