Differentiable vectors and unitary representations of Fréchet–Lie supergroups
Differentiable vectors and unitary representations of Fréchet–Lie supergroups
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Fréchet-Lie 超群的可微向量和酉表示
DOI：
10.1007/s00209-012-1142-5
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发表时间：
2012
影响因子：
0.8
通讯作者：
Hadi Salmasian
中科院分区：
文献类型：
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作者：
K. Neeb;Hadi Salmasian
一个局部凸李群G具有Trotter性质，如果，对于每 $$x_1, x_2 \in \mathfrak{g }$$， $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$的紧子集上一致成立 $$\mathbb{R }$$． 所有的局部指数李群都具有这一性质，而且紧致光滑流形上主束的自同构群也具有这一性质。本文的一个关键结果是，如果G具有Trotter性质， $$\pi : G \rightarrow {\mathrm{GL}}(V)$$ 是G在局部凸空间上的连续表示，并且 $$v \in V$$ 一个向量是这样的吗 $$\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v$$ 存在于每一个 $$x \in \mathfrak{g }$$，然后是地图 $$\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v$$ 是线性的。利用这一结果，我们得出，对于可度量的局部凸空间上的局部指数fr<s:1>切-李群G的表示，的空间 $$\mathcal{C }^{k}$$向量与算子的k倍积的公域重合 $$\overline{\mathtt{d}\pi }(x)$$． 对于Hilbert空间上的酉表示，局部指数性的假设可以被削弱为Trotter性质。作为一个应用，我们证明了平滑（响应）。frsamet - lie超群的解析酉表示 $$(G,\mathfrak{g })$$ 其中G具有Trotter性质，即算子的公域 $$\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}$$ 总是可以扩展到平滑的空间。（解析）G的向量。
A locally convex Lie group G has the Trotter property if, for every $$x_1, x_2 \in \mathfrak{g }$$, $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$holds uniformly on compact subsets of $$\mathbb{R }$$. All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if G has the Trotter property, $$\pi : G \rightarrow {\mathrm{GL}}(V)$$ is a continuous representation of G on a locally convex space, and $$v \in V$$ is a vector such that $$\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v$$ exists for every $$x \in \mathfrak{g }$$, then the map $$\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v$$ is linear. Using this result we conclude that, for a representation of a locally exponential Fréchet–Lie group G on a metrizable locally convex space, the space of $$\mathcal{C }^{k}$$-vectors coincides with the common domain of the k-fold products of the operators $$\overline{\mathtt{d}\pi }(x)$$. For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Fréchet–Lie supergroups $$(G,\mathfrak{g })$$ where G has the Trotter property, the common domain of the operators of $$\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}$$ can always be extended to the space of smooth (resp., analytic) vectors for G.