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 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

Solving the two-dimensional constant quantum Yang-Baxter equation

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

Physics for algebraists

non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction

129

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

Hopf algebras

Quantum groups for ring theorists

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