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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© Copyright 1994
American Mathematical Society