Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces

2004/03/17 by B. Enriquez, Benjamin Enriquez, Pavel Etingof +6
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA

paper · pdf · doi:10.48550/arxiv.math/0403283

36 pages

openalex publication_date 2004/03/17 · arxiv created 2004/06/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Poisson-Lie (PL) dynamical r-matrices are generalizations of dynamical r-matrices, where the base is a Poisson-Lie group. We prove analogues of basic results for these r-matrices, namely constructions of (quasi)Poisson groupoids and of Poisson homogeneous spaces. We introduce a class of PL dynamical r-matrices, associated to nondegenerate Lie bialgebras with a splitting; this is a generalization of trigonometric r-matrices with an abelian base. We prove a composition theorem for PL dynamical r-matrices, and construct quantizations of the polarized PL dynamical r-matrices. This way, we obtain quantizations of Poisson homogeneous structures on G/L (G a semisimple Lie group, L a Levi subgroup), thereby generalizing earlier constructions.

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