Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups

Details Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups

1997-08-16

arxiv.org/abs/q-alg/9708015v1

Mathematics

Quantum Algebra

55 pages, amstex

Scientific paper

10.1007/s002200050437

The quantum dynamical Yang-Baxter (QDYB) equation is a useful generalization of the quantum Yang-Baxter (QYB) equation introduced by Gervais, Neveu, and Felder. The QDYB equation and its quasiclassical analogue (the classical dynamical Yang-Baxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder. In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical Yang-Baxter equation, obtained in our previous paper q-alg/9703040. All solutions we found can be obtained from Felder's elliptic solution by a limiting process and gauge transformations. Fifteen years ago the quantum Yang-Baxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum Yang-Baxter equation. In this paper we propose a similar language, originating from Felder's ideas, which we found to be adequate for the dynamical Yang-Baxter equation. This is the language of dynamical quantum groups (or $\h$-Hopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper q-alg/9703040.

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