Schatten classes and traces on compact groups

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian, then it follows readily that$d_{m,n}(G)$is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group$G$: for any nonopen closed subgroup$H$of$G$, the sets$\{g\in G: g^k\in H\}$for both$k=m$and$k=n$have Haar measure$0$. Indeed, we show that if a compact group$G$satisfies this condition and if$d_{m,n}(G)\gt 0$, then the identity component of$G$is abelian.

It is shown that every compact group G is a -simplicial space where Q is any field of characteristic zero.As a consequence it follows that G satisfies a variation of the Lefschetz fixed point theorem.It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2].The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order.It is shown that every compact group G satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.1. Theorem.Let G be a compact group.Then G is an inverse limit of Lie groups G = proj lim GM and if f : G" -> G is a homomorphism belonging to this inverse system, then there is a commutative diagram of covering compact Lie groups Presented to the Society, January 18,

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

This paper explores further the connection between Langlands duality and T-duality for compact simple Lie groups, which appeared in work of Daenzer-Van Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to isomorphisms of twisted K-groups, but that these K-groups are trivial except in the simplest case of SU(2) and SO(3). Along the way we compute explicitly the map on $H^3$ induced by a covering of compact simple Lie groups, which is either 1 or 2 depending in a complicated way on the type of the groups involved. We also give a new method for computing twisted K-theory using the Segal spectral sequence, giving simpler computations of certain twisted K-theory groups of compact Lie groups relevant for D-brane charges in WZW theories and rank-level dualities. Finally we study a duality for orientifolds based on complex Lie groups with an involution.

In this paper, we determine explicitly the spectra of the Dirichlet problems of some domains in simply connected compact simple Lie groups. As their applications, combining results of Hoffman [6] and Mori [10], we can state some stability conditions of these domains for the standard minimal isometric immersions into unit spheres.

Homogeneous Einstein [formula omitted]-metrics on compact simple Lie groups and spheres

There exists an infinite family of p p -compact groups whose Weyl groups correspond to the finite p p -adic pseudoreflection groups G ( q , r , n ) G(q,r,n) of family 2a in the Clark-Ewing list. In this paper we study these p p -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical J J -homomorphism. Finally, we also describe a faithful complexification homomorphism from these p p -compact groups to the p p -completion of unitary compact Lie groups.

We classify elementary abelian 2-subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian \(p\)-subgroups of compact simple Lie groups (equivalently, complex linear algebraic simple groups) of adjoint type started in Griess (Geom Dedicata 39(3):253–305, 1991).

In this paper the structure of all open semigroups in any compact topological group is determined. We can then extend this result to a classification of the angular semigroups of any abelian topological group H which contains a compact open subgroup K. Coupling this with the results for Euclidean n-space, we obtain an essentially complete description of the angular semigroups of an arbitrary locally compact group. For compact groups, the result is remarkably simple.

Since the solution of the celebrated Hilbert’s Fifth Problem, one knows that a locally euclidean topological group is a Lie group. In accordance with Hilbert’s original idea of getting rid of the differentiability hypothesis in Lie theory, one can look for a homotopy analogue of this result. Unfortunately the most straightforward idea is false: Hilton, Roitberg and Stasheff have exhibited a topological group having the homotopy type of a compact manifold and which is not homotopy equivalent to any Lie group (see [Sta69]). Nevertheless, the problem of finding a homotopy characterisation of Lie groups remains open and has been intensively investigated for the last three decades. Obviously, the first step of such a programme is a better understanding of the homotopy properties of Lie groups. One has to restrict this study to the class of compact connected Lie groups: the Iwasawa decomposition tells us that any connected Lie group has the homotopy type of a compact one.

(II) fedN(X) = ^ e ± i ^ i J k is a tempered distributional function of t. The classical trace formula says that the singular support of (II) is contained in the length spectrum of X. Moreover, under suitable hypotheses on geodesic flow, the trace formula gives considerable information about the singularities in (II). (See [DG and C].) There is a fairly straightforward (and not terribly interesting) generalization of the trace formula to vector bundles. (See, for instance, the introduction to [DG].) We will be concerned in this article with a much more subtle generalization inspired by recent articles of Hogreve, Potthoff, and Schrader [HPS], and Schrader and Taylor [ST] in Communications in Mathematical Physics. Let G be a compact Lie group and ir: P —> X a principle G-bundle with connection. Given a finite-dimensional unitary representation, p, of G we will denote by Ep the vector bundle over X associated with p and by Dp the associated connection. Now consider a ladder {pe, e = 1,2,...} of irreducible representations of G. (This means that the maximal weight of pe is e times the maximal weight of p\.) For given e let Afc,e? # = 1, 2, O, . . . be the spectrum of the Laplace operator on C(Epe): Ae = D*peDpe + e .

Optimization problems over compact Lie groups have been extensively studied due to their broad applications in linear programming and optimal control. This paper analyzes least square problems over a noncompact Lie group, the symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of control over dynamical transformations in classical mechanics and quantum optics. The critical topology for minimizing the Frobenius distance from a target symplectic transformation is solved. It is shown that the critical points include a unique local minimum and a number of saddle points. The topology is more complicated than those of previously studied problems on compact Lie groups such as the orthogonal and unitary groups because the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group brings significant nonlinearity to the problem. Nonetheless, the lack of traps guarantees the global convergence of local optimization algorithms.

Determination of compact Lie groups with the Borsuk-Ulam property

Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on compact Lie groups, we prove that, under certain algebraic and topological conditions, the cohomology associated to left-invariant elliptic, and even hypocomplex, involutive structures on compact Lie groups can be computed by using only Lie algebras, thus reducing the analytical problem to a purely algebraic one. The main tool is the Leray spectral sequence that connects the result obtained Chevalley and Eilenberg to a result by Bott on the Dolbeault cohomology of a homogeneous manifold.

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing