Quantization of a Poisson structure on products of principal affine\n spaces

2018/07/25 by Victor Mouquin, Mouquin, Victor
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1807.09843

openalex publication_date 2018/07/25 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

We give the analogue for Hopf algebras of the polyuble Lie bialgebra\nconstruction by Fock and Rosli. By applying this construction to the\nDrinfeld-Jimbo quantum group, we obtain a deformation quantization\n\ℂ_ hslash[(N backslash G)^m] of a Poisson structure \π^(m) on\nproducts (N backslash G)^m of principal affine spaces of a connected and\nsimply connected complex semisimple Lie group G. The Poisson structure\n\π^(m) descends to a Poisson structure \π_m on products (B backslash\nG)^m of the flag variety of G which was introduced and studied by the Lu and\nthe author. Any ample line bundle on (B backslash G)^m inherits a natural\nflat Poisson connection, and the corresponding graded Poisson algebra is\nquantized to a subalgebra of \ℂ_ hslash[(N backslash G)^m].\n We define the notion of a strongly coisotropic subalgebra in a Hopf algebra,\nand explain how strong coisotropicity guarantees that any homogeneous\ncoordinate ring of a homogeneous space of a Poisson Lie group can be quantized\nin the sense of Ciccoli, Fioresi, and Gavarini.\n

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