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ZERO-DIVISORS AND IDEMPOTENTS IN QUANDLE RINGS

Bardakov, Valeriy G. 大阪大学 DOI:10.18910/88489

2022.07

概要

The paper develops the theory of quandle rings introduced by the authors in a recent work. Orderability of quandles is investigated and many interesting examples of such quandles are given. It is proved that quandle rings of left or right orderable quandles which are semi-latin have no zero-divisors. Idempotents in quandle rings of certain interesting quandles are computed and used to determine sets of maximal quandles in these rings. Understanding of idempotents is further applied to determine automorphism groups of these quandle rings. Also, commutator width of quandle rings is introduced and computed in a few cases.

参考文献

[1] A.A. Albert: On the power-associativity of rings, Summa Brasil. Math. 2 (1948), 21–32.

[2] N. Andruskiewitsch and M. Gran˜a: From racks to pointed Hopf algebras, Adv. Math. 178 (2003), 177–243.

[3] V.G. Bardakov: Computation of commutator length in free groups, Algebra Log. 39 (2000), 395–440.

[4] V.G. Bardakov, P. Dey and M. Singh: Automorphism groups of quandles arising from groups, Monatsh. Math. 184 (2017), 519–530.

[5] V.G. Bardakov, T. Nasybullov and M. Singh: Automorphism groups of quandles and related groups, Monatsh. Math. 189 (2019), 1–21.

[6] V.G. Bardakov, I.B.S. Passi and M. Singh: Quandle rings, J. Algebra Appl. 18 (2019), 1950157, 23 pp.

[7] V.G. Bardakov, M. Singh and M. Singh: Free quandles and knot quandles are residually finite, Proc. Amer. Math. Soc. 147 (2019), 3621–3633.

[8] V.G. Bardakov, M. Singh and M. Singh: Link quandles are residually finite, Monatsh. Math. 191 (2020), 679–690.

[9] M. Bonatto: Principal and doubly homogeneous quandles, Monatsh. Math. 191 (2020) 691–717.

[10] S. Boyer, D. Rolfsen and B. Wiest: Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), 243–288.

[11] J.S. Carter: A survey of quandle ideas; in Introductory lectures on knot theory, Ser. Knots Everything 46, World Sci. Publ., Hackensack, NJ, 2012, 22–53.

[12] J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito: Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), 3947–3989.

[13] B. Deroin, A. Navas and C. Rivas: Groups, Orders, and Dynamics, arXiv:1408.5805v2.

[14] M. Eisermann: Yang-Baxter deformations of quandles and racks, Algebr. Geom. Topol. 5 (2005), 537–562.

[15] M. Elhamdadi, J. Macquarrie and R. Restrepo: Automorphism groups of quandles, J. Algebra Appl. 11(2012), 1250008, 9 pp.

[16] M. Elhamdadi, N. Fernando and B. Tsvelikhovskiy: Ring theoretic aspects of quandles, J. Algebra 526(2019), 166–187.

[17] R. Fenn and C. Rourke: Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), 343–406.

[18] D. Joyce: An algebraic approach to symmetry with applications to knot theory, PhD thesis, University of Pennsylvania, 1979.

[19] D. Joyce: A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37–65.

[20] I. Kaplansky: Fields and Rings, Second edition, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, Ill.-London, 1972.

[21] S. Kamada: Knot invariants derived from quandles and racks; in Invariants of knots and 3-manifolds (Kyoto, 2001), Geom. Topol. Monogr., 4, Geom. Topol. Publ., Coventry, 2002, 103–117 (electronic).

[22] S. Kamada: Kyokumen musubime riron (Surface-knot theory), (in Japanese), Springer Gendai Sugaku Series 16 (2012), Maruzen Publishing Co. Ltd.

[23] O. Loos: Reflexion spaces and homogeneous symmetric spaces, Bull. Amer. Math. Soc. 73 (1967) 250–253.

[24] S.V. Matveev: Distributive groupoids in knot theory, (Russian) Mat. Sb. (N.S.) 119(161) (1982), 78–88, 160.

[25] R.B. Mura and A. Rhemtulla: Orderable Groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, Inc., New York-Basel, 1977.

[26] S. Nelson: The combinatorial revolution in knot theory, Notices Amer. Math. Soc. 58 (2011), 1553–1561.

[27] D.S. Passman: The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley- Interscience, New York-London-Sydney, 1977.

[28] B. Perron and D. Rolfsen: On orderability of fibred knot groups, Math. Proc. Cambridge Philos. Soc. 135(2003), 147–153.

[29] E.N. Poroshenko: Commutator width of the elements in a free metabelian Lie algebra, Algebra Logic 53(2014), 377–396.

[30] V.A. Roman’kov: The commutator width of some relatively free Lie algebras and nilpotent groups, Sib. Math. J. 57 (2016), 679–695

[31] M. Szymik: Quandle cohomology is a Quillen cohomology. Trans. Amer. Math. Soc. 371 (2019), 5823– 5839.

[32] L. Vendramin: Doubly transitive groups and cyclic quandles, J. Math. Soc. Japan. 69 (2017), 1051–1057.

[33] A.A. Vinogradov: On the free product of ordered groups, (Russian) Mat. Sbornik N.S. 25(67) (1949), 163–168.

[34] K.A. Zˇ hevlakov, A.M. Slin’ko, I.P. Sˇestakov and A.I. Sˇirsˇov: Rings That Are Nearly Associative, (in Russian) Nauka, Moscow, 1978.

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