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
Fulltext PDF
Abstract 
References 
Similar Articles 
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.
 1.
N. Ben David and Y. Ginosar, On groups of central type, nondegenerate and bijective cohomology classes, Israel J. Math. to appear, ArXiv: 0704.2516v1 [math.GR].
 2.
Ferran
Cedó, Eric
Jespers, and Jan
Okniński, The GelfandKirillov dimension of
quadratic algebras satisfying the cyclic condition, Proc. Amer. Math. Soc. 134 (2006), no. 3, 653–663 (electronic). MR 2180881
(2006g:16051), http://dx.doi.org/10.1090/S0002993905080032
 3.
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
(94a:17006), http://dx.doi.org/10.1007/BFb0101175
 4.
Pavel
Etingof and Shlomo
Gelaki, A method of construction of finitedimensional triangular
semisimple Hopf algebras, Math. Res. Lett. 5 (1998),
no. 4, 551–561. MR 1653340
(99i:16069), http://dx.doi.org/10.4310/MRL.1998.v5.n4.a12
 5.
Pavel
Etingof, Travis
Schedler, and Alexandre
Soloviev, Settheoretical solutions to the quantum YangBaxter
equation, Duke Math. J. 100 (1999), no. 2,
169–209. MR 1722951
(2001c:16076), http://dx.doi.org/10.1215/S001270949910007X
 6.
Tatiana
GatevaIvanova, A combinatorial approach to the settheoretic
solutions of the YangBaxter equation, J. Math. Phys.
45 (2004), no. 10, 3828–3858. MR 2095675
(2005h:16077), http://dx.doi.org/10.1063/1.1788848
 7.
Tatiana
GatevaIvanova and Shahn
Majid, Matched pairs approach to set theoretic solutions of the
YangBaxter equation, J. Algebra 319 (2008),
no. 4, 1462–1529. MR 2383056
(2009a:16079), http://dx.doi.org/10.1016/j.jalgebra.2007.10.035
 8.
Tatiana
GatevaIvanova and Michel
Van den Bergh, Semigroups of 𝐼type, J. Algebra
206 (1998), no. 1, 97–112. MR 1637256
(99h:20090), http://dx.doi.org/10.1006/jabr.1997.7399
 9.
B.
Huppert, Endliche Gruppen. I, Die Grundlehren der
Mathematischen Wissenschaften, Band 134, SpringerVerlag, BerlinNew York,
1967 (German). MR 0224703
(37 #302)
 10.
Eric
Jespers and Jan
Okniński, Monoids and groups of 𝐼type, Algebr.
Represent. Theory 8 (2005), no. 5, 709–729. MR 2189580
(2007b:20071), http://dx.doi.org/10.1007/s1046800503427
 11.
Eric
Jespers and Jan
Okniński, Noetherian semigroup algebras, Algebras and
Applications, vol. 7, Springer, Dordrecht, 2007. MR 2301033
(2007k:16001)
 12.
Christian
Kassel, Quantum groups, Graduate Texts in Mathematics,
vol. 155, SpringerVerlag, New York, 1995. MR 1321145
(96e:17041)
 13.
JiangHua
Lu, Min
Yan, and YongChang
Zhu, On the settheoretical YangBaxter equation, Duke Math.
J. 104 (2000), no. 1, 1–18. MR 1769723
(2001f:16076), http://dx.doi.org/10.1215/S0012709400104115
 14.
Donald
Passman, Permutation groups, W. A. Benjamin, Inc., New
YorkAmsterdam, 1968. MR 0237627
(38 #5908)
 15.
Derek
J. S. Robinson, A course in the theory of groups, 2nd ed.,
Graduate Texts in Mathematics, vol. 80, SpringerVerlag, New York,
1996. MR
1357169 (96f:20001)
 16.
Wolfgang
Rump, A decomposition theorem for squarefree unitary solutions of
the quantum YangBaxter equation, Adv. Math. 193
(2005), no. 1, 40–55. MR 2132760
(2005k:81132), http://dx.doi.org/10.1016/j.aim.2004.03.019
 17.
Wolfgang
Rump, Braces, radical rings, and the quantum YangBaxter
equation, J. Algebra 307 (2007), no. 1,
153–170. MR 2278047
(2007m:16065), http://dx.doi.org/10.1016/j.jalgebra.2006.03.040
 18.
C.
N. Yang, Some exact results for the manybody problem in one
dimension with repulsive deltafunction interaction, Phys. Rev. Lett.
19 (1967), 1312–1315. MR 0261870
(41 #6480)
 1.
 N. Ben David and Y. Ginosar, On groups of central type, nondegenerate and bijective cohomology classes, Israel J. Math. to appear, ArXiv: 0704.2516v1 [math.GR].
 2.
 F. Cedó, E. Jespers and J. Okniński, The GelfandKirillov dimension of quadratic algebras satisfying the cyclic condition, Proc. AMS 134 (2005), 653663. MR 2180881 (2006g:16051)
 3.
 V. G. Drinfeld, On unsolved problems in quantum group theory. Quantum Groups, Lecture Notes in Math. 1510, SpringerVerlag, Berlin, 1992, 18. MR 1183474 (94a:17006)
 4.
 P. Etingof and S. Gelaki, A method of construction of finitedimensional triangular semisimple Hopf algebras, Mathematical Research Letters 5 (1998), 551561. MR 1653340 (99i:16069)
 5.
 P. Etingof, T. Schedler and A. Soloviev, Settheoretical solutions to the quantum YangBaxter equation, Duke Math. J. 100 (1999), 169209. MR 1722951 (2001c:16076)
 6.
 T. GatevaIvanova, A combinatorial approach to the settheoretic solutions of the YangBaxter equation, J. Math. Phys. 45 (2004), 38283858. MR 2095675 (2005h:16077)
 7.
 T. GatevaIvanova and S. Majid, Matched pairs approach to set theoretic solutions of the YangBaxter equation, J. Algebra 319 (2008), no. 4, 14621529. MR 2383056
 8.
 T. GatevaIvanova and M. Van den Bergh, Semigroups of type, J. Algebra 206 (1998), 97112. MR 1637256 (99h:20090)
 9.
 B. Huppert, Endliche Gruppen I, SpringerVerlag, Berlin, 1967. MR 0224703 (37:302)
 10.
 E. Jespers and J. Okniński, Monoids and Groups of Type, Algebr. Represent. Theory 8 (2005), 709729. MR 2189580 (2007b:20071)
 11.
 E. Jespers and J. Okniński, Noetherian Semigroup Algebras, Springer, Dordrecht, 2007. MR 2301033 (2007k:16001)
 12.
 C. Kassel, Quantum Groups, Graduate Text in Mathematics 155, SpringerVerlag, New York, 1995. MR 1321145 (96e:17041)
 13.
 JiangHua Lu, Min Yan and YongChang Zhu, On the settheoretical YangBaxter equation, Duke Math. J. 104 (2000),153170. MR 1769723 (2001f:16076)
 14.
 D. S. Passman, Permutation Groups, Benjamin, New York, 1968. MR 0237627 (38:5908)
 15.
 D. K. Robinson, A course in the theory of groups, second edition, SpringerVerlag, New York, 1996. MR 1357169 (96f:20001)
 16.
 W. Rump, A decomposition theorem for squarefree unitary solutions of the quantum YangBaxter equation, Adv. Math. 193 (2005), 4055. MR 2132760 (2005k:81132)
 17.
 W. Rump, Braces, radical rings, and the quantum YangBaxter equation, J. Algebra 307 (2007), 153170. MR 2278047 (2007m:16065)
 18.
 C.N. Yang, Some exact results for the manybody problem in one dimension with repulsive deltafunction interaction, Phys. Rev. Lett. 19 (1967), 13121315. MR 0261870 (41:6480)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
81R50,
20F29,
20B35,
20F16
Retrieve articles in all journals
with MSC (2010):
81R50,
20F29,
20B35,
20F16
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
PII:
S 00029947(09)049277
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
