Journal of Algebraic Combinatorics

, Volume 29, Issue 3, pp 315–335

The crystal commutor and Drinfeld’s unitarized R-matrix

Article

DOI: 10.1007/s10801-008-0137-0

Cite this article as:
Kamnitzer, J. & Tingley, P. J Algebr Comb (2009) 29: 315. doi:10.1007/s10801-008-0137-0

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 categoryQuantum groupR-matrixCrystal basis

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

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