Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

TL;DR

This paper introduces a new associative star-product on a subspace of quantized functions related to semisimple Lie algebras, enabling progress on Kostant's problem and Poisson space quantization.

Contribution

It constructs a specific star-product on a subspace of quantized functions and proves its isomorphism to a locally finite module, advancing understanding of Kostant's problem and Poisson homogeneous space quantization.

Findings

The algebra (F',*) is isomorphic to the locally finite part of an irreducible module.

The star-product has limiting properties that solve Kostant's problem in certain cases.

Provides a U_q(g)-invariant quantization of Poisson homogeneous spaces.

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 $F'$ 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 $(F',\star)$ 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 $\lambda$.