Cofinite integral braces and flat manifolds. (English) Zbl 07841520
Summary: Braces are ring-like structures equivalent to groups \(G\) with a \(G\)-module structure such that the identity map is a 1-cocycle. By comparison, crystallographic groups give rise to 1-cocycles into the Euclidean space on which they act. Associated with every torsion-free crystallographic group is a closed connected flat Riemannian manifold. The Calabi construction for these manifolds is reinterpreted and extended to cofinite integral braces, which can be conceived as analogues and refinements of crystallographic groups. It is shown that all three-dimensional Bieberbach groups and most of the two-dimensional crystallographic groups are adjoint groups of cofinite integral braces. For arbitrary cofinite (e.g., finite) braces, the transfer map from the adjoint group into the socle is shown to be a brace morphism, which leads to surprising connections between group-theoretic and brace-theoretic invariants.
MSC:
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 20H15 | Other geometric groups, including crystallographic groups |
| 57N45 | Flatness and tameness of topological manifolds |
Software:
CARATReferences:
| [1] | Acri, E.; Lutowski, R.; Vendramin, L., Retractability of solutions to the Yang-Baxter equation and p-nilpotency of skew braces, Int. J. Algebra Comput., 30, 91-115, 2020 · Zbl 1436.16044 |
| [2] | Akgün, Ö.; Mereb, M.; Vendramin, L., Enumeration of set-theoretic solutions to the Yang-Baxter equation, Math. Comput., 91, 1469-1481, 2022 · Zbl 1504.16058 |
| [3] | Angiono, I.; Galindo, C.; Vendramin, L., Hopf braces and Yang-Baxter operators, Proc. Am. Math. Soc., 145, 5, 1981-1995, 2017 · Zbl 1392.16032 |
| [4] | Auslander, L., Examples of locally affine spaces, Ann. Math., 64, 255-259, 1956 · Zbl 0072.39703 |
| [5] | Auslander, L.; Kuranishi, M., On the holonomy group of locally Euclidean spaces, Ann. Math., 65, 411-415, 1957 · Zbl 0079.38304 |
| [6] | Auslander, L.; Markus, L., Holonomy of flat affinely connected manifolds, Ann. Math., 62, 139-151, 1955 · Zbl 0065.37603 |
| [7] | Bachiller, D., Counterexample to a conjecture about braces, J. Algebra, 453, 160-176, 2016 · Zbl 1338.16022 |
| [8] | Bachiller, D., Classification of braces of order \(p^3\), J. Pure Appl. Algebra, 219, 8, 3568-3603, 2015 · Zbl 1312.81099 |
| [9] | Benoist, Y., Une nilvariété non affine, J. Differ. Geom., 41, 21-52, 1995 · Zbl 0828.22023 |
| [10] | Bieberbach, L., Über die bewegungsgruppen der euklidischen Räume, Math. Ann., 70, 297-336, 1911 · JFM 42.0144.02 |
| [11] | Bieberbach, L., Über die bewegungsgruppen der euklidischen Räume, Math. Ann., 72, 400-412, 1912 · JFM 43.0186.01 |
| [12] | Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H., Crystallographic Groups of Four-Dimensional Space, 1978, J. Wiley and Sons: J. Wiley and Sons New York · Zbl 0381.20002 |
| [13] | Calabi, E., Closed, locally Euclidean, 4-dimensional manifolds, Bull. Am. Math. Soc., 63, 1957, Abstract 295, 135 |
| [14] | Caranti, A.; Dalla Volta, F.; Sala, M., Abelian regular subgroups of the affine group and radical rings, Publ. Math. (Debr.), 69, 297-308, 2006 · Zbl 1123.20002 |
| [15] | Castelli, M.; Catino, F.; Pinto, G., Indecomposable involutive set-theoretic solutions of the Yang-Baxter equation, J. Pure Appl. Algebra, 223, 4477-4493, 2019 · Zbl 1412.16023 |
| [16] | Catino, F.; Colazzo, I.; Stefanelli, P., On regular subgroups of the affine group, Bull. Aust. Math. Soc., 91, 1, 76-85, 2015 · Zbl 1314.20001 |
| [17] | 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 |
| [18] | Cedó, F.; Jespers, E.; del Río, Á., Involutive Yang-Baxter groups, Trans. Am. Math. Soc., 362, 5, 2541-2558, 2010 · Zbl 1188.81115 |
| [19] | Charlap, L. S., Bieberbach Groups and Flat Manifolds, 1966, Springer-Verlag: Springer-Verlag N.Y. |
| [20] | Childs, L. N., Fixed-point free endomorphisms and Hopf Galois structures, Proc. Am. Math. Soc., 141, 4, 1255-1265, 2013 · Zbl 1269.12003 |
| [21] | Cid, C.; Schulz, T., Computation of five- and six-dimensional Bieberbach groups, Exp. Math., 10, 109-115, 2001 · Zbl 0980.20043 |
| [22] | Coxeter, H. S.M.; Moser, W. O.J., Generators and Relations for Discrete Groups, 1957, Springer Verlag: Springer Verlag Heidelberg · Zbl 0077.02801 |
| [23] | Drinfeld, V. G., On some unsolved problems in quantum group theory, (Quantum Groups. Quantum Groups, Leningrad, 1990. Quantum Groups. Quantum Groups, Leningrad, 1990, LNM, vol. 1510, 1992, Springer-Verlag: Springer-Verlag Berlin), 1-8 · Zbl 0765.17014 |
| [24] | 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 |
| [25] | Featherstonhaugh, S. C.; Caranti, A.; Childs, L. N., Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Am. Math. Soc., 364, 7, 3675-3684, 2012 · Zbl 1287.12002 |
| [26] | Fedoroff, E. S., Reguläre Plan- und Raumtheilung, Abh. Bayer. Akad. Wiss. II. Kl., 20, 465-588, 1900 · JFM 31.0479.01 |
| [27] | Gateva-Ivanova, T.; Cameron, P., Multipermutation solutions of the Yang-Baxter equation, Commun. Math. Phys., 309, 583-621, 2012 · Zbl 1247.81211 |
| [28] | Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of I-type, J. Algebra, 206, 97-112, 1998 · Zbl 0944.20049 |
| [29] | Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comput., 86, 2519-2534, 2017 · Zbl 1371.16037 |
| [30] | Hall, M., The Theory of Groups, 1959, The Macmillan Company: The Macmillan Company New York · Zbl 0084.02202 |
| [31] | Hantzsche, W.; Wendt, H., Dreidimensionale euklidische Raumformen, Math. Ann., 110, 593-611, 1935 · JFM 60.0517.02 |
| [32] | Henry, N. F.M.; Lonsdale, K., International Tables for X-Ray Crystallography 1, 1952, Kynoch Press: Kynoch Press Birmingham, England |
| [33] | Hertweck, M., A counterexample to the isomorphism problem for integral group rings, Ann. Math., 154, 115-138, 2001 · Zbl 0990.20002 |
| [34] | Hillman, J. A., Flat 4-manifold groups, N.Z. J. Math., 24, 29-40, 1995 · Zbl 0870.57024 |
| [35] | Lutowski, R.; Szczepański, A., Minimal non-solvable Bieberbach groups |
| [36] | Promislow, S. D., A simple example of a torsion-free, nonunique product group, Bull. Lond. Math. Soc., 20, 302-304, 1988 · Zbl 0662.20022 |
| [37] | Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, GTM, vol. 149, 2006, Springer-Verlag: Springer-Verlag New York · Zbl 1106.51009 |
| [38] | Ratcliffe, J. G.; Tschantz, S. T., Fibered orbifolds and crystallographic groups, Algebraic Geom. Topol., 10, 1627-1664, 2010 · Zbl 1245.57026 |
| [39] | Ratcliffe, J. G.; Tschantz, S. T., The Calabi construction for compact flat orbifolds, Topol. Appl., 178, 87-106, 2024 · Zbl 1327.57026 |
| [40] | Reiner, I., Maximal Orders, 2003, Clarendon Press: Clarendon Press Oxford · Zbl 1024.16008 |
| [41] | Rump, W., A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math., 193, 40-55, 2005 · Zbl 1074.81036 |
| [42] | Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 153-170, 2007 · Zbl 1115.16022 |
| [43] | Rump, W., Generalized radical rings, unknotted biquandles, and quantum groups, Colloq. Math., 109, 85-100, 2007 · Zbl 1117.81084 |
| [44] | Rump, W., Classification of cyclic braces, J. Pure Appl. Algebra, 209, 671-685, 2007 · Zbl 1170.16031 |
| [45] | Rump, W., The brace of a classical group, Note Mat., 34, 1, 115-144, 2014 · Zbl 1344.14029 |
| [46] | Rump, W., Classification of non-degenerate involutive set-theoretic solutions to the Yang-Baxter equation with multipermutation level two, Algebr. Represent. Theory, 25, 1293-1307, 2022 · Zbl 1513.81087 |
| [47] | Rump, W., Construction of finite braces, Ann. Comb., 23, 391-416, 2019 · Zbl 1458.20025 |
| [48] | du Sautoy, M. P.F.; McDermott, J. J.; Smith, G. C., Zeta functions of crystallographic groups and analytic continuation, Proc. Lond. Math. Soc., 79, 511-534, 1999 · Zbl 1039.11056 |
| [49] | Schattschneider, D., The plane symmetry groups: their recognition and notation, Am. Math. Mon., 85, 439-450, 1978 · Zbl 0381.20036 |
| [50] | Serre, J.-P., Local Fields, 1979, Springer Verlag: Springer Verlag New York · Zbl 0423.12016 |
| [51] | Thurston, W. P., The Geometry and Topology of Three-Manifolds, Lecture Notes, 1980, Princeton, Electronic version 1.1 - March 2002 |
| [52] | Vasquez, A. T., Flat Riemannian manifolds, J. Differ. Geom., 4, 367-382, 1970 · Zbl 0209.25402 |
| [53] | Watters, J. F., On the adjoint group of a radical ring, J. Lond. Math. Soc., 43, 725-729, 1968 · Zbl 0162.04801 |
| [54] | Weinstein, A.; Xu, P., Classical solutions of the quantum Yang-Baxter equation, Commun. Math. Phys., 148, 2, 309-343, 1992 · Zbl 0849.17015 |
| [55] | Wolf, J. A., Spaces of Constant Curvature, 2011, AMS Chelsea Publ.: AMS Chelsea Publ. Providence, R.I. · Zbl 1216.53003 |
| [56] | Yau, S.-T., Compact flat Riemannian manifolds, J. Differ. Geom., 6, 395-402, 1972 · Zbl 0238.53025 |
| [57] | Zimmermann, B., On the Hantzsche-Wendt manifold, Monatshefte Math., 110, 321-327, 1990 · Zbl 0717.57005 |
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.
