Quantum dynamical Yang-Baxter equation over a nonabelian base

Details Quantum dynamical Yang-Baxter equation over a nonabelian base Quantum dynamical Yang-Baxter equation over a nonabelian base

2001-04-06

arxiv.org/abs/math/0104071v2

Commun.Math.Phys. 226 (2002) 475-495

Mathematics

Quantum Algebra

23 pages, minor changes made, final version to appear in Comm. Math. Phys

Scientific paper

10.1007/s002200200621

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra $\frakg =\frakh \oplus \frakm$, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix $r: \frakh^* \lon \wedge^2 \frakg$ corresponds to a Poisson manifold $\frakh^* \times G$. A special type of quantizations of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang-Baxter equation (or Gervais-Neveu-Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed.

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