Xu's quantization conjecture for Lie bialgebroids

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Let (E,E)(E,E^*) be a Lie bialgebroid. A quantization is a topological deformation of the standard associative bialgebroid structure on the universal enveloping algebra UR(E)\mathcal{U}_{R}(E), with RC(M)R\equiv\mathscr C^\infty(\mathscr{M}), whose semiclassicalisation reproduces the given Lie bialgebroid structure; such a quantum object is called a quantum groupoid. Xu's conjecture. Every Lie bialgebroid admits a quantization as a quantum groupoid. Quantization is known for Lie bialgebroids associated with Poisson manifolds, regular triangular Lie algebroids, and generic triangular Lie bialgebroids, but the conjecture remains open for a generic Lie bialgebroid.

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Sources & referencesView supporting material

Primary source

Kevin Morand, “A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids”, arXiv:2102.07593 (2022).

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