Bernstein's theorem for compact groups

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie bialgebra. This result is extended in Liu et al. (1998) to a classification of the Poisson homogeneous spaces of a Poisson groupoid (G⇉P, πG) via a special class of Dirac structures in the Courant algebroid defined by the corresponding Lie bialgebroid. It is hence natural to ask what corresponds via these classifications, or extensions of them, to arbitrary Dirac structures in the Drinfel'd double or the Courant algebroid associated to a Poisson group(oid). We show in this thesis that there is a bigger class of Dirac structures that fits in this correspondence. They correspond naturally to Dirac homogeneous spaces of the Poisson group(oid). Dirac Lie groups and Dirac groupoids have been defined by Ortiz (2009) as a generalization of Poisson Lie groups, Poisson groupoids and presymplectic groupoids. We prove in an alternative manner the existence of a natural Lie bialgebra associated to a Dirac Lie group, and we find good candidates for the infinitesimal data of a Dirac groupoid; a square of morphisms of Lie algebroids associated to the multiplicative Dirac structure. These objects generalize simultaneously the Lie bialgebroid of Poisson groupoids and the IM-2-forms associated to presymplectic groupoids (Bursztyn et al. (2004)). We show also the existence of a Courant algebroid associated to these algebroids. This Courant algebroid structure is induced in a natural way by the ambient standard Courant algebroid TG ×G T*G and is isomorphic to the Courant algebroid AG ×P A*G in the Poisson case. We show that Dirac homogeneous spaces of Dirac Lie groups (respectively Dirac groupoids) correspond to certain classes of Dirac structures in the Drinfel'd double of the Lie bialgebra (respectively in the Courant algebroid). These results generalize the classification theorems in Drinfel'd (1993) and Liu et al. (1998). Along the way, we study involutive multiplicative foliations on Lie groupoids and show when and how there is a natural groupoid structure defined on the leaf spaces in the set of objects and the set of units of the Lie groupoid. This is a fact that is always true in the Lie group case, where a multiplicative foliation is automatically the left and right invariant image of an ideal in the Lie algebra, i.e., the leaves are the cosets of a normal subgroup of the Lie group.

On the absolute convergence of Vilenkin-Fourier series

We are concerned with conditions under which a locally compact group $G$ has a maximal compact normal subgroup $K$ and whether or not $G/K$ is a Lie group. If $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group, then $G$ is pro-Lie. If in $G$ there is a collection of closed normal subgroups $\{ {H_\alpha }\}$ such that $\cap {H_\alpha } = e$ and $G/{H_\alpha }$ is a Lie group for each $\alpha$, then $G$ is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.

Suppose that G is a locally compact group, that \({\Gamma}\) is a discrete, finitely generated group, and that $$\varphi\colon G \longrightarrow \Gamma$$ is an ‘abstract’ surjective homomorphism. We are interested in conditions which imply that \({\varphi}\) is automatically continuous. We obtain a complete answer to this question in the case where G is a topologically finitely generated locally compact abelian group or an almost connected Lie group. In these two cases the well-known structure theory for such groups G leads quickly to a solution. The question becomes much more difficult if one assumes only that G is a locally compact group. This leads to interesting questions about normal subgroups in infinite products and in ultraproducts. Łos’ theorem, the solution of the 5th Hilbert problem, and recent results by Nikolov–Segal can be combined to answer the question.

PROOF. Replacing G by a finite covering we may and will assume that G is a direct product of simple Lie groups, none of which is compact. The group In of isometries of the Euclidean n-space is known to be the semidirect product of the orthogonal group and R , RI being the normal subgroup. Therefore the projection of In onto the orthogonal group is a differentiable group morphism. Let f be a differentiable morphism from G into In. Then f followed by the projection of In onto the orthogonal group is a differentiable morphism. If this composition were nontrivial it would be nontrivial on some simple factor of G. Therefore, there would be a noncompact simple Lie group contained in the orthogonal group. This is impossible because a semisimple Lie subgroup of a compact Lie group is closed, hence compact (cf. [2, p. 128D. Thus, the image of f is contained in RW. This implies that G has a nontrivial abelian factor, unless f is the trivial morphism.

The Burnside ring A{G) of an arbitrary compact group G is defined in analogy to torn Dieck's definition for a compact Lie group.A topology is introduced on the set of conjugacy classes of closed subgroups, and a theorem of Dress on the topology of the prime spectrum of A(G) for profinite G is improved and extended to this general setting.This paper is condensed from part of my doctoral thesis [5], written under the supervision of Professor T. torn Dieck, to whom I am indebted for encouragement and advice.1.The Burnside ring and its additive structure.Suppose G is a compact group, and define a G-manifold as a differentiable manifold M together with a continuous left G-action such that, for each g G G, the translation g : M - M is a diffeomorphism.Examples are real G-representations and manifolds invariantly imbedded in such.Theorem 1.1.If G is a compact group, M a compact effective G-manifold, then G is a Lie group and M is a differentiable G-manifold.Proof.Since M is compact and locally connected, it has finitely many connected components, say M = Ji"A/,.For 1 < / < n, set G, = {g G GI g(M) = A/,}.This is a subgroup of G, and, since for any w, G M, the map /: G- M via f(g) = g-m is continuous, while G, = f~x(M), G, is clopen in G. Thus H = n"G, is a clopen normal subgroup.Now set H = {h E H\im G M, h-m = m], so that H is a normal subgroup of H and H/H acts effectively on A/,.Then since M is connected, Montgomery-Zippin [7, p. 208, Theorem 2] shows /////, to be a Lie group.Hence H/C\"xHi is again a Lie group.But since G operates effectively.Pi?//, = {h G //|Vw G M, h m = m) = {e}, so that H is a Lie group, and with it G since it is open in G.That M is then a differentiable G-manifold follows from [7, p. 208, Theorem 3].Now for any G-space X, let t(X) = {g G G|Vx G X,g x = x) be that part of G which acts trivially.Then for a G-manifold M, the G-action is derived from an effective G/i(A/)-action via the projection G-G/i(A/).Hence G/t(M) is a Lie group.

The eigenmodes of the Poincar\'e dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group $\pi_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck(\tilde{M})$, acting on the universal cover $\tilde{M}$. ($\pi_1(M)$, $\tilde{M}$) are known to be the binary icosahedral group ${\cal H}_3$ and the sphere $S^3$ respectively. Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck(\tilde{M}) \sim {\cal H}^r_3$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${\cal H}^r_3$ as normal subgroup and from a second group ${\cal H}^c_3$ which provides the icosahedral symmetries of $M$. Based on F. Klein's fundamental icosahedral ${\cal H}_3$-invariant, we construct a novel hermitian ${\cal H}_3$-invariant polynomial (generalized Casimir) operator ${\cal K}$. Its eigenstates with eigenvalues $\kappa$ quantize a complete orthogonal basis on Poincar\'{e}'s dodecahedral 3-manifold. The eigenstates of lowest degree $\lambda=12$ are 12 partners of Klein's invariant polynomial. The analysis has applications in cosmic topology \cite{LA},\cite{LE}. If the Poincar\'{e} 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of ${\cal K}$.

Slice theorem and orbit type stratification in infinite dimensions

Chapter 3 - Lie Groups

Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T T [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the n n -dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to L log ⁡ L L\log L . In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

In what follows, G will denote a compact, connected Lie group.Definition. A complex-valued function f ε L2(G) lies in the space A(G) if it can be written in the formwhereThe A(G) norm of f is the infimum of all sums (2) with respect to all decompositions (1).

If K is a closed subgroup of a compact group G, the probability that a randomly chosen pair of elements from K and G commute is denoted by Pr(K, G). Say that a subgroup K ≤ G is ϵ-central in G if Pr(〈g〉, G) ≥ ϵ for any g in K. Here 〈g〉 denotes the monothetic subgroup generated by g ∈ G. Our main result is that if K is ϵ-central in G, then there is an ϵ-bounded number e and a normal subgroup T ≤ G such that both the index [G:T] and the order of the commutator subgroup [Ke, T] are finite and ϵ-bounded. In particular, if G is a compact group for which there is ϵ > 0 such that Pr(〈g〉, G) ≥ ϵ for any g ∈ G, then there is an ϵ-bounded number e and a normal subgroup T such that both the index [G:T] and the order of [Ge, T] are finite and ϵ-bounded.

Exponential Radicals of Solvable Lie Groups

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.