Polyadic Hopf Algebras and Quantum Groups

Keywords: polyadic field, polyadic algebra, bialgebra, Hopf algebra, antipode, braid equation, braiding, R-matrix, Yang-Baxter equation, mediality, co-medaility, M-matrix, quasitriangularity


This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions (“arity freedom principle”). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but “quantized”. The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity,  quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). We propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.


Download data is not yet available.


M.E. Sweedler, Hopf Algebras. (W.A. Benjamin, New York, 1969).

E. Abe, Hopf Algebras, (Cambridge Univ. Press, Cambridge, 1980).

D.E. Radford, Hopf Algebras, (World Scientific, Hackensack, 2012).

V.G. Drinfeld, in Proceedings of the ICM, Berkeley, edited by A. Gleason (AMS, Phode Island, 1987), pp. 798.

S. Shnider, and S. Sternberg, Quantum Groups, (International Press, Boston, 1993).

V. Chari, and A. Pressley, A Guide to Quantum Groups, (Cambridge University Press, Cambridge, 1996).

C. Kassel, Quantum Groups, (Springer-Verlag, New York, 1995).

S. Majid, Foundations of Quantum Group Theory, (Cambridge University Press, Cambridge, 1995).

G. Karaali, Commun. Algebra, 36, 4341 (2008), https://doi.org/10.1080/00927870802182424.

S. Duplij, J. Math. Physics, Analysis, Geometry, 15(1), 3-56 (2019), (to appear), https://arxiv.org/abs/1703.10132.

J.A. de Azcarraga, and J. M. Izquierdo, J. Phys. A, 43, 293001 (2010), https://doi.org/10.1088/1751-8113/43/29/293001.

P.W. Michor, and A.M. Vinogradov, Rend. Sem. Mat. Univ. Pol. Torino, 54, 373 (1996), http://www.seminariomatematico.polito.it/rendiconti/cartaceo/54-4/373.pdf.

M. Goze, and M. Rausch de Traubenberg, J. Math. Phys. 50, 063508 (2009), https://doi.org/10.1063/1.3152631.

S. Duplij, in: Exotic Algebraic and Geometric Structures in Theoretical Physics, edited by S. Duplij (Nova Publishers, New York, 2018), pp. 251, arXiv: math.RT/1308.4060, https://arxiv.org/abs/1308.4060.

M. Markl, S. Shnider, and J. Stasheff, Operads in Algebra, Topology and Physics, (AMS, Providence, 2002).

J.-L. Loday, and B. Vallette, Algebraic operads, (Springer, Heidelberg, 2012).

W. Dörnte, Mathematische Zeitschrift, 29, 1–19 (1929), https://doi.org/10.1007/BF01180515.

G. Crombez, Abh. Math. Semin. Univ. Hamb. 37, 180 (1972), https://doi.org/10.1007/BF02999695.

J.J. Leeson, and A.T. Butson, Algebra Univers. 11, 42 (1980), https://doi.org/10.1007/BF02483082.

S. Duplij, p-Adic Numbers, Ultrametric Analysis and Appl. 9, 267-291 (2017), https://arxiv.org/abs/1707.00719.

J. Ušan, Mathematica Moravica, Special vol., 203 (2003), http://www.moravica.ftn.kg.ac.rs/Special/n-Groups_in_the_Light_of_Natural_Operations-v2006.pdf.

S. Duplij, and W. Werner, Structure of unital 3-fields, (2015), https://arxiv.org/abs/1505.04393v1.

Takeo Yokonuma, Tensor Spaces and Exterior Algebra, (AMS, Providence, 1992).

S. Duplij, in: Symmetry in Nonlinear Mathematical Physics, edited by A.G. Nikitin, and V.M. Boyko (Institute of Mathematics, Kiev, 2001), pp. 25.

S. Duplij, in: Exotic Algebraic and Geometric Structures in Theoretical Physics, edited by S. Duplij (Nova Publishers, New York, 2018), pp. 309.

T. Evans, Duke Math J. 30, 331 (1963), https://doi.org/10.1215/S0012-7094-63-03035-7.

V.D. Belousov, n-Ary Quasigroups, (Shtintsa, Kishinev, 1972).

C.W. Curtis, and I. Reiner, Representation theory of finite groups and associative algebras, (AMS, Providence, 1962).

A.A. Kirillov, Elements of the Theory of Representations, (Springer-Verlag, Berlin, 1976).

M. Hazewinkel, N. Gubareni, and V.V. Kirichenko, Algebras, Rings and Modules: Lie Algebras and Hopf Algebras, (AMS, Providence, 2010).

A. Borowiec, W. Dudek, and S. Duplij, Commun. Algebra, 34, 1651 (2006), https://doi.org/10.1080/00927870500542564.

M.E. Sweedler, J. Algebra, 8, 262 (1968), https://doi.org/10.1016/0021-8693(68)90059-8.

R. G. Heyneman, and M. E. Sweedler, J. Algebra 13, 192 (1969), https://doi.org/10.1016/0021-8693(69)90071-4.

Y. Kashina, S. Montgomery, and S.-H. Ng, Israel J. Math. 188, 57 (2012), https://doi.org/10.1007/s11856-011-0092-7.

M. Aguiar, and A. Lauve, Algebra Number Theory, 9, 547 (2015), https://doi.org/10.2140/ant.2015.9.547.

E.L. Post, Trans. Amer. Math. Soc. 48, 208 (1940), https://doi.org/10.2307/1990085.

W.A. Dudek, Discuss. Math., Gen. Algebra Appl. 27, 199 (2007), http://dx.doi.org/10.7151/dmgaa.

E. Heine, Handbuch der Kugelfunktionan, (Reimer, Berlin, 1878).

V. Kac, and P. Cheung, Quantum calculus, (Springer, New York, 2002).

S. Duplij, Pure Math. Appl. 9, 283 (1998), http://duplij.univer.kharkov.ua/puma.pdf.gz.

S. Duplij and W. Marcinek, Nucl. Phys. Proc. Suppl. 102, 293 (2001), Int. Conference on Supersymmetry and Quantum Field Theory: D.V. Volkov Memorial Conference, (Kharkov, Ukraine, 2000).

S. Duplij, and W. Marcinek, J. Math. Phys. 43, 3329 (2002), https://doi.org/10.1063/1.1473681.

S. Duplij, and W. Marcinek, in: Exotic Algebraic and Geometric Structures in Theoretical Physics, edited by S. Duplij, (Nova Publishers, New York, 2018), pp. 15.

S. Duplij, and F. Li, Czech. J. Phys. 51, 1306 (2001), https://doi.org/10.1023/A:1013313802053.

F. Li, and S. Duplij, Commun. Math. Phys. 225, 191 (2002), https://doi.org/10.1007/s002201000576.

K. Szlach´anyi, in: Operator Algebras and Quantum Field Theory, edited by S. Doplicher, R. Longo, J.E. Roberts, and L. Zsid´o (International Press, New York, 1996), p. 221.

A.M. Gal’mak, n-Ary Groups, Part 1, (Gomel University, Gomel, 2003).

V.G. Drinfeld, Leningrad Math. J. 1, 321 (1989), http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=10&what=fullt&option_lang=rus. (in Russian)

L.A. Lambe, and D.E. Radford, Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach, (Kluwer, Dordrecht, 1997).

V.G. Drinfeld, in: Problems of Modern Quantum Field Theory, edited by A.A. Belavin, A.V. Klimyk, and A.B. Zamolodchikov (Springer-Verlag, Heidelberg, 1989), pp. 1.

0 article
How to Cite
Duplij, S. (2021). Polyadic Hopf Algebras and Quantum Groups. East European Journal of Physics, (2), 5-50. https://doi.org/10.26565/2312-4334-2021-2-01