Journal of Algebraic Combinatorics

, Volume 29, Issue 3, pp 315–335

The crystal commutor and Drinfeld’s unitarized R-matrix

Article

Abstract

Drinfeld defined a unitarized R-matrix for any quantum group \(U_{q}(\mathfrak {g})\) . This gives a commutor for the category of \(U_{q}(\mathfrak {g})\) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives \(U_{q}(\mathfrak {g})\) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer’s construction agrees with Drinfeld’s commutor. We then describe the action of Drinfeld’s commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor.

Keywords

Coboundary category Quantum group R-matrix Crystal basis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bakalov, B., Kirillov, A.: Lectures on Tensor Categories and Modular Functors. American Mathematical Society, Providence (2001) MATHGoogle Scholar
  2. 2.
    Berenstein, A., Zwicknagl, S.: Braided symmetric and exterior algebras. To appear in Trans. Am. Math. Soc. math.QA/0504155
  3. 3.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994) MATHGoogle Scholar
  4. 4.
    Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1(6), 1419–1457 (1990) MathSciNetGoogle Scholar
  5. 5.
    Henriques, A., Kamnitzer, J.: Crystals and coboundary categories. Duke Math. J. 132(2), 191–216 (2006). math.QA/0406478 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kashiwara, M.: On crystal bases of the q-analogue of the universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kirillov, A.N., Reshetikhin, N.: q-Weyl group and a multiplicative formula for universal R-matrices. Commun. Math. Phys. 134(2), 421–431 (1990) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Levendorskii, S.Z., Soibelman, Ya.S.: The quantum Weyl group and a multiplicative formula for the R-matrix of a simple Lie algebra. Funct. Anal. Appl. 25(2), 143–145 (1991) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol. 110. Birkhäuser, Boston (1993) MATHGoogle Scholar
  10. 10.
    Lusztig, G.: Canonical Bases arising from quantized enveloping algebras. II. Common trends in mathematics and quantum field theories (Kyoto, 1990). Program. Theor. Phys. Suppl. 102, 175–201 (1991) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.American Institute of MathematicsPalo AltoUSA
  2. 2.Department of MathematicsUC BerkeleyBerkeleyUSA

Personalised recommendations