The crystal commutor and Drinfeld's unitarized R -matrix
Joel Kamnitzer1
and Peter Tingley2
1American Institute of Mathematics Palo Alto CA USA
2UC Berkeley Department of Mathematics Berkeley CA USA
2UC Berkeley Department of Mathematics Berkeley CA USA
DOI: 10.1007/s10801-008-0137-0
Abstract
Drinfeld defined a unitarized R-matrix for any quantum group U q(\mathfrak g) U_{q}(\mathfrak {g}) . This gives a commutor for the category of U q(\mathfrak g) U_{q}(\mathfrak {g}) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives U q(\mathfrak g) 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.
Pages: 315–335
Keywords: keywords coboundary category; quantum group; R-matrix; crystal basis
Full Text: PDF
References
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2. Berenstein, A., Zwicknagl, S.: Braided symmetric and exterior algebras. To appear in Trans. Am. Math. Soc.
3. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)
4. Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1(6), 1419-1457 (1990)
5. Henriques, A., Kamnitzer, J.: Crystals and coboundary categories. Duke Math. J. 132(2), 191-216 (2006).
6. Kashiwara, M.: On crystal bases of the q-analogue of the universal enveloping algebras. Duke Math. J. 63(2), 465-516 (1991)
7. Kirillov, A.N., Reshetikhin, N.: q-Weyl group and a multiplicative formula for universal R-matrices.