On Frobenius algebras

and the quantum Yang-Baxter equation

Authors:
K. I. Beidar, Y. Fong and A. Stolin

Journal:
Trans. Amer. Math. Soc. **349** (1997), 3823-3836

MSC (1991):
Primary 81R50; Secondary 16L60

DOI:
https://doi.org/10.1090/S0002-9947-97-01808-4

MathSciNet review:
1401512

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya.

**1.**K. I. Beidar, Y. Fong and A. Stolin,*On Frobenius Algebras and Quantum Yang-Baxter Equation*, Preprint, TRITA-MAT-1994-0040, Royal Inst. of Technology, Stockholm, Sweden, 1994.**2.**A. Braun,*On Artin's Theorem and Azumaya Algebras*, J. Algebra**77**(1982), 323-332. MR**83m:16005****3.**F. De Meyer and E. Ingraham,*Separable Algebras over Commutative Rings*, Lecture Notes in Math. N 181, Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR**43:6199****4.**V. G. Drinfel'd,*Hamiltonian Structures on Lie Groups, Lie Bialgebras and the Geometrical Meaning of the Classical Yang-Baxter Equation*, Soviet Math. Dokl.**27**(1983), 68-71. MR**84i:58044****5.**V. G. Drinfel'd,*Quantum Groups*, Proc. ICM-86 at Berkeley (A. Gleasson, ed.), vol. 1, AMS, 1987, pp. 798-820. MR**89f:17017****6.**S. Eilenberg and T. Nakayama,*On the Dimension of Modules and Algebras, II*, Nagoya Math. J.**9**(1955), 1-16. MR**17:453a****7.**L. A. Lambe and D. E. Radford,*Algebraic Aspects of the Quantum Yang-Baxter Equation*, J. Algebra.**154**(1992), 228-288. MR**94b:17026****8.**R. G. Larson and M. E. Sweedler,*An Associative Orthogonal Bilinear Form for Hopf Algebras*, Amer. J. Math.**91**(1969), 75-94. MR**39:1523****9.**S. Majid,*Quasitriangular Hopf Algebras and Yang-Baxter Equations*, International J. Modern Physics, A.**5**(1990), 1-91. MR**90k:16008****10.**B. Pareigis,*When Hopf Algebras are Frobenius Algebras*, J. Algebra**18**(1971), 588-596. MR**43:6242****11.**L. H. Rowen,*Ring Theory, II*, Academic Press, Inc., San Diego, 1988. MR**89h:16002****12.**A. A. Stolin,*On Rational Solutions of Yang-Baxter Equation for*, Math. Scand.**69**(1991), 59-80. MR**93d:17023****13.**A. A. Stolin,*On Constant Solutions of Yang-Baxter Equation for*, Math. Scand.**69**(1991), 81-90. MR**93b:17053**

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Additional Information

**K. I. Beidar**

Affiliation:
Department of Mathematics, National Cheng-Kung University, Tainan, Taiwan

Email:
beidar@mail.ncku.edu.tw; fong@mail.ncku.edu.tw

**A. Stolin**

Affiliation:
Department of Mathematics, University of Göteborg, S-41296 Göteborg, Sweden

Email:
astolin@math.chalmers.se

DOI:
https://doi.org/10.1090/S0002-9947-97-01808-4

Keywords:
Quantum Yang-Baxter equation,
Frobenius algebra,
Azumaya algebra.

Received by editor(s):
August 4, 1995

Received by editor(s) in revised form:
March 7, 1996

Additional Notes:
We would like to express our gratitude to the Swedish Academy of Science for the support of the visit of the first author to Sweden, during which this paper was finished. We are thankful to Professor B. Pareigis for fruitful discussions.

Article copyright:
© Copyright 1997
American Mathematical Society