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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: 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 $ \mathcal{G}$ of $ I$-type is a group isomorphic to a subgroup of $ \mathrm{Fa}_n\rtimes \mathrm{Sym}_n$ so that the projection onto the first component is a bijective map, where $ \mathrm{Fa}_n$ is the free abelian group of rank $ n$ and $ \mathrm{Sym}_{n}$ is the symmetric group of degree $ n$. The projection of $ \mathcal{G}$ onto the second component $ \mathrm{Sym}_n$ 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.


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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
PII: S 0002-9947(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