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

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- by David E. Radford PDF
- Trans. Amer. Math. Soc.
**343**(1994), 455-477 Request permission

## 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.

## References

- V. G. Drinfel′d,
*Quantum groups*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR**934283** - N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev,
*Quantization of Lie groups and Lie algebras*, Algebra i Analiz**1**(1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J.**1**(1990), no. 1, 193–225. MR**1015339** - Robert G. Heyneman and David E. Radford,
*Reflexivity and coalgebras of finite type*, J. Algebra**28**(1974), 215–246. MR**346001**, DOI 10.1016/0021-8693(74)90035-0 - Jarmo Hietarinta,
*All solutions to the constant quantum Yang-Baxter equation in two dimensions*, Phys. Lett. A**165**(1992), no. 3, 245–251. MR**1169634**, DOI 10.1016/0375-9601(92)90044-M
—, - Larry A. Lambe and David E. Radford,
*Algebraic aspects of the quantum Yang-Baxter equation*, J. Algebra**154**(1993), no. 1, 228–288. MR**1201922**, DOI 10.1006/jabr.1993.1014 - Richard Gustavus Larson,
*Characters of Hopf algebras*, J. Algebra**17**(1971), 352–368. MR**283054**, DOI 10.1016/0021-8693(71)90018-4 - Shahn Majid,
*Doubles of quasitriangular Hopf algebras*, Comm. Algebra**19**(1991), no. 11, 3061–3073. MR**1132774**, DOI 10.1080/00927879108824306
—, - Shahn Majid,
*Quasitriangular Hopf algebras and Yang-Baxter equations*, Internat. J. Modern Phys. A**5**(1990), no. 1, 1–91. MR**1027945**, DOI 10.1142/S0217751X90000027 - David E. Radford,
*Minimal quasitriangular Hopf algebras*, J. Algebra**157**(1993), no. 2, 285–315. MR**1220770**, DOI 10.1006/jabr.1993.1102 - David E. Radford,
*Solutions to the quantum Yang-Baxter equation and the Drinfel′d double*, J. Algebra**161**(1993), no. 1, 20–32. MR**1245841**, DOI 10.1006/jabr.1993.1203
M. E. Sweedler, - David N. Yetter,
*Quantum groups and representations of monoidal categories*, Math. Proc. Cambridge Philos. Soc.**108**(1990), no. 2, 261–290. MR**1074714**, DOI 10.1017/S0305004100069139

*Solving the two-dimensional constant quantum Yang-Baxter equation*, preprint, 1992.

*Physics for algebraists*:

*non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction*, J. Algebra

**129**(1990), 1-91.

*Hopf algebras*, Math. Lecture Notes Ser., Benjamin, New York, 1969. P. Smith,

*Quantum groups for ring theorists*, preprint.

## Additional Information

- © Copyright 1994 American Mathematical Society
- 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