Counterexample to a conjecture about braces. (English) Zbl 1338.16022
The paper provides a counterexample for a conjecture which arose in connection with braces, a ring-like structure generalizing radical rings. The adjoint group of a brace is connected with the additive group by a 1-cocycle, and conversely, every bijective 1-cocycle comes from a brace. For a finite brace, the adjoint group is solvable. So the question arose whether conversely, every finite solvable group is the adjoint group of a brace. For \(\mathbb R\)-linear braces, a similar question is Milnor’s 1977 conjecture, equivalent to the assertion that every simply connected solvable Lie group is isomorphic to the adjoint group of an \(\mathbb R\)-brace [J. W. Milnor, Adv. Math. 25, 178-187 (1977; Zbl 0364.55001)]. Milnor proved that the Lie groups in question are exactly the connected Lie groups which admit a free action of the affine group on a finite space. Such groups are solvable. However, the converse was believed to be true until Y. Benoist constructed a counterexample [J. Differ. Geom. 41, No. 1, 21-52 (1995; Zbl 0828.22023)] by means of a filiform Lie algebra of dimension 11, which led to a nilvariety without an affine structure. The paper extends this to the discrete case where some battle with the characteristic has to be fought.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 16N20 | Jacobson radical, quasimultiplication |
| 16T25 | Yang-Baxter equations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
braces; IYB groups; bijective 1-cocycles; radical rings; Hopf-Galois extensions; nilpotent groups; Lie algebras; LSA structures; adjoint groupsReferences:
| [1] | Auslander, L., Simply transitive groups of affine motions, Amer. J. Math., 99, 809-826 (1977) · Zbl 0357.22006 |
| [2] | Bachiller, D.; Cedó, F.; Jespers, E., Solutions of the Yang-Baxter equation associated with a left brace (10 Mar 2015) |
| [3] | Ben David, N.; Ginosar, Y., On groups of I-type and involutive Yang-Baxter groups (24 Apr 2014) |
| [4] | Benoist, Y., Une nilvariété non affine, J. Differential Geom., 41, 21-52 (1995) · Zbl 0828.22023 |
| [5] | Burde, D., Affine structures on nilmanifolds, Int. J. Math., 7, 5, 599-616 (1996) · Zbl 0868.57034 |
| [6] | Byott, N. P., Uniqueness of Hopf structures of separable field extensions, Comm. Algebra, 24, 3217-3228 (1996), 3705 · Zbl 0878.12001 |
| [7] | Catino, F.; Rizzo, R., Regular subgroups of the affine group and radical circle algebras, Bull. Aust. Math. Soc., 79, 103-107 (2009) · Zbl 1184.20001 |
| [8] | Cedó, F.; Jespers, E.; del Río, Á., Involutive Yang-Baxter groups, Trans. Amer. Math. Soc., 362, 2541-2558 (2010) · Zbl 1188.81115 |
| [9] | Cedó, F.; Jespers, E.; Okniński, J., Braces and the Yang-Baxter equation, Comm. Math. Phys., 327, 101-116 (2014) · Zbl 1287.81062 |
| [10] | Cedó, F.; Jespers, E.; Okniński, J., Nilpotent groups of class three and braces, Publ. Mat., 60, 55-79 (2016) · Zbl 1345.16039 |
| [11] | Corwin, L. J.; Greenleaf, F. P., Representations of Nilpotent Lie Groups and Their Applications. Part I, Cambridge Stud. Adv. Math., vol. 18 (1990), Cambridge University Press: Cambridge University Press Cambridge, viii+269 · Zbl 0704.22007 |
| [12] | Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 169-209 (1999) · Zbl 0969.81030 |
| [13] | Featherstonhaugh, S. C., Abelian Hopf Galois structures on Galois field extensions of prime power order (2013), University of Albany: University of Albany State University of New York, USA, Ph.D. thesis · Zbl 1287.12002 |
| [14] | Featherstonhaugh, S. C.; Caranti, A.; Childs, L. N., Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc., 364, 7, 3675-3684 (July 2012) · Zbl 1287.12002 |
| [15] | Eisele, F., On the IYB-property in some solvable groups, Arch. Math., 101, 309-318 (2013) · Zbl 1288.20004 |
| [16] | Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of \(I\)-type, J. Algebra, 206, 97-112 (1998) · Zbl 0944.20049 |
| [17] | Khukhro, E. I., \(p\)-Automorphisms of Finite \(p\)-Groups, London Math. Soc. Lecture Note Ser., vol. 246 (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0897.20018 |
| [18] | Milnor, J., On fundamental groups of complete affinely flat manifolds, Adv. Math., 25, 178-187 (1977) · Zbl 0364.55001 |
| [19] | Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 153-170 (2007) · Zbl 1115.16022 |
| [20] | Rump, W., The brace of a classical group, Note Mat., 34, 1, 115-144 (2014) · Zbl 1344.14029 |
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.
