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Representation Theory

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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
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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.

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Additional Information

Peter Tingley
Affiliation: Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
MR Author ID: 679482

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.