Hilbert-Lie群的协变射影表示
Covariant projective representations of Hilbert-Lie groups
AI总结：
本文研究Hilbert-Lie群的酉表示，刻画有界表示的离散可分解性，并利用协变性发展微扰理论，最后扩展到射影表示并显式确定中心扩张。
摘要：
Hilbert-Lie群是其李代数为实Hilbert空间且其内积在伴随作用下不变的Lie群。这些无穷维Lie群是与紧Lie群最接近的亲属。这里我们从多个角度研究这些群的酉表示。首先，我们处理范数连续（也称为有界）表示：它们对于单群是众所周知的，但一般情形更为复杂。我们的第一个主要结果是关于所有有界表示的离散可分解性的刻画，该刻画以余根集合的有界性为条件。我们还证明了如果余根集合无界，则存在类型II和III的有界表示。其次，我们利用关于单参数自同构群的协变性来实现某种正则性。这里我们发展了基于半Lie群的微扰理论，将问题归结为固定一个“极大环面”的情形，从而可以研究相容的权分解。第三，我们将上下文扩展到对单参数自同构群协变的射影表示。这里重要的表示族来源于“有界极值权”，对于这些表示，相应的中心扩张可以显式确定，连同所有存在协变扩张的单参数群。
英文摘要：
Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study unitary representations of these groups from various perspectives. First, we address norm-continuous, also called bounded, representations: they are well-known for simple groups, but the general picture is more complicated. Our first main result is a characterization of the discrete decomposability of all bounded representations in terms of boundedness of the set of coroots. We also show that bounded representations of type II and III exist if the set of coroots is unbounded. Second, we use covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half Lie groups that reduces matters to the case where a ``maximal torus'' is fixed, so that compatible weight decompositions can be studied. Third, we extend the context to projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from ``bounded extremal weights'', and for these the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists.
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