## Involutive Yang-Baxter groups

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- by Ferran Cedó, Eric Jespers and Ángel del Río PDF
- Trans. Amer. Math. Soc.
**362**(2010), 2541-2558 Request permission

## 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.## References

- N. Ben David and Y. Ginosar, On groups of central type, non-degenerate and bijective cohomology classes, Israel J. Math. to appear, ArXiv: 0704.2516v1 [math.GR].
- Ferran Cedó, Eric Jespers, and Jan Okniński,
*The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition*, Proc. Amer. Math. Soc.**134**(2006), no. 3, 653–663. MR**2180881**, DOI 10.1090/S0002-9939-05-08003-2 - V. G. Drinfel′d,
*On some unsolved problems in quantum group theory*, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 1–8. MR**1183474**, DOI 10.1007/BFb0101175 - Pavel Etingof and Shlomo Gelaki,
*A method of construction of finite-dimensional triangular semisimple Hopf algebras*, Math. Res. Lett.**5**(1998), no. 4, 551–561. MR**1653340**, DOI 10.4310/MRL.1998.v5.n4.a12 - Pavel Etingof, Travis Schedler, and Alexandre Soloviev,
*Set-theoretical solutions to the quantum Yang-Baxter equation*, Duke Math. J.**100**(1999), no. 2, 169–209. MR**1722951**, DOI 10.1215/S0012-7094-99-10007-X - Tatiana Gateva-Ivanova,
*A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation*, J. Math. Phys.**45**(2004), no. 10, 3828–3858. MR**2095675**, DOI 10.1063/1.1788848 - Tatiana Gateva-Ivanova and Shahn Majid,
*Matched pairs approach to set theoretic solutions of the Yang-Baxter equation*, J. Algebra**319**(2008), no. 4, 1462–1529. MR**2383056**, DOI 10.1016/j.jalgebra.2007.10.035 - Tatiana Gateva-Ivanova and Michel Van den Bergh,
*Semigroups of $I$-type*, J. Algebra**206**(1998), no. 1, 97–112. MR**1637256**, DOI 10.1006/jabr.1997.7399 - B. Huppert,
*Endliche Gruppen. I*, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703**, DOI 10.1007/978-3-642-64981-3 - Eric Jespers and Jan Okniński,
*Monoids and groups of $I$-type*, Algebr. Represent. Theory**8**(2005), no. 5, 709–729. MR**2189580**, DOI 10.1007/s10468-005-0342-7 - Eric Jespers and Jan Okniński,
*Noetherian semigroup algebras*, Algebra and Applications, vol. 7, Springer, Dordrecht, 2007. MR**2301033** - Christian Kassel,
*Quantum groups*, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR**1321145**, DOI 10.1007/978-1-4612-0783-2 - Jiang-Hua Lu, Min Yan, and Yong-Chang Zhu,
*On the set-theoretical Yang-Baxter equation*, Duke Math. J.**104**(2000), no. 1, 1–18. MR**1769723**, DOI 10.1215/S0012-7094-00-10411-5 - Donald Passman,
*Permutation groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0237627** - Derek J. S. Robinson,
*A course in the theory of groups*, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR**1357169**, DOI 10.1007/978-1-4419-8594-1 - Wolfgang Rump,
*A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation*, Adv. Math.**193**(2005), no. 1, 40–55. MR**2132760**, DOI 10.1016/j.aim.2004.03.019 - Wolfgang Rump,
*Braces, radical rings, and the quantum Yang-Baxter equation*, J. Algebra**307**(2007), no. 1, 153–170. MR**2278047**, DOI 10.1016/j.jalgebra.2006.03.040 - C. N. Yang,
*Some exact results for the many-body problem in one dimension with repulsive delta-function interaction*, Phys. Rev. Lett.**19**(1967), 1312–1315. MR**261870**, DOI 10.1103/PhysRevLett.19.1312

## 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
- MR Author ID: 94560
- Email: efjesper@vub.ac.be
**Ángel del Río**- Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
- MR Author ID: 288713
- Email: adelrio@um.es
- 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. - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**362**(2010), 2541-2558 - MSC (2010): Primary 81R50, 20F29, 20B35, 20F16
- DOI: https://doi.org/10.1090/S0002-9947-09-04927-7
- MathSciNet review: 2584610