Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 005, 26 pages      arXiv:2010.01060      https://doi.org/10.3842/SIGMA.2021.005
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

Quantum Groups for Restricted SOS Models

Giovanni Felder a and Muze Ren b
a) Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
b) Department of Mathematics, University of Geneva, 2-4 rue du Lièvre, c.p. 64, 1211 Geneva 4, Switzerland

Received October 05, 2020, in final form January 05, 2021; Published online January 12, 2021

Abstract
We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of $\pi$-graded vector spaces for a groupoid $\pi$.

Key words: elliptic quantum groups; dynamical $R$-matrices; groupoid grading; RSOS models.

pdf (503 kb)   tex (39 kb)  

References

  1. Aganagic M., Okounkov A., Elliptic stable envelopes, J. Amer. Math. Soc. 34 (2021), 79-133, arXiv:1604.00423.
  2. Andersen H.H., Paradowski J., Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995), 563-588.
  3. Andrews G.E., Baxter R.J., Forrester P.J., Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), 193-266.
  4. Baxter R.J., One-dimensional anisotropic Heisenberg chain, Ann. Physics 70 (1972), 323-337.
  5. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  6. Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.
  7. Bethe H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205-226.
  8. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  9. Costello K., Witten E., Yamazaki M., Gauge theory and integrability, I, ICCM Not. 6 (2018), 46-119, arXiv:1709.09993.
  10. Costello K., Witten E., Yamazaki M., Gauge theory and integrability, II, ICCM Not. 6 (2018), 120-146, arXiv:1802.01579.
  11. Date E., Jimbo M., Miwa T., Okado M., Fusion of the eight vertex SOS model, Lett. Math. Phys. 12 (1986), 209-215.
  12. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  13. Etingof P., Latour F., The dynamical Yang-Baxter equation, representation theory, and quantum integrable systems, Oxford Lecture Series in Mathematics and its Applications, Vol. 29, Oxford University Press, Oxford, 2005.
  14. Etingof P., Schiffmann O., Lectures on the dynamical Yang-Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), London Math. Soc. Lecture Note Ser., Vol. 290, Cambridge University Press, Cambridge, 2001, 89-129, arXiv:math.QA/9908064.
  15. Etingof P., Varchenko A., Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys. 192 (1998), 77-120, arXiv:q-alg/9703040.
  16. Etingof P., Varchenko A., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys. 196 (1998), 591-640, arXiv:q-alg/9708015.
  17. Etingof P., Varchenko A., Exchange dynamical quantum groups, Comm. Math. Phys. 205 (1999), 19-52, arXiv:math.QA/9801135.
  18. Faddeev L., Integrable models in $(1+1)$-dimensional quantum field theory, in Recent Advances in Field Theory and Statistical Mechanics (Les Houches, 1982), North-Holland, Amsterdam, 1984, 561-608.
  19. Felder G., Conformal field theory and integrable systems associated to elliptic curves, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 1247-1255, arXiv:hep-th/9407154.
  20. Felder G., Elliptic quantum groups, in XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 211-218, arXiv:hep-th/9412207.
  21. Felder G., Tarasov V., Varchenko A., Solutions of the elliptic qKZB equations and Bethe ansatz. I, in Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser. 2, Vol. 180, Amer. Math. Soc., Providence, RI, 1997, 45-75, arXiv:q-alg/9606005.
  22. Felder G., Varchenko A., Algebraic Bethe ansatz for the elliptic quantum group $E_{\tau,\eta}({\rm sl}_2)$, Nuclear Phys. B 480 (1996), 485-503, arXiv:q-alg/9605024.
  23. Felder G., Varchenko A., On representations of the elliptic quantum group $E_{\tau,\eta}({\rm sl}_2)$, Comm. Math. Phys. 181 (1996), 741-761, arXiv:q-alg/9601003.
  24. Felder G., Varchenko A., Resonance relations for solutions of the elliptic QKZB equations, fusion rules, and eigenvectors of transfer matrices of restricted interaction-round-a-face models, Commun. Contemp. Math. 1 (1999), 335-403, arXiv:math.QA/9901111.
  25. Gautam S., Toledano-Laredo V., Elliptic quantum groups and their finite-dimensional representations, arXiv:1707.06469.
  26. Gervais J.-L., Neveu A., Novel triangle relation and absence of tachyons in Liouville string field theory, Nuclear Phys. B 238 (1984), 125-141.
  27. Huse D.A., Exact exponents for infinitely many new multicritical points, Phys. Rev. B 30 (1984), 3908-3915.
  28. Inozemtsev V.I., On the connection between the one-dimensional $S=1/2$ Heisenberg chain and Haldane-Shastry model, J. Statist. Phys. 59 (1990), 1143-1155.
  29. Jimbo M., Konno H., Odake S., Shiraishi J., Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4 (1999), 303-327, arXiv:q-alg/9712029.
  30. Jimbo M., Kuniba A., Miwa T., Okado M., The $A^{(1)}_n$ face models, Comm. Math. Phys. 119 (1988), 543-565.
  31. Jimbo M., Miwa T., Okado M., Solvable lattice models related to the vector representation of classical simple Lie algebras, Comm. Math. Phys. 116 (1988), 507-525.
  32. Kalmykov A., Safronov P., A categorical approach to dynamical quantum groups, arXiv:2008.09081.
  33. Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  34. Kirillov A.N., Reshetikhin N.Yu., Representations of the algebra ${U}_q({\rm sl}(2)),q$-orthogonal polynomials and invariants of links, in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 285-339.
  35. Kirillov A.N., Reshetikhin N.Yu., Formulas for multiplicities of occurence of irreducible components in the tensor product of representations of simple Lie algebras, J. Math. Sci. 80 (1996), 1768-1772.
  36. Konno H., Elliptic quantum groups: representations and related geometry, SpringerBriefs in Mathematical Physics, Vol. 37, Springer, Singapore, 2020.
  37. Kulish P.P., Reshetikhin N.Yu., Sklyanin E.K., Yang-Baxter equations and representation theory. I, Lett. Math. Phys. 5 (1981), 393-403.
  38. Lundström P., The category of groupoid graded modules, Colloq. Math. 100 (2004), 195-211.
  39. Lusztig G., Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
  40. McGuire J.B., Study of exactly soluble one-dimensional $N$-body problems, J. Math. Phys. 5 (1964), 622-636.
  41. Sawin S.F., Quantum groups at roots of unity and modularity, J. Knot Theory Ramifications 15 (2006), 1245-1277, arXiv:math.QA/0308281.
  42. Shibukawa Y., Hopf algebroids and rigid tensor categories associated with dynamical Yang-Baxter maps, J. Algebra 449 (2016), 408-445.
  43. Stokman J., Reshetikhin N., $N$-point spherical functions and asymptotic boundary KZB equations, arXiv:2002.02251.
  44. Turaev V., Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter & Co., Berlin, 1994.
  45. Turaev V., Virelizier A., Monoidal categories and topological field theory, Progress in Mathematics, Vol. 322, Birkhäuser/Springer, Cham, 2017.
  46. Verlinde E., Fusion rules and modular transformations in $2$D conformal field theory, Nuclear Phys. B 300 (1988), 360-376.
  47. Yang C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.

Previous article  Next article  Contents of Volume 17 (2021)