Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebras B ∨ of sl(N ) the explicit expressions are obtained for the twist element F, universal R-matrix and the corresponding canonical element T . It is shown that the twisted Hopf algebra U F (B ∨ ) is self dual. The cohomological properties of the involved Lie bialgebras are studied to justify the existence of a contraction from the Dinfeld-Jimbo quantization to the jordanian one. The construction of the twist is generalized to a certain type of inhomogenious Lie algebras.

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and their Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.

We prove that the reflection equation (RE) algebra L R associated with a finite dimensional representation of a quasitriangular Hopf algebra H is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that L R is a module algebra over the twisted tensor square H R ⊗ H and the double D(H). We define FRT-and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.

Let G be a simple complex classical group and g its Lie algebra. Let U (g) be the Drinfeld-Jimbo quantization of the universal enveloping algebra U (g). We construct an explicit U (g)-equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers.