Primitive Roots of Unity in H-Manifolds

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.

Homotopy uniqueness of classifying spaces of compact connected lie groups at primes dividing the order of the weyl group

Introduction .220 Chapter I. A fixed point theorem for finite diffeomorphism groups generated by reflections 221 Chapter II. Geometric weight system and a fixed point theorem for compact connected Lie groups.... 231 1. Actions of compact simple Lie groups of rank 1.232 2. Basic properties of the geometric weight system.237 3. A fixed point theorem for compact connected Lie groups.240 Chapter III. Classification of the types of principal isotropy subgroups for actions of compact connected Lie groups on acyclic manifolds .242 1. Classification of principal orbit types for actions of simple compact connected Lie groups on acyclic manifolds .243 2. Classification of principal orbit types for actions of general compact connected Lie groups .247 References ...... 255

Bu tez bes bolumden olusmaktadir. Birinci bolum giris kismina ayrilmistir. Ikinci bolumde tez icin gerekli olan tarif ve teoremler verilmistir. Ucuncu bolumde, homotopi ve demet teorisi ele alinmistir. Dorduncu bolumde, Lie gruplari ve manifoldlar incelenerek bazi sonuclar takdim edilmistir. Son olarak besinci bolumde ise Kompakt Irtibatli Lie gruplari uzerinde demetler teskil edilerek bazi karakterizasyonlar verilmis ve Homotopi Normalite tarif edilerek demet morfizmi altinda homotopi normalligin korundugu gosterilmistir.Abstract This thesis consists of five chapters. The first chapter has been devoted to the introduction. In the second chapter, it was given fundamental definitions and theorems that will be needed for this thesis. In the third chapter, homotopy and sheaf theory were included. In the fourth chapter, investigating Lie groups some corollaries were established. Finally in the fifth chapter, by constructing sheaves on compact connected Lie groups, some characterizations were given and by defining homotopy normality, it was shown that homotopy normality is preserved under sheaves morphisms.

This chapter is about structure theory for compact Lie groups, and a certain amount of representation theory is needed for the development. The first section gives examples of group representations and shows how to construct new representations from old ones by using tensor products and the symmetric and exterior algebras.In the abstract representation theory for compact groups, the basic result is Schur’s Lemma, from which the Schur orthogonality relations follow. A deeper result is the Peter-Weyl Theorem, which guarantees a rich supply of irreducible representations. From the Peter-Weyl Theorem it follows that any compact Lie group can be realized as a group of real or complex matrices.The Lie algebra of a compact Lie group admits an invariant inner product, and consequently such a Lie algebra is reductive. From Chapter I it is known that a reductive Lie algebra is always the direct sum of its center and its commutator subalgebra. In the case of the Lie algebra of a compact connected Lie group, the analytic subgroups corresponding to the center and the commutator subalgebra are closed. Consequently the structure theory of compact connected Lie groups in many respects reduces to the semisimple case.If T is a maximal torus of a compact connected Lie group G, then each element of G is conjugate to a member of T. It follows that the exponential map for G is onto and that the centralizer of a torus is always connected. The analytically defined Weyl group W(G, T) is the quotient of the normalizer of T by the centralizer of T, and it coincides with the Weyl group of the underlying root system.Weyl ’s Theorem says that the fundamental group of a compact semsimple Lie group G is finite. Hence the universal covering group of G is compact.KeywordsCompact GroupInvariant SubspaceWeyl GroupMaximal TorusCartan SubalgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The first main results are that simply connected compact semisimple Lie groups are in one-one correspondence with abstract Cartan matrices and their associated Dynkin diagrams and that the outer automorphisms of such a group correspond exactly to automorphisms of the Dynkin diagram. The remainder of the first section prepares for the definition of a reductive Lie group: A compact connected Lie group has a complexification that is unique up to holomorphic isomorphism. A semisimple Lie group of matrices is topologically closed and has finite center.Reductive Lie groups G are defined as 4-tuples (G, K,θ, B) satisfying certain compatibility conditions. Here G is a Lie group, K is a compact subgroup, B is an involution of the Lie algebra go of G, and B is a bilinear form on go. Examples include semisimple Lie groups with finite center, any connected closed linear group closed under conjugate transpose, and the centralizer in a reductive group of a θ stable abelian subalgebra of the Lie algebra. The involution θ, which is called the “Cartan involution” of the Lie algebra, is the differential of a global Cartan involution Θ of G. In terms of Θ, G has a global Cartan decomposition that generalizes the polar decomposition of matrices.A number of properties of semisimple Lie groups with finite center generalize to reductive Lie groups. Among these are the conjugacy of the maximal abelian subspaces of the —1 eigenspace p0 of θ, the theory of restricted roots, the Iwasawa decomposition, and properties of Cartan subalgebras. The chapter addresses also some properties not discussed in Chapter VI, such as the K Ap K decomposition and the Bruhat decomposition. Here A p is the analytic subgroup corresponding to a maximal abelian subspace of p0.The degree of disconnectedness of the subgroup M p = Z K (A p) controls the disconnectedness of many other subgroups of G. The most complete description of M p is in the case that G has a complexification, and then serious results from Chapter V about representation theory play a decisive role.Parabolic subgroups are closed subgroups containing a conjugate of M p A p N p. They are parametrized up to conjugacy by subsets of simple restricted roots. A Cartan subgroup is defined to be the centralizer of a Cartan subalgebra. It has only finitely many components, and each regular element of G lies in one and only one Cartan subgroup of G. When G has a complexification, the component structure of Cartan subgroups can be identified in terms of the elements that generate M p.A reductive Lie group G that is semisimple has the property that G/K admits a complex structure with G acting holomorphically if and only if the centralizer in go of the center of the Lie algebra to of K is just to. In this case, G/K may be realized as a bounded domain in some ℂn by means of the Harish-Chandra decomposition. The proof of the Harish-Chandra decomposition uses facts about parabolic subgroups. The spaces G/K of this kind may be classified easily by inspection of the classification of simple real Lie algebras in Chapter VI.KeywordsParabolic SubgroupCartan SubalgebraCartan SubgroupIwasawa DecompositionParabolic SubalgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Actions of compact Lie groups on spaces X X with H ∗ ( X , Q ) ≅ Q [ x 1 , … , x n ] / I 0 {H^{\ast }}(X,{\mathbf {Q}}) \cong {\mathbf {Q}}[{x_1}, \ldots ,{x_n}]/{I_0} , Q ∈ I 0 Q \in {I_0} a definite quadratic form, deg ⁡ x i = 2 \deg {x_i} = 2 , are considered. It is shown that the existence of an effective action of a compact Lie group G G on such an X X implies χ ( X ) ≡ O ( | W G | ) \chi (X) \equiv O(|WG|) , where χ ( X ) \chi (X) is the Euler characteristic of X X and | W G | |WG| means the order of the Weyl group of G G . Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree N t ( G / T ) {N_t}(G/T) is equal to dim ⁡ G \dim G if G G is a compact connected Lie group and T ⊂ G T \subset G its maximal torus. Effective actions of compact connected Lie groups K K on G / T G/T with dim ⁡ K = dim ⁡ G \dim K = \dim G are completely classified.

Let M be a manifold (= connected, separable, locally euclidean space) of dimension n + 1, and G a compact connected Lie group acting on M in such a way that there is at least one n-dimensional orbit.2 In this paper, we show that the space of orbits M/G is homeomorphic to one of (i) a circle, (ii) an open interval, (iii) a half-open interval, or (iv) a closed interval. Moreover, there exists a subgroup N C G such that in (i) and (ii) M is homeomorphic to (G/N) X (M/G). In (iii) there is a subgroup K D N such that K/N is an r-sphere for some 0 < r < n, and such that M is homeomorphic to (G/N) X (M/G) with G/N identified to G/K over the end point. In (iv), a similar situation holds for subgroups K1 and K2 so that K1/N and K2/N are spheres (not necessarily of the same dimension). This, then, completely determines all manifolds admitting such a transformation group, and, moreover, the group G must act on M in the obvious way. (Thus, the operation of G is equivalent to G acting differentiably). In particular, this will show that the only two-dimensional manifolds which admit a compact connected Lie group operating non-trivially are as follows: (1) the torus, (2) the infinite circular cylinder, (3) the plane, (4) the (open) Moebius strip, (5) the sphere, (6) the Klein bottle, and (7) the projective plane: that is, precisely the Klein spaces. Each of these admits a circle group of operators, and in fact only the circle group operates effectively and non-transitively (except, of course, the trivial-i.e., one point-Lie group). A further application determines all compact, connected and effective Lie groups which can operate on a 3-dimensional manifold. All those 3-dimensional manifolds which admit a compact Lie group acting in such a way that there is a 2-dimensional orbit are determined. There are just 15 such. The author wishes to express his thanks to Dr. P. Conner, Profs. D. Montgomery, M. Goto, and A. L. Shields for their interest and suggestions.

Maxwell’s equations for free fields are studied on the underlying C∞ manifold of a compact Lie group. The formulation is in terms of exterior differential forms as given by Wheeler. It is found that on a compact connected Lie group there are no free electromagnetic fields. The results obtained are essentially a physical interpretation of the well-known theorem that the second Betti number of a compact semisimple Lie group is zero.

Given a compact connected Lie group G with Lie algebra , we prove the connectedness of the fibres for the following two maps: 1. defined by g ↦ π(Ad(g)A), , where is a Cartan subalgebra of and is the orthogonal projection with respect to the G-invariant inner product ⟨·, ·⟩ on . 2. ξ A,C : G → ℝ defined by g ↦ ⟨Ad (g)A, C⟩, . As a corollary of (2), we obtain a new proof of the convexity theorem in Tam [T.Y. Tam, Convexity of generalized numerical range associated with a compact Lie group, J. Austral. Math. Soc. 70 (2002), pp. 57–66].

STABLE ERGODICITY OF SKEW EXTENSIONS BY COMPACT LIE GROUPS

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.

Compact Lie groups are ‘perfect’ entities in modern mathematics because: (a) They are differentiable (and even analytical) manifolds of finite dimension and therefore can be analysed with the help of the most powerful tool of mathematics: ordinary and partial differential operators (equations); (b) Denoting by £(G) or L(G) = T e (G) the tangent space to G in unity e, we find that L(G) can be identified with the set of left-invariant vector fields on G (or with the set of appropriate differential operators of order one). Thus L(G) naturally becomes a Lie algebra: on L(G) there exists a structure [·, ·] which satisfies the axioms of Lie algebra (see below); (c) This rich algebraic structure, in turn, makes it possible to apply the powerful algebraic tools (theory of Lie algebras); (d) An abstract (finite dimensional) Lie algebra g defines (up to an isomorphism) a simply connected Lie group G such that £(G) = g. Therefore investigations of the Lie group G can be to large extent replaced by investigations of its Lie algebra g; (e) Compactness of a Lie group G makes it possible to use the theory of representations of compact groups developed by Weyl and Peter. This theory is extraordinarily beautiful and simple: it reduces to large extend to the spectral theory of compact operators; (f) And finally every complex, compact (connected) commutative Lie group is a torus. Moreover, as it was shown by E. Cartan, every compact Lie group contains a maximal torus T (any other such a torus is conjugated T 1 = gTg −1 ); and any point g 0 of the group G belongs to some maximal torus: G = U g gTg −1. This fundamental Cartan theorem makes it possible to apply to T (and thus to the whole of G) the theory of representations of abelian groups: every representation of the torus T decomposes into the finite (and orthogonal) sum of unit representations (over ℂ), the so called weights. Thus the knowledge of these weights is the fundamental element of the theory of representations of compact Lie groups.

In this paper, we study the geometry of Lie groups with bi-invariant Finsler metrics. We first show that every compact Lie group admits a bi-invariant Finsler metric. Then, we prove that every compact connected Lie group is a symmetric Finsler space with respect to the bi-invariant absolute homogeneous Finsler metric. Finally, we show that if G is a Lie group endowed with a bi-invariant Finsler metric, then, there exists a bi-invariant Riemanninan metric on G such that its Levi-Civita connection coincides the connection of F.

When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this chapter, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance.