Quasi Three-parametric \(R\)-matrix and Quantum Supergroups \(GL_{p,q}(1/1)\) and \(U_{p,q}[\textit{gl}(1/1)]\)

Nguyen Thi Hong Van, Nguyen Anh Ky


An overparametrized (three-parametric) R-matrix satisfying a graded Yang-Baxter equation is introduced. It turns out that such an overparametrization is very helpful. Indeed, this R-matrix with one of the parameters being auxiliary, thus, reducible to a two-parametric R-matrix, allows the construction of quantum supergroups GLp,q(1/1) and Up,q[gl(1/1)] which, respectively, are two-parametric deformations of the supergroup GL(1/1) and the universal enveloping algebra U[gl(1/1)]. These two-parametric quantum deformations GLpq(1/1) and Upq[gl(1/1)], to our knowledge, are constructed for the first time via the present approach. The quantum deformation Up,q[gl(1/1)] obtained here is a true two-parametric deformation of Drinfel’d-Jimbo’s type, unlike some other one obtained previously elsewhere.


Quantum supergroup, R-matrix, Drinfel'd-Jimbo deformation, Multi- parametric quantum deformation

Full Text:



G. Aad et al. (ATLAS collaboration), Phys. Lett. B716, 1 (2012) [arXiv:1207.7214 [hep-ex]].

S. Chatrchyan et al. (CMS collaboration), Phys. Lett. B716, 30 (2012) [arXiv:1207.7235 [hep-ex]].

Nguyen Anh Ky and Nguyen Thi Hong Van, “Was the Higgs boson discovered?”, Commun. Phys. 25,

(2015)[arXiv:1503.08630 [hep-ph]].

Ho Kim Quang and Pham Xuan Yem, “Elementary particles and their interactions: concepts and

phenomena”, Springer-Verlag, Berlin, 1998.

S. Willenbrock, “Symmetries of the standard model”, hep-ph/0410370.

M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018).

L. Faddeev, N. Reshetikhin and L. Takhtajan, Algebra and Analys 1, 178 (1987).

Yu. Manin, “Quantum groups and non-commutative geometry””, Centre des Recherchers

Math ́ematiques, Montr ́eal, 1988.

S. Gomez, M. Ruiz-Altaba and G, Sierra, “Quantum groups in two-dimensional physics”, Cambridge

university press, Cambridge, 1996.

V. Drinfel’d, “Quantum groups”, J. Sov. Math., 41, 898 (1988); Zap. Nauch. Semin. 155, 18 (1986);

also in Proceedings of the International Congress of Mathematicians, Berkeley 1986, vol 1, The American

Mathematical Society, Providence, RI, 1987, pp. 798 - 820.

M. Jimbo, Lett. Math. Phys. 10, 63 (1985); ibit 11, 247 (1986).

Yu. Manin, Commun. Math. Phys. 123, 169 (1989).

P. Kulish and N. Reshetikhin, Lett. Math. Phys 18, 143 (1989).

J. Schmidke, S. Volos and B. Zumino, Z. Phys. C 48, 249 (1990).

M. Chaichian and P. Kulish, Phys. Lett. 234B, 72 (1990).

N. Reshetikhin, Lett. Math. Phys. 20, 331 (1990).

A. Schirrmacher, J. Wess and B. Zumino, Z. Phys. C 49, 317 (1991).

L. Dabrowski and L.-y. Wang, Phys. Lett. 266B, 51 (1991).

H. Hinrichsen and V. Rittenberg, Phys. Lett. 275B, 350 (1992) [hep-th/9110074].

Nguyen Anh Ky, J. Phys. A 29, 1541 (1996) [math.QA/9909067].

V. Dobrev and E. Tahri, Int. J. Mod. Phys. A 13, 4339 (1998).

Nguyen Anh Ky, J. Math. Phys. 41, 6487 (2000) [math.QA/0005122].

Nguyen Anh Ky, J. Phys. A 34, 7881 (2001) [math.QA/0104105].

Naihong Hu1, Marc Rosso, Honglian Zhang1, Commun. Math. Phys. 278, 453 (2008).

Yun Gao1, Naihong Hu, and Honglian Zhang, J. Math. Phys. 56, 011704 (2015).

Naihuan Jing1 and Honglian Zhang, J. Math. Phys. 57, 091702 (2016).

Nguyen Anh Ky and Nguyen Thi Hong Van, “A two-parametric deformation of U[sl(2)], its represen-

tations and complex “spin””, math.QA/0506539.

A. Kundu, Phys. Rev. Lett. 82, 3936 (1999).

A Jellal, Mod. Phys. Lett. A 17, 701 (2002).

A. Algin and B Deriven, J. Phys. A 38, 5945 (2005).

Nguyen Anh Ky, J. Math. Phys. 35, 2583 (1994) [hep-th/9305183].

Nguyen Anh Ky and N. Stoilova, J. Math. Phys. 36, 5979 (1995) [hep-th/9411098].

DOI: https://doi.org/10.15625/0868-3166/29/4/14009 Display counter: Abstract : 239 views. PDF : 67 views.


  • There are currently no refbacks.

Editorial Office:

Communications in Physics

1st Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam

Tel: (+84) 024 3791 7102 

Email: cip@vjs.ac.vn

Copyright by