×

Bijective 1-cocycles, braces, and non-commutative prime factorization. (English) Zbl 1508.16043

The algebraic structure of the brace, a generalisation of the classical Jacobson radical rings, has been introduced by the author in [J. Algebra 307, No. 1, 153–170 (2007; Zbl 1115.16022)]. One of the motivations for studying braces is their interplay with non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation, a fundamental equation of mathematical physics (see [V. G. Drinfel’d, Lect. Notes Math. 1510, 1–8 (1992; Zbl 0765.17014)]).
In the paper under review, the author focuses on quasirings, namely additive abelian groups \(\left(A,+\right)\) endowed with a multiplication (written as juxtaposition) satisfying the following identities \begin{align*} 0a &=0\\ a\left(b + c\right) &= ab + ac\\ \left(ab + a + b\right)c &= a\left(bc\right) + ac + bc, \end{align*} for all \(a,b,c\in A\). By defining the operation \(\circ\) given by \(a\circ b:= ab + a + b\), one has that \(\left(A, \circ \right)\) is a monoid; if \(\left(A, \circ \right)\) is a group, then \(A\) is a brace. Consistently with braces, quasirings are equivalent to bijective \(1\)-cocycles \(M\to A\) from a monoid \(M\) onto an \(M\)-module \(A\).
The author shows that a specific class of lattice-ordered quasirings characterises the divisor groups of non-commutative smooth algebraic curves. Moreover, the adjoint monoid structure extends the multiplication of fractional ideals of a hereditary noetherian ring to the set of all divisors. Besides, he provides a description of the multiplication of divisors as an extension of the functional representation of fractional ideals given in [the author and Y. Yang, J. Algebra 468, 214–252 (2016; Zbl 1400.20058)].

MSC:

16T25 Yang-Baxter equations
14A22 Noncommutative algebraic geometry
06F05 Ordered semigroups and monoids
20M30 Representation of semigroups; actions of semigroups on sets
11M55 Relations with noncommutative geometry
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
20F36 Braid groups; Artin groups
05E18 Group actions on combinatorial structures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang-Baxter operators, Proc. Amer. Math. Soc. 145 (2017), 1981-1995. · Zbl 1392.16032
[2] D. Bachiller, Counterexample to a conjecture about braces, J. Algebra 453 (2016), 160-176. · Zbl 1338.16022
[3] D. Bachiller, F. Cedó, E. Jespers and J. Okniński, Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation, Trans. Amer. Math. Soc. 370 (2018), 4881-4907. · Zbl 1431.16035
[4] A. Bigard, K. Keimel et S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math. 608, Springer, Berlin, 1977. · Zbl 0384.06022
[5] G. Birkhoff, Lattice-ordered groups, Ann. of Math. 43 (1942), 298-331. · Zbl 0060.05808
[6] S. Boyer, D. Rolfsen and B. Wiest, Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), 243-288. · Zbl 1068.57001
[7] E. Brieskorn und K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245-271. · Zbl 0243.20037
[8] A. Brumer, Structure of hereditary orders, Bull. Amer. Math. Soc. 69 (1963), 721-724. · Zbl 0113.26002
[9] A. Brumer, Addendum to “Structure of hereditary orders”, Bull. Amer. Math. Soc. 70 (1964), 185. · Zbl 0113.26002
[10] J. S. Carter, M. Elhamdadi and M. Saito, Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles, Fund. Math. 184 (2004), 31-54. · Zbl 1067.57006
[11] F. Catino, I. Colazzo and P. Stefanelli, On regular subgroups of the affine group, Bull. Austral. Math. Soc. 91 (2015), 76-85. · Zbl 1314.20001
[12] F. Catino and R. Rizzo, Regular subgroups of the affine group and radical circle algebras, Bull. Austral. Math. Soc. 79 (2009), 103-107. · Zbl 1184.20001
[13] F. Cedó, E. Jespers and J. Okniński, Braces and the Yang-Baxter equation, Comm. Math. Phys. 327 (2014), 101-116. · Zbl 1287.81062
[14] F. Cedó, E. Jespers and Á. del Río, Involutive Yang-Baxter groups, Trans. Amer. Math. Soc. 362 (2010), 2541-2558. · Zbl 1188.81115
[15] L. N. Childs, Fixed-point free endomorphisms and Hopf Galois structures, Proc. Amer. Math. Soc. 141 (2013), 1255-1265. · Zbl 1269.12003
[16] F. Chouraqui, Garside groups and Yang-Baxter equation, Comm. Algebra 38 (2010), 4441-4460. · Zbl 1216.16023
[17] P. Conrad, Right-ordered groups, Michigan Math. J. 6 (1959), 267-275. · Zbl 0099.01703
[18] M. R. Darnel, Theory of Lattice-Ordered Groups, Monogr. Textbooks Pure Appl. Math. 187, Dekker, New York, 1995. · Zbl 0810.06016
[19] P. Dehornoy, Groupes de Garside, Ann. Sci. École Norm. Sup. (4) 35 (2002), 267-306. · Zbl 1017.20031
[20] P. Dehornoy, Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs, Adv. Math. 282 (2015), 93-127. · Zbl 1326.20039
[21] P. Dehornoy, F. Digne, E. Godelle, D. Krammer and J. Michel, Foundations of Gar-side Theory, EMS Tracts in Math. 22, Eur. Math. Soc., Zürich, 2015. · Zbl 1370.20001
[22] P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. (3) 79 (1999), 569-604. · Zbl 1030.20021
[23] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302. · Zbl 0238.20034
[24] V. G. Drinfeld, On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, Berlin, 1992, 1-8. · Zbl 0765.17014
[25] D. Eisenbud and J. C. Robson: Hereditary Noetherian prime rings, J. Algebra 16 (1970), 86-104. · Zbl 0211.05701
[26] P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209. · Zbl 0969.81030
[27] M. A. Farinati and J. García Galofre, A differential bialgebra associated to a set theoretical solution of the Yang-Baxter equation, J. Pure Appl. Algebra 220 (2016), 3454-3475. · Zbl 1347.16036
[28] S. C. Featherstonhaugh, A. Caranti and L. N. Childs, Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364 (2012), 3675-3684. · Zbl 1287.12002
[29] F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235-254. · Zbl 0194.03303
[30] T. Gateva-Ivanova and P. Cameron, Multipermutation solutions of the Yang-Baxter equation, Comm. Math. Phys. 309 (2012), 583-621. · Zbl 1247.81211
[31] T. Gateva-Ivanova and M. Van den Bergh, Semigroups of I-type, J. Algebra 206 (1998), 97-112. · Zbl 0944.20049
[32] L. Guarnieri and L. Vendramin, Skew braces and the Yang-Baxter equation, Math. Comp. 86 (2017), 2519-2534. · Zbl 1371.16037
[33] M. Harada, Hereditary orders, Trans. Amer. Math. Soc. 107 (1963), 273-290. · Zbl 0113.26001
[34] M. Harada, Structure of hereditary orders over local rings, J. Math. Osaka City Univ. 14 (1963) 1-22. · Zbl 0152.02003
[35] M. Harada, Multiplicative ideal theory in hereditary orders, J. Math. Osaka City Univ. 14 (1963), 83-106. · Zbl 0152.02005
[36] H. Jacobinski, Two remarks about hereditary orders, Proc. Amer. Math. Soc. 28 (1971), 1-8. · Zbl 0216.06501
[37] E. Jespers and J. Okniński, Monoids and groups of I-type, Algebras Represent. Theory 8 (2005), 709-729. · Zbl 1091.20024
[38] R. P. Kent IV and D. Peifer, A geometric and algebraic description of annular braid groups, Int. J. Algebra Comput. 12 (2002), 85-97. · Zbl 1010.20024
[39] D. Kussin, Weighted noncommutative regular projective curves, J. Noncommut. Geom. 10 (2016), 1465-1540. · Zbl 1357.14007
[40] V. Lebed and L. Vendramin, Cohomology and extensions of braces, Pacific J. Math. 284 (2016), 191-212. · Zbl 1357.20009
[41] V. Lebed and L. Vendramin, Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. Math. 304 (2017), 1219-1261. · Zbl 1356.16027
[42] H. Lenzing and I. Reiten, Hereditary Noetherian categories of positive Euler charac-teristic, Math. Z. 254 (2006), 133-171. · Zbl 1105.18010
[43] M. W. Liebeck, C. E. Praeger and J. Saxl, Transitive subgroups of primitive permu-tation groups, J. Algebra 234 (2000), 291-361. · Zbl 0972.20001
[44] J.-H. Lu, M. Yan and Y.-C. Zhu, On the set-theoretical Yang-Baxter equation, Duke Math. J. 104 (2000), 1-18. · Zbl 0960.16043
[45] G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623-653. · Zbl 0526.22015
[46] J. McCammond, Dual euclidean Artin groups and the failure of the lattice property, J. Algebra 437 (2015), 308-343. · Zbl 1343.20039
[47] J. McCammond and R. Sulway, Artin groups of Euclidean type, Invent. Math. 210 (2017), 231-282. · Zbl 1423.20032
[48] H. Meng, A. Ballester-Bolinches and R. Esteban-Romero, Left braces and the quantum Yang-Baxter equation, Proc. Edinburgh Math. Soc. 62 (2019), 595-608. · Zbl 1471.17030
[49] I. Reiner, Maximal Orders, corrected reprint of the 1975 original, London Math. Soc. Monogr. (N.S.) 28, Clarendon Press, Oxford, 2003. · Zbl 1024.16008
[50] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295-366. · Zbl 0991.18009
[51] C. Rourke and B. Wiest, Order automatic mapping class groups, Pacific J. Math. 194 (2000), 209-227. · Zbl 1016.57015
[52] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005), 40-55. · Zbl 1074.81036
[53] W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), 153-170. · Zbl 1115.16022
[54] W. Rump, Classification of cyclic braces, II, Trans. Amer. Math. Soc. 372 (2019), 305-328. · Zbl 1417.81140
[55] W. Rump, Generalized radical rings, unknotted biquandles, and quantum groups, Col-loq. Math. 109 (2007), 85-100. · Zbl 1117.81084
[56] W. Rump, The brace of a classical group, Note Mat. 34 (2014), no. 1, 115-144. · Zbl 1344.14029
[57] W. Rump, Right -groups, geometric Garside groups, and solutions of the quantum Yang-Baxter equation, J. Algebra 439 (2015), 470-510. · Zbl 1360.20030
[58] W. Rump, Decomposition of Garside groups and self-similar L-algebras, J. Algebra 485 (2017), 118-141. · Zbl 1434.20025
[59] W. Rump, Set-theoretic solutions to the Yang-Baxter equation, skew-braces, and re-lated near-rings, J. Algebra Appl. 18 (2019), no. 8, art. 1950145, 22 pp. · Zbl 1429.16024
[60] W. Rump, Classification of the affine structures of a generalized quaternion group of order ≥ 32, J. Group Theory 23 (2020), 847-869. · Zbl 1485.20066
[61] W. Rump and Y. C. Yang, Hereditary arithmetics, J. Algebra 468 (2016), 214-252. · Zbl 1400.20058
[62] J. Y. Shi, The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups, Lecture Notes in Math. 1179, Springer, Berlin, 1986. · Zbl 0582.20030
[63] H. Short and B. Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. 46 (2000), 279-312. · Zbl 1023.57013
[64] A. Smoktunowicz, On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation, Trans. Amer. Math. Soc. 370 (2018), 6535-6564. · Zbl 1440.16040
[65] J. T. Stafford and R. B. Warfield Jr., Constructions of hereditary Noetherian rings and simple rings, Proc. London Math. Soc. 51 (1985), 1-20. · Zbl 0573.16007
[66] B. Stenström, Rings of Quotients, Springer, New York, 1975. · Zbl 0296.16001
[67] J. Tate and M. Van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), 619-647. · Zbl 0876.17010
[68] A. Weinstein and P. Xu, Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. 148 (1992), 309-343. · Zbl 0849.17015
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.