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Twisted affine Lie superalgebra of type Q and quantization of its enveloping superalgebra

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Abstract

We introduce a new quantum group which is a quantization of the enveloping superalgebra of a twisted affine Lie superalgebra of type Q. We study generators and relations for superalgebras in the finite and twisted affine cases, and also universal central extensions. Afterwards, we apply the FRT formalism to a certain solution of the quantum Yang–Baxter equation to define that new quantum group and we study some of its properties. We construct a functor of Schur–Weyl type which connects it to affine Hecke–Clifford algebras and prove that it provides an equivalence between two categories of modules.

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References

  1. Arakawa T.: Functor and finite dimensional representations of Yangian. Commun. Math. Phys. 205(1), 1–18 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brundan J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \({\mathfrak{q}(n)}\) . Adv. Math. 182(1), 28–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, H., Guay, N.: Central extensions of matrix Lie superalgebras over \({\mathbb{Z}/2\mathbb{Z}}\) -graded algebras, preprint

  4. Chari V., Pressley A.: Quantum affine algebras and affine Hecke algebras. Pacific J. Math. 174(2), 295–326 (1996)

    MathSciNet  MATH  Google Scholar 

  5. Curtis C., Reiner I.: Methods of representation theory—With applications to finite groups and orders. Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1981)

    Google Scholar 

  6. Drinfeld V.: Degenerate affine Hecke algebras and Yangians. (Russian) Funktsional. Anal. i Prilozhen 20(1), 69–70 (1986)

    MathSciNet  Google Scholar 

  7. Frisk A., Mazorchuck V.: Regular strongly typical blocks of \({\mathcal{O}^{\mathfrak{q}}}\) . Commun. Math. Phys. 291(2), 533–542 (2009)

    Article  MATH  Google Scholar 

  8. Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantization of Lie groups and Lie algebras (Russian), Algebra i Analiz 1 (1989), no. 1, 178–206; translation in Leningrad Math. J. 1(1), 193–225 (1990)

  9. Frappat L., Sciarrino A., Sorba P.: Dictionary on Lie algebras and superalgebras. Academic Press, Inc., San Diego, CA (2000)

    MATH  Google Scholar 

  10. Geck M., Pfeiffer G.: Characters of Finite Coxeter Groups and Iwahori-Hecke algebras, London Mathematical Society Monographs, New Series, 21. The Clarendon Press, Oxford University Press, New York (2000)

    Google Scholar 

  11. Grantcharov D., Jung J.H., Kang S.-J., Kim M.: Highest weight modules over quantum queer superalgebra \({U_q(\mathfrak{q}(n))}\) . Commun. Math. Phys. 296(3), 827–860 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grantcharov D., Jung J.H., Kang S.-J., Kashiwara M., Kim M.: Quantum Queer Superalgebra and Crystal Bases. Proc. Japan Acad. Ser. A Math. Sci. 86(10), 177–182 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gorelik M., Serganova V.: On representations of the affine superalgebra \({\mathfrak{q}(n)^{(2)} }\) . Mosc. Math. J. 8(1), 91–109 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Guay N.: Affine Yangians and deformed double current algebras in type A. Adv. Math. 211(2), 436–484 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guay N.: Quantum algebras and symplectic reflection algebras for wreath products. Represent. Theory 14, 148–200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hill D., Kujawa J., Sussan J.: Degenerate affine Hecke–Clifford algebras and type Q Lie superalgebras. Math. Z. 268(3–4): 1091–1158

  17. Hopkins, M.J., Molev, A.I.: A q-analogue of the centralizer construction and skew representations of the quantum affine algebra, SIGMA Symmetry Integrability. Geom. Methods Appl. 2(092) (2006)

  18. Iohara K., Koga Y.: Central extensions of Lie superalgebras. Comment. Math. Helv. 76(1), 110–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ivanov I.T., Uglov D.B.: sl(N) Onsager’s algebra and integrability. J. Stat. Phys. 82(1–2), 87–113 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Jones A., Nazarov M.: Affine sergeev algebra and q-analogues of the young symmetrizers for projective representations of the symmetric group. Proc. Lond. Math. Soc. 78(3), 481–512 (1999)

    Article  MathSciNet  Google Scholar 

  21. Kac V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  MATH  Google Scholar 

  22. Khongsap T.: Hecke–Clifford algebras and spin Hecke algebras III—the trigonometric type. J. Algebra 322(8), 2731–2750 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khongsap T., Wang W.: Hecke–Clifford algebras and spin Hecke algebras I—the classical affine type. Transform. Groups 13(2), 389–412 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Khongsap T., Wang W.: Hecke–Clifford algebras and spin Hecke algebras II—the rational double affine type. Pacific J. Math. 238(1), 73–103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Khongsap, T., Wang, W.: Hecke–Clifford algebras and spin Hecke algebras IV—odd double affine type, SIGMA Symmetry Integrability. Geom. Methods Appl. 5(012) (2009)

  26. Leites, D., Serganova, V.: Defining relations for classical Lie superalgebras. I. Superalgebras with Cartan matrix or Dynkin-type diagram, Topological and geometrical methods in field theory. (Turku, 1991), 194201, World Sci. Publ., River Edge (1992)

  27. Nazarov, M.: Yangians of the “strange” Lie superalgebras, in Quantum groups (Leningrad, 1990), 90–97, Lecture Notes in Maths 1510, Springer, Berlin (1992)

  28. Nazarov M.: Yangian of the queer Lie superalgebra. Commun. Math. Phys. 208(1), 195–223 (1999)

    Article  MATH  Google Scholar 

  29. Nazarov, M., Sergeev, A.: Centralizer construction of the Yangian of the queer Lie superalgebra, Studies in Lie theory, 417–441, Progr. Math., 243, Birkhäuser Boston, Boston (2006)

  30. Olshanski G.: Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra. Lett. Math. Phys. 24(2), 93–102 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Penkov I., Serganova V.: Characters of finite-dimensional irreducible \({\mathfrak{q}(n)}\) -modules. Lett. Math. Phys. 40(2), 147–158 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Penkov I., Serganova V.: Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G = Q(N), Algebraic geometry. 7. J. Math. Sci. 84(5), 1382–1412 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Sergeev A.N.: Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m) and Q(n). (Russian) Mat. Sb. (N.S.) 123(165)(3), 422–430 (1984)

    MathSciNet  Google Scholar 

  34. Stukopin, V.: Yangian of the Strange Lie Superalgebra of Q n-1 Type, Drinfel’d Approach, SIGMA Symmetry Integrability Geom. Methods Appl. 3(069) (2007)

  35. Yamane H.: Quantized enveloping algebras associated with simple Lie superalgebras and their universal R-matrices. Publ. Res. Inst. Math. Sci. 30(1), 15–87 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, AB, T6G 2G1, Canada

    Hongjia Chen & Nicolas Guay

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  1. Hongjia Chen
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  2. Nicolas Guay
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Correspondence to Nicolas Guay.

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Chen, H., Guay, N. Twisted affine Lie superalgebra of type Q and quantization of its enveloping superalgebra. Math. Z. 272, 317–347 (2012). https://doi.org/10.1007/s00209-011-0935-2

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