  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.

Quantum Algebra;Quantum Algebra