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

MathSciNet review:
1201324

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Abstract: Let 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 of the bialgebra 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 is a pointed bialgebra, and we determine all solutions *R* to the quantum Yang-Baxter equation when is pointed and (with a few technical exceptions when *k* has characteristic 2). Extensions of such solutions to the quantum plane are studied.

**[1]**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****[2]**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****[3]**Robert G. Heyneman and David E. Radford,*Reflexivity and coalgebras of finite type*, J. Algebra**28**(1974), 215–246. MR**0346001****[4]**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**, 10.1016/0375-9601(92)90044-M**[5]**-,*Solving the two-dimensional constant quantum Yang-Baxter equation*, preprint, 1992.**[6]**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**, 10.1006/jabr.1993.1014**[7]**Richard Gustavus Larson,*Characters of Hopf algebras*, J. Algebra**17**(1971), 352–368. MR**0283054****[8]**Shahn Majid,*Doubles of quasitriangular Hopf algebras*, Comm. Algebra**19**(1991), no. 11, 3061–3073. MR**1132774**, 10.1080/00927879108824306**[9]**-,*Physics for algebraists*:*non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction*, J. Algebra**129**(1990), 1-91.**[10]**Shahn Majid,*Quasitriangular Hopf algebras and Yang-Baxter equations*, Internat. J. Modern Phys. A**5**(1990), no. 1, 1–91. MR**1027945**, 10.1142/S0217751X90000027**[11]**David E. Radford,*Minimal quasitriangular Hopf algebras*, J. Algebra**157**(1993), no. 2, 285–315. MR**1220770**, 10.1006/jabr.1993.1102**[12]**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**, 10.1006/jabr.1993.1203**[13]**M. E. Sweedler,*Hopf algebras*, Math. Lecture Notes Ser., Benjamin, New York, 1969.**[14]**P. Smith,*Quantum groups for ring theorists*, preprint.**[15]**David N. Yetter,*Quantum groups and representations of monoidal categories*, Math. Proc. Cambridge Philos. Soc.**108**(1990), no. 2, 261–290. MR**1074714**, 10.1017/S0305004100069139

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

Article copyright:
© Copyright 1994
American Mathematical Society