Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



A formula for the $ R$-matrix using a system of weight preserving endomorphisms

Author: Peter Tingley
Journal: Represent. Theory 14 (2010), 435-445
MSC (2010): Primary 17B37; Secondary 16Txx
Published electronically: June 3, 2010
MathSciNet review: 2652074
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a formula for the universal $ R$-matrix of the quantized universal enveloping algebra $ U_q(\mathfrak{g}).$ This is similar to a previous formula due to Kirillov-Reshetikhin and Levendorskii-Soibelman, except that where they use the action of the braid group element $ T_{w_0}$ on each representation $ V$, we show that one can instead use a system of weight preserving endomorphisms. One advantage of our construction is that it is well defined for all symmetrizable Kac-Moody algebras. However, we have only established that the result is equal to the universal $ R$-matrix in finite type.

References [Enhancements On Off] (What's this?)

  • [CP] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994. MR 1300632 (95j:17010)
  • [HK] A. Henriques and J. Kamnitzer, Crystals and coboundary categories, Duke Math. J., 132 (2006) no. 2, 191-216. MR 2219257 (2007m:17020)
  • [KT] J. Kamnitzer and P. Tingley. The crystal commutor and Drinfeld's unitarized $ R$-matrix. J. Algebraic Combin. 29 (2009), no. 3, 315-335; arXiv:math/0707.2248v2. MR 2496310
  • [K] M. Kashiwara, On crystal bases of the $ q$-analogue of the universal enveloping algebras, Duke Math. J., 63 (1991), no. 2, 465-516. MR 1115118 (93b:17045)
  • [KR] A. N. Kirillov and N. Reshetikhin, $ q$-Weyl group and a multiplicative formula for universal $ R$-matrices, Comm. Math. Phys. 134 (1990), no. 2, 421-431. MR 1081014 (92c:17023)
  • [LS] S. Z. Levendorskii and Ya. S. Soibelman, The quantum Weyl group and a multiplicative formula for the $ R$-matrix of a simple Lie algebra, Funct. Anal. Appl. 25 (1991), no. 2, 143-145. MR 1142216 (93a:17017)
  • [L] G. Lusztig. Introduction to quantum groups, Birkhäuser Boston Inc. 1993. MR 1227098 (94m:17016)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 17B37, 16Txx

Retrieve articles in all journals with MSC (2010): 17B37, 16Txx

Additional Information

Peter Tingley
Affiliation: Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): February 24, 2008
Published electronically: June 3, 2010
Additional Notes: This work was supported by the RTG grant DMS-0354321.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society