Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Ternary Distributive Structures and Quandles

  • Received : 2015.02.27
  • Accepted : 2015.06.17
  • Published : 2016.03.23
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Abstract

We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from 3-Lie algebras are provided. We also describe ternary distributive algebraic structures coming from groups and give examples from vector spaces whose bases are elements of a finite ternary distributive set. We introduce a cohomology theory that is analogous to Hochschild cohomology and relate it to a formal deformation theory of these structures.

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References

  1. Arnlind J., Makhlouf A., Silvestrov S., Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras, J. Math. Phys., 51(4)(2010), 043515, 11 pp. https://doi.org/10.1063/1.3359004
  2. Ammar F., Mabrouk S., Makhlouf, A., Representations and cohomology of n-ary multiplicative Hom-Nambu-Lie algebras, J. Geom. Phys., 61(10)(2011), 1898-1913, https://doi.org/10.1016/j.geomphys.2011.04.022
  3. Ataguema H., Makhlouf A., Deformations of ternary algebras, Journal of Generalized Lie Theory and Applications, 1(2007), 41-45. https://doi.org/10.4303/jglta/S070104
  4. Ataguema H., Makhlouf A., Notes on cohomologies of ternary algebras of associative type, Journal of Generalized Lie Theory and Applications, 3(3)(2009), 154-174.
  5. Ataguema H., Makhlouf A., Silvestrov S., Generalization of n-ary Nambu algebras and beyond, J. Math. Phys., 50(8)(2009), 083501, 15 pp. https://doi.org/10.1063/1.3167801
  6. Borowiec A., Dudek W. A. and Duplij S., Basis concepts of ternary Hopf algebras, Journal of Kharkov National University, ser. Nuclei, Particles and Fields, 529(15)(2001), 22-29.
  7. Biyogmam G. R., Lie central triple racks, Int. Electron. J. Algebra, 17(2015), 58-65. https://doi.org/10.24330/ieja.266212
  8. Biyogmam G. R., A study of n-subracks, Quasigroups Related Systems, 21(1)(2013), 19-28.
  9. Biyogmam G. R., Lie n-racks, C. R. Math. Acad. Sci. Paris, 349(17-18)(2011), 957-960. https://doi.org/10.1016/j.crma.2011.07.019
  10. Carlsson R., Cohomology of associative triple systems, Proc. Amer. Math. Soc., 60(1976), 1-7. https://doi.org/10.1090/S0002-9939-1976-0430026-X
  11. Carter S., Crans A., Elhamdadi M., and Saito M., Cohomology of categorical selfdistributivity, J. Homotopy Relat. Struct., 3(1),(2008), 13-63.
  12. Carter S., Crans A., Elhamdadi M., and Saito M., Cohomology of the adjoint of Hopf algebras, J. Gen. Lie Theory Appl., 2(1)(2008), 19-34. https://doi.org/10.4303/jglta/S070102
  13. Carter S., Crans A., Elhamdadi M., Karadayi, E., and Saito M., Cohomology of Frobenius algebras and the Yang-Baxter equation, Commun. Contemp. Math., 10(2008), suppl. 1, 791-814. https://doi.org/10.1142/S0219199708003022
  14. Carter J. S., Jelsovsky D., Kamada S., Langford L., Saito M., Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc., 355(2003), 3947-3989. https://doi.org/10.1090/S0002-9947-03-03046-0
  15. de Azca rraga J. A., Izquierdo J. M., n-ary algebras: a review with applications, J. Phys., A43(2010), 293001-1-117. https://doi.org/10.1088/1751-8113/43/29/293001
  16. Duplij, S., Ternary Hopf algebras. In Symmetry in nonlinear mathematical physics, Part 1, 2 (Kyiv, 2001), Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Kiev, 2002, 439-448.
  17. Elhamdadi M., MacQuarrie J. and Restrepo R., Automorphism groups of quandles, J. Algebra Appl., 11(1)(2012), 1250008 (9 pages). https://doi.org/10.1142/S0219498812500089
  18. Fenn R., and Rourke C., Racks and links in codimension two, J. Knot Theory Ramifications, 1(1992), 343-406. https://doi.org/10.1142/S0218216592000203
  19. Filippov V. T., n-Lie algebras, Siberian Math. J., 26(1985), 879-891.
  20. Gerstenhaber M., On the deformation of rings and algebras, Ann. of Math., (2) 79(1964), 59-103. https://doi.org/10.2307/1970484
  21. Goze, M. and Rausch de Traubenberg, M., Hopf algebras for ternary algebras. J. Math. Phys., 50(6)(2009), 1089-7658.
  22. Harris B., Cohomology of Lie triple systems and Lie algebras with involution, Trans. Amer. Math. Soc., 98(1961), 148-162. https://doi.org/10.1090/S0002-9947-1961-0120313-0
  23. Hestenes M. R., A ternary algebra with applications to matrices and linear transformations, Arch. Rational Mech. Anal., 11(1962), 138-194. https://doi.org/10.1007/BF00253936
  24. Jackobson, N., Lie and Jordan triple systems, Amer. J. Math., 71,(1949), 149-170. https://doi.org/10.2307/2372102
  25. Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23(1982), 37-65. https://doi.org/10.1016/0022-4049(82)90077-9
  26. Lister, W. G., Ternary rings, Trans. Amer. Math. Soc., 154(1971), 37-55. https://doi.org/10.1090/S0002-9947-1971-0272835-6
  27. Loos, O., Assoziative tripelsysteme, Manuscripta Math., 7(1972), 103-112. https://doi.org/10.1007/BF01679707
  28. Matveev, S., Distributive groupoids in knot theory, (Russian) Mat. Sb. (N.S.), 119(1)(1982), 78-88.
  29. Niebrzydowski, M., On some ternary operations in knot theory, Fund. Math., 225(2014), 259-276. https://doi.org/10.4064/fm225-1-12
  30. Nijenhuis A., Richardson R. W. Jr., Deformations of Lie algebra structures, J. Math. Mech., 17(1967), 89-105.
  31. Okubo S., Triple products and Yang-Baxter equation. II. Orthogonal and symplectic ternary systems, J. Math. Phys., 34(7)(1993), 3292-3315. https://doi.org/10.1063/1.530077
  32. Okubo S., Triple products and Yang-Baxter equation. I. Octonionic and quaternionic triple systems, J. Math. Phys., 34(7)(1993), 3273-3291. https://doi.org/10.1063/1.530076
  33. Przytycki J., Distributivity versus Associativity in the Homology Theory of Algebraic structures, Demonstratio Mathematica, Vol. XLIV No 4 (2011).
  34. Seibt P., Cohomology of algebras and triple systems, Comm. Algebra, 3(12)(1975), 1097-1120. https://doi.org/10.1080/00927877508822090
  35. Takhtajan L., Higher order analog of Chevalley-Eilenberg complex and deformation theory of n-algebras, Algebra i Analiz 6 (1994(2)) 262-272; translation in St. Petersburg Math. J., 6(2)(1995), 429-438.
  36. Yamaguti K., On representations of Jordan triple systems, Kumamoto J. Sci. Ser. A, 5(1962), 171-184.
  37. Zekovic, B., Ternary Hopf algebras, Algebra Discrete Math., 3(2005), 96-106.