arXiv 0908.03492009-08math.QAmath.RT

Poisson齐性空间的等变量子化与Kostant问题

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

E. Karolinsky, A. Stolin, V. Tarasov

备注：: 20 pages; introduction revised in v6

摘要：

设$\mathfrak g$是有限维分裂半单李代数，$λ$是$\mathfrak g$的权.设$F$是连通单连通群$G$上对应于$\mathfrak g$的量子正则函数的代数。本文引入了$F$的一个子空间$F'$（它不一定是$F$的子代数），并利用所谓的约化融合元赋予它一个结合$\star$ -积.证明了代数$(F',\star)$与$(L(λ))_{fin}$同构，其中$L(λ)$是不可约的最高权$\check{U}_q\mathfrak g$ -模，“$fin$”代表关于$\check{U}_q\mathfrak g$的伴随作用的局部有限元的子代数。引入的$\star$ -产品有一些限制性的性质，使我们能够证明Kostant的问题，$\check{U}_q\mathfrak g$在某些情况下。我们提醒读者，这意味着$(L(λ))_{fin}$与$L(λ)$中的$\check{U}_q\g$的图像一致。我们还注意到，如果$λ$是这样的，$<λ,α_i^\vee>=0$对于一些简单的根$α_i$和一般的，那么$(F,\star)$是一个$\check{U}_q\mathfrak g$不变的量化的泊松齐次空间$G/K$，其中$K$是稳定的$λ$。

英文摘要：

Let $\mathfrak g$ be a finite dimensional split semisimple Lie algebra and $λ$ 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 $(L(λ))_{fin}$, where $L(λ)$ 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 $(L(λ))_{fin}$ coincides with the image of $\check{U}_q\g$ in $L(λ)$. We also note that if $λ$ is such that $<λ,α_i^\vee>=0$ for some simple roots $α_i$ 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 $λ$.

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摘要：

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