Advanced SearchSearch Tips
Ternary Distributive Structures and Quandles
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Kyungpook mathematical journal
  • Volume 56, Issue 1,  2016, pp.1-27
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2016.56.1.1
 Title & Authors
Ternary Distributive Structures and Quandles
Elhamdadi, Mohamed; Green, Matthew; Makhlouf, Abdenacer;
  PDF(new window)
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.
 Cited by
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. crossref(new window)

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, crossref(new window)

Ataguema H., Makhlouf A., Deformations of ternary algebras, Journal of Generalized Lie Theory and Applications, 1(2007), 41-45. crossref(new window)

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.

Ataguema H., Makhlouf A., Silvestrov S., Generalization of n-ary Nambu algebras and beyond, J. Math. Phys., 50(8)(2009), 083501, 15 pp. crossref(new window)

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.

Biyogmam G. R., Lie central triple racks, Int. Electron. J. Algebra, 17(2015), 58-65. crossref(new window)

Biyogmam G. R., A study of n-subracks, Quasigroups Related Systems, 21(1)(2013), 19-28.

Biyogmam G. R., Lie n-racks, C. R. Math. Acad. Sci. Paris, 349(17-18)(2011), 957-960. crossref(new window)

Carlsson R., Cohomology of associative triple systems, Proc. Amer. Math. Soc., 60(1976), 1-7. crossref(new window)

Carter S., Crans A., Elhamdadi M., and Saito M., Cohomology of categorical selfdistributivity, J. Homotopy Relat. Struct., 3(1),(2008), 13-63.

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

de Azca rraga J. A., Izquierdo J. M., n-ary algebras: a review with applications, J. Phys., A43(2010), 293001-1-117. crossref(new window)

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.

Elhamdadi M., MacQuarrie J. and Restrepo R., Automorphism groups of quandles, J. Algebra Appl., 11(1)(2012), 1250008 (9 pages). crossref(new window)

Fenn R., and Rourke C., Racks and links in codimension two, J. Knot Theory Ramifications, 1(1992), 343-406. crossref(new window)

Filippov V. T., n-Lie algebras, Siberian Math. J., 26(1985), 879-891.

Gerstenhaber M., On the deformation of rings and algebras, Ann. of Math., (2) 79(1964), 59-103. crossref(new window)

Goze, M. and Rausch de Traubenberg, M., Hopf algebras for ternary algebras. J. Math. Phys., 50(6)(2009), 1089-7658.

Harris B., Cohomology of Lie triple systems and Lie algebras with involution, Trans. Amer. Math. Soc., 98(1961), 148-162. crossref(new window)

Hestenes M. R., A ternary algebra with applications to matrices and linear transformations, Arch. Rational Mech. Anal., 11(1962), 138-194. crossref(new window)

Jackobson, N., Lie and Jordan triple systems, Amer. J. Math., 71,(1949), 149-170. crossref(new window)

Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23(1982), 37-65. crossref(new window)

Lister, W. G., Ternary rings, Trans. Amer. Math. Soc., 154(1971), 37-55. crossref(new window)

Loos, O., Assoziative tripelsysteme, Manuscripta Math., 7(1972), 103-112. crossref(new window)

Matveev, S., Distributive groupoids in knot theory, (Russian) Mat. Sb. (N.S.), 119(1)(1982), 78-88.

Niebrzydowski, M., On some ternary operations in knot theory, Fund. Math., 225(2014), 259-276. crossref(new window)

Nijenhuis A., Richardson R. W. Jr., Deformations of Lie algebra structures, J. Math. Mech., 17(1967), 89-105.

Okubo S., Triple products and Yang-Baxter equation. II. Orthogonal and symplectic ternary systems, J. Math. Phys., 34(7)(1993), 3292-3315. crossref(new window)

Okubo S., Triple products and Yang-Baxter equation. I. Octonionic and quaternionic triple systems, J. Math. Phys., 34(7)(1993), 3273-3291. crossref(new window)

Przytycki J., Distributivity versus Associativity in the Homology Theory of Algebraic structures, Demonstratio Mathematica, Vol. XLIV No 4 (2011).

Seibt P., Cohomology of algebras and triple systems, Comm. Algebra, 3(12)(1975), 1097-1120. crossref(new window)

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.

Yamaguti K., On representations of Jordan triple systems, Kumamoto J. Sci. Ser. A, 5(1962), 171-184.

Zekovic, B., Ternary Hopf algebras, Algebra Discrete Math., 3(2005), 96-106.