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Communications in Mathematical Physics

, Volume 349, Issue 1, pp 271–283 | Cite as

A New Braid-like Algebra for Baxterisation

  • N. CrampeEmail author
  • L. Frappat
  • E. Ragoucy
  • M. Vanicat
Article

Abstract

We introduce a new Baxterisation for R-matrices that depend separately on two spectral parameters. The Baxterisation is based on a new algebra, close to but different from the braid group. We study representations of this new algebra on the vector space \({(\mathbb{C}^m)^{\otimes n}}\), when the generators act locally. The ones for \({m=2}\) are completely classified. We also introduce some representations for generic m: they allow us to recover the R-matrix of the multi-species generalization of the totally asymmetric simple exclusion process with different hopping rates.

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References

  1. 1.
    Kulish P.P., Reshetikhin N.Y., Sklyanin E.K.: Yang–Baxter equation and representation theory: I. Lett. Math. Phys. 5, 393 (1981)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Kulish P.P., Sklyanin E.K.: Solutions of the Yang–Baxter equation. J. Sov. Math. 19, 1596 (1982)CrossRefzbMATHGoogle Scholar
  3. 3.
    Jimbo M.: Quantum R matrix for the generalized Toda system. Commun. Math. Phys. 102, 537 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Jones, V.F.R.: Baxterisation. Int. J. Mod. Phys. B 4, 701 (1990), proceedings of “Yang–Baxter equations, conformal invariance and integrability in statistical mechanics and field theory”, Canberra (1989)Google Scholar
  5. 5.
    Jimbo M.: A q-difference analogue of \({U(gl(n+1))}\), Hecke algebra and the Yang–Baxter equation. Lett. Math. Phys. 11, 247 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Isaev, A.P.: Quantum groups and Yang–Baxter equations. Max–Planck Institut für Mathematik (2004). Max-Planck Institute preprint MPI 04-132Google Scholar
  7. 7.
    Cheng Y., Ge M.L., Xue K.: Yang–Baxterization of braid group representations. Commun. Math. Phys. 136, 195 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Zhang R.B., Gould M.D., Bracken A.J.: From representations of the braid group to solutions of the Yang–Baxter equation. Nucl. Phys. B 354, 625 (1991)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Li Y.-Q.: Yang Baxterization. J. Math. Phys. 34, 757 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boukraa, S., Maillard, J.M.: Let’s Baxterise. J. Stat. Phys. 102, 641 (2001). arXiv:hep-th/0003212
  11. 11.
    Arnaudon D., Chakrabarti A., Dobrev V.K., Mihov S.G.: Spectral decomposition and Baxterisation of exotic bialgebras and associated noncommutative geometries. Int. J. Mod. Phys. A 18, 4201 (2003)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Kulish, P.P., Manojlović, N., Nagy, Z.: Symmetries of spin systems and Birman–Wenzl–Murakami algebra. J. Math. Phys. 51, 043516 (2010). arXiv:0910.4036 [nlin.SI]
  13. 13.
    Fonseca T., Frappat L., Ragoucy E.: R matrices of three-state Hamiltonians solvable by coordinate Bethe ansatz. J. Math. Phys 56, 013503 (2015). arXiv:1406.3197
  14. 14.
    Drinfel’d V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419 (1990)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Drinfel’d V.G.: Structure of quasitriangular quasi-Hopf algebras. Funct. Anal. Appl. 26, 63 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Sklyanin E.K., Takhtadzhyan L.A., Faddeev L.D.: Quantum inverse problem method. I. Theor. Math. Phys 40, 688 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hietarinta J.: Solving the two-dimensional constant quantum Yang–Baxter equation. J. Math. Phys. 34, 1725 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Cantini, L.: Algebraic Bethe Ansatz for the two species ASEP with different hopping rates. J. Phys. A 41, 095001 (2008). arXiv:0710.4083
  19. 19.
    Jones V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Derrida B., Evans M., Hakim V., Pasquier V.: Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26, 1493 (1993)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Blythe, R.A., Evans, M.R.: Nonequilibrium steady states of matrix product form: a Solver’s guide. J. Phys. A 40, R333 (2007). arXiv:0706.1678
  22. 22.
    Sasamoto T., Wadati M.: Stationary state of integrable systems in matrix product form. J. Phys. Soc. Japan 66, 2618 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Crampe, N., Ragoucy E., Vanicat, M.: Integrable approach to simple exclusion processes with boundaries. Review and Progress. J. Stat. Mech. P11032 (2014). arXiv:1408.5357
  24. 24.
    Arita, C., Mallick, K.: Matrix product solution to an inhomogeneous multi-species TASEP. J. Phys. A 46, 085002 (2013). arXiv:1209.1913

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.L2C, UMR 5221 CNRS-Université de MontpellierMontpellierFrance
  2. 2.LAPTh, CNRS–Université de Savoie Mont BlancAnnecy-le-Vieux CedexFrance

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