Semibounded representations and invariant cones in infinite dimensional Lie algebras
Semibounded representations and invariant cones in infinite dimensional Lie algebras
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无限维李代数中的半有界表示和不变锥
DOI：
10.1142/s1793744210000132
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
K. Neeb
中科院分区：
文献类型：
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作者：
K. Neeb
可能是无限维的李群$G$的酉表示称为半有界，如果来自导出表示的相应运算符$i\dd\pi（x）$从上到下在李代数$\g$的某个非空开子集上一致有界。在本文的第一部分中，我们解释了这个概念如何导致一个富有成效的相互作用的领域之间的无限维凸性，李理论，辛几何（动量映射）和复杂的分析。在这里，李代数中的开不变锥起着中心作用，半有界表示与$C^*$-代数有着有趣的联系，这与有限维李群的群$C^*$-代数的经典用法有很大不同。第二部分是专门详细讨论半有界表示的圆的群，Virasoro组，metaplectic表示的玻色子福克空间和自旋表示的费米子福克空间。
A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semi-bounded if the corresponding operators $i\dd\pi(x)$ from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra $\g$. In the first part of the present paper we explain how this concept leads to a fruitful interaction between the areas of infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps) and complex analysis. Here open invariant cones in Lie algebras play a central role and semibounded representations have interesting connections to $C^*$-algebras which are quite different from the classical use of the group $C^*$-algebra of a finite dimensional Lie group. The second half is devoted to a detailed discussion of semibounded representations of the diffeomorphism group of the circle, the Virasoro group, the metaplectic representation on the bosonic Fock space and the spin representation on fermionic Fock space.