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

Radford, David E.

doi:10.1090/S0002-9947-1994-1201324-2

Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras
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by David E. Radford

Trans. Amer. Math. Soc. 343 (1994), 455-477

DOI: https://doi.org/10.1090/S0002-9947-1994-1201324-2

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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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