标题： Hilbert-Lie群的协变投影表示 显示英文标题

标题： Covariant projective representations of Hilbert-Lie groups

摘要： 希尔伯特-李群是李代数为实希尔伯特空间的李群，其标量积在伴随作用下是不变的。 这些无限维李群是紧致李群最接近的亲戚。 在这里，我们从不同角度研究这些群的酉表示。 首先，我们讨论范数连续的，也称为有界的表示：它们对于简单群来说是众所周知的，但一般情况更为复杂。 我们的第一个主要结果是用余根集的有界性来表征所有有界表示的离散可分解性。 我们还证明，如果余根集是无界的，则存在类型的II和III有界表示。 其次，我们利用一个单参数自同构群的协变性来实现某些正则性。 在这里，我们开发了一种基于半李群的扰动理论，将问题简化到固定一个“极大环面”的情况，从而可以研究相容的权分解。 第三，我们将上下文扩展到对一个单参数自同构群具有协变性的投影表示。 在这里，重要的表示族来自于“有界极值权”，对于这些表示，可以显式地确定相应的中心扩张，以及所有存在协变扩张的单参数群。 显示英文摘要

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.