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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras


Author: David E. Radford
Journal: Trans. Amer. Math. Soc. 343 (1994), 455-477
MSC: Primary 17B37; Secondary 16W30
DOI: https://doi.org/10.1090/S0002-9947-1994-1201324-2
MathSciNet review: 1201324
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Abstract: Let $ R:M \otimes M \to M \otimes M$ be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. We introduce a quotient $ {A^{{\text{red}}}}(R)$ of the bialgebra $ A(R)$ constructed by Fadeev, Reshetihkin and Takhtajan, whose characteristics seem to more faithfully reflect properties R possesses as a linear operator. We characterize all R such that $ {A^{{\text{red}}}}(R)$ is a pointed bialgebra, and we determine all solutions R to the quantum Yang-Baxter equation when $ {A^{{\text{red}}}}(R)$ is pointed and $ \dim M = 2$ (with a few technical exceptions when k has characteristic 2). Extensions of such solutions to the quantum plane are studied.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1201324-2
Article copyright: © Copyright 1994 American Mathematical Society