Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Abstract

Let $\mathfrak{g}$ be a finite dimensional split semisimple Lie algebra and $\lambda$ a weight of $\mathfrak{g}$. Let $F$ be the algebra of quantized regular functions on the connected simply connected group $G$ corresponding to $\mathfrak{g}$. In the present paper we introduce a certain subspace $F^{\prime }$ of $F$ (which is not necessary a subalgebra of $F$) and endow it with an associative $\star$-product using the so-called reduced fusion element. We prove that the algebra $(F^{\prime },\star )$ is isomorphic to $(L(\lambda ){)}_{fin}$, where $L(\lambda )$ is the irreducible highest weight ${\check{U}}_q\mathfrak{g}$-module and "$fin$" stands for the subalgebra of the locally finite elements with respect to the adjoint action of ${\check{U}}_q\mathfrak{g}$. The introduced $\star$-product has some limiting properties what enables us to prove Kostant's problem for ${\check{U}}_q\mathfrak{g}$ in certain cases. We remind the reader that this means that $(L(\lambda ){)}_{fin}$ coincides with the image of $\check{U}_q\g$ in $L(\lambda )$. We also note that if $\lambda$ is such that $<\lambda ,{\alpha }_i^{\vee }>=0$ for some simple roots ${\alpha }_i$ and generic otherwise, then $(F,\star )$ is a ${\check{U}}_q\mathfrak{g}$-invariant quantization of the Poisson homogeneous space $G/K$, where $K$ is the stabilizer of $\lambda$.

20 pages; introduction revised in v6

References in corpus (5)

• Deformation quantization of Poisson manifolds, I
• On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization
• A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups
• On classification of dynamical r-matrices
• Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras