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Representation Theory
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
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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
Email: ptingley@math.mit.edu

DOI: http://dx.doi.org/10.1090/S1088-4165-2010-00378-7
PII: S 1088-4165(2010)00378-7
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.