×

Construction of finite braces. (English) Zbl 1458.20025

Summary: Affine structures of a group \(G\) (= braces with adjoint group \(G\)) are characterized equationally without assuming further invertibility conditions. If \(G\) is finite, the construction of affine structures is reduced to affine structures of \(p,q\)-groups. The delicate relationship between finite solvable groups and involutive Yang-Baxter groups is further clarified by showing that much of an affine structure is already inherent in the Sylow system of the group. Semidirect products of braces are modified (shifted) in two ways to handle affine structures of semidirect products of groups. As an application, two of Vendramin’s conjectures on affine structures of \(p,q\)-groups are verified. A further example illustrates what happens beyond semidirect products.

MSC:

20E22 Extensions, wreath products, and other compositions of groups
20F16 Solvable groups, supersolvable groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
53B05 Linear and affine connections
16T25 Yang-Baxter equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Auslander, L.: The structure of complete locally affine manifolds. Topology 3(suppl.1), 131-139 (1964) · Zbl 0136.43102
[2] Auslander, L.: Simply transitive groups of affine motions. Amer. J. Math. 99(4), 809-826 (1977) · Zbl 0357.22006 · doi:10.2307/2373867
[3] Auslander, L., Markus, L.: Holonomy of flat affinely connected manifolds. Ann. of Math. (2) 62, 139-151 (1955) · Zbl 0065.37603 · doi:10.2307/2007104
[4] Bachiller, D.: Classification of braces of order \[p^3\] p3. J. Pure Appl. Algebra 219(8), 3568-3603 (2015) · Zbl 1312.81099 · doi:10.1016/j.jpaa.2014.12.013
[5] Bachiller, D.: Study of the Algebraic Structure of Left Braces and the Yang-Baxter Equation. Ph.D. Thesis. Univ. Autònoma de Barcelona, Barcelona (2016)
[6] Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics, Vol. 608. Springer-Verlag, Berlin-New York (1977) · Zbl 0384.06022
[7] Cedó, F., Jespers, E., del Río, Á.: Involutive Yang-Baxter groups. Trans. Amer. Math. Soc. 362(5), 2541-2558 (2010) · Zbl 1188.81115 · doi:10.1090/S0002-9947-09-04927-7
[8] Caranti, A., Dalla Volta, F., Sala, M.: Abelian regular subgroups of the affine group and radical rings. Publ. Math. Debrecen 69(3), 297-308 (2006) · Zbl 1123.20002
[9] Catino, F., Rizzo, R.: Regular subgroups of the affine group and radical circle algebras. Bull. Aust. Math. Soc. 79(1), 103-107 (2009) · Zbl 1184.20001 · doi:10.1017/S0004972708001068
[10] Charlap, L.S.: Bieberbach Groups and Flat Manifolds. Universitext. Springer-Verlag, New York (1986) · Zbl 0608.53001 · doi:10.1007/978-1-4613-8687-2
[11] Cohn, H., Kumar, A.: Metacommutation of Hurwitz primes. Proc. Amer. Math. Soc. 143(4), 1459-1469 (2015) · Zbl 1309.11080 · doi:10.1090/S0002-9939-2014-12358-6
[12] Conway, J.H., Smith, D.A.: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A K Peters, Ltd., Natick, MA (2003) · Zbl 1098.17001 · doi:10.1201/9781439864180
[13] Dehornoy, P.: Groupes de Garside. Ann. Sci. École Norm. Sup. (4) 35(2), 267-306 (2002) · Zbl 1017.20031 · doi:10.1016/S0012-9593(02)01090-X
[14] Dehornoy, P., Paris, L.: Gaussian groups and Garside groups, two generalisations of Artin groups. Proc. London Math. Soc. (3) 79(3), 569-604 (1999) · Zbl 1030.20021 · doi:10.1112/S0024611599012071
[15] Drinfel’d, V.G.: On some unsolved problems in quantum group theory. In: Kulish, P.P. (ed.) Quantum Groups (Leningrad, 1990), pp. 1-8. Lecture Notes in Math., Vol. 1510. Springer-Verlag, Berlin (1992) · Zbl 0765.17014
[16] Etingof, P., Schedler, T., Soloviev, A.: Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Math. J. 100(2), 169-209 (1999) · Zbl 0969.81030 · doi:10.1215/S0012-7094-99-10007-X
[17] Forsyth, A., Gurev, J., Shrima, S.: Metacommutation as a group action on the projective line over \[{\mathbb{F}}_p\] Fp. Proc. Amer. Math. Soc. 144(11), 4583-4590 (2016) · Zbl 1351.11076 · doi:10.1090/proc/13126
[18] Gateva-Ivanova, T.: Noetherian properties of skew polynomial rings with binomial relations. Trans. Amer. Math. Soc. 343(1), 203-219 (1994) · Zbl 0807.16026 · doi:10.1090/S0002-9947-1994-1173854-3
[19] Gateva-Ivanova, T.: Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity. Adv. Math. 230(4-6), 2152-2175 (2012) · Zbl 1267.81209 · doi:10.1016/j.aim.2012.04.016
[20] Gateva-Ivanova, T., Van den Bergh, M.: Semigroups of I-type. J. Algebra 206(1), 97-112 (1998) · Zbl 0944.20049 · doi:10.1006/jabr.1997.7399
[21] Guarnieri, L., Vendramin, L.: Skew braces and the Yang-Baxter equation. Math. Comp. 86(307), 2519-2534 (2017) · Zbl 1371.16037 · doi:10.1090/mcom/3161
[22] Hall, P.: On the Sylow systems of a soluble group. Proc. London Math. Soc. (2) 43(5), 316-323 (1937) · JFM 63.0069.04
[23] Hall, P.: A characteristic property of soluble groups. J. London Math. Soc. 12(3), 198-200 (1937) · Zbl 0016.39204 · doi:10.1112/jlms/s1-12.2.198
[24] Jespers, E., Okniński, J.: Monoids and groups of I-type. Algebr. Represent. Theory 8(5), 709-729 (2005) · Zbl 1091.20024 · doi:10.1007/s10468-005-0342-7
[25] Kim, H.: Complete left-invariant affine structures on nilpotent Lie groups. J. Differential Geom. 24(3), 373-394 (1986) · Zbl 0591.53045 · doi:10.4310/jdg/1214440553
[26] Milnor, J.: On fundamental groups of complete affinely flat manifolds. Adv. Math. 25(2), 178-187 (1977) · Zbl 0364.55001 · doi:10.1016/0001-8708(77)90004-4
[27] Picantin, M.: The center of thin Gaussian groups. J. Algebra 245(1), 92-122 (2001) · Zbl 1002.20022 · doi:10.1006/jabr.2001.8894
[28] Rump, W.: A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation. Adv. Math. 193(1), 40-55 (2005) · Zbl 1074.81036 · doi:10.1016/j.aim.2004.03.019
[29] Rump, W.: Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307(1), 153-170 (2007) · Zbl 1115.16022 · doi:10.1016/j.jalgebra.2006.03.040
[30] Rump, W.: Classification of cyclic braces. J. Pure Appl. Algebra 209(3), 671-685 (2007) · Zbl 1170.16031 · doi:10.1016/j.jpaa.2006.07.001
[31] Rump, \[W.: L\] L-algebras, self-similarity, and \[l\] l-groups. J. Algebra 320(6), 2328-2348 (2008) · Zbl 1158.06009 · doi:10.1016/j.jalgebra.2008.05.033
[32] Rump, W.: Semidirect products in algebraic logic and solutions of the quantum Yang-Baxter equation. J. Algebra Appl. 7(4), 471-490 (2008) · Zbl 1153.81505 · doi:10.1142/S0219498808002904
[33] Rump, W.: The brace of a classical group. Note Mat. 34(1), 115-144 (2014) · Zbl 1344.14029
[34] Rump, W.: Decomposition of Garside groups and self-similar \[L\] L-algebras. J. Algebra 485, 118-141 (2017) · Zbl 1434.20025 · doi:10.1016/j.jalgebra.2017.04.023
[35] Segal, D.: Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff-Witt theorem. J. Algebra 164(3), 750-772 (1994) · Zbl 0831.17001 · doi:10.1006/jabr.1994.1088
[36] Tate, J., Van den Bergh, M.: Homological properties of Sklyanin algebras. Invent. Math. 124(1-3), 619-647 (1996) · Zbl 0876.17010 · doi:10.1007/s002220050065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.