The aim of this talk is the study of quantization of coisotropic Lie subalgebras of Lie bialgebras. A coisotropic Lie subalgebra c of a Lie bialgebra g is a Lie subalgebra which is also a Lie coideal. The problem of quantization of coisotropic Lie subalgebra was set forth by V. Drinfeld, in his study of quantization of Poisson homogeneous spaces G/C. These problems are closely related by the duality principle established by N. Ciccoli et F. Gavarini. In this talk, we search for an answer to this quantization problem in the universal setting. We will first recall the universal quantization of Lie bialgebras, then we will try to construct step by step a universal quantization of coisotropic Lie subalgebras. This will allow us to find an obstruction to the universal quantization using a third order universal quantization given by V. Drinfeld. Finally, we will show that this obstruction vanishes for simple Lie algebras.
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