Semibounded Representations and Invariant Cones in Infinite Dimensional Lie Algebras

Abstract

For the Lie algebra g of a connected infinite-dimensional Lie group G, there is a natural duality between so-called semi-equicontinuous weak- * -closed convex Ad * (G)-invariant subsets of the dual space g ′ and Ad(G)-invariant lower semicontinuous positively homogeneous convex functions on open convex cones in g. In this survey, we discuss various aspects of this duality and some of its applications to a more systematic understanding of open invariant cones and convexity properties of coadjoint orbits. In particular, we show that root decompositions with respect to elliptic Cartan subalgebras provide powerful tools for important classes of infinite Lie algebras, such as completions of locally finite Lie algebras, Kac-Moody algebras and twisted loop algebras with infinite-dimensional range spaces. We also formulate various open problems. Mathematics Subject Classification: 22E65, 22E45

Schur–Weyl Theory for C*‐algebras

“…Another intrinsically defined object is the full momentum set I A π λ . As we have seen in [26,Theorem 7.1], it does not separate the unitary representations of U(A) obtained by restricting inequivalent algebra representations with the same kernel, and such representations exist for separable C * -algebras not of type I [11, Theorem 9.1], [37]. Remark 4.10 (a) However, [17,Theorem 1.1] asserts that the normal subgroup of asymptotically inner automorphisms of a separable C * -algebra A acts transitively on Ext(I π ) for any irreducible representation π of A.…”

“…The broad issue of developing a systematic approach to the classification of representations of the unitary groups of C * -algebras was raised in [33], and this is precisely the problem which we address in the present work-by using different tools, however: namely, the momentum sets of unitary representations (cf. [22], [24], [25], and [26]). For a Banach-Lie group G and a normcontinuous unitary representation π : G → U(H), there are two variants of the momentum set.…”

Classification of Positive Energy Representations of the Virasoro Group

“…be the reproducing kernel of H ρ μ , as defined in (20). By Proposition 3.9(iv), K ρ μ is holomorphic.…”

On differentiable vectors for representations of infinite dimensional Lie groups

“…It is a rule of thumb that complex analytic methods apply well to bounded and semibounded unitary representations, where the latter class is defined by the condition that the operators id π(x) are uniformly bounded above on some open subset of the Lie algebra (see [45] for a survey on semibounded representations). Beyond semibounded representations one has to rely on real analytic techniques.…”

“…Even more important is the restricted symplectic group Sp res (H) consisting of all elements for which g g − 1 is Hilbert-Schmidt (cf. [45]). The latter group has a by far richer (projective) representation theory than the former.…”