Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G^∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G^∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G, then h is the super Lie algebra of a sub-super Lie group of G. Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G. We also show that if H is a closed sub-super Lie group of a super Lie group G, then G→G/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G^∞ category.     

On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps \(P_k:G\rightarrow G:g\mapsto g^k\). We give criteria for the density of \(P_k(G)\) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if \(\mathrm{Reg}(G)\) is the set of regular elements of G, then \(P_k(G)\cap \mathrm{Reg}(G)\) is closed in \(\mathrm{Reg}(G)\). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps \(P_k\). In linear Lie groups, weak exponentiality reduces to the density of \(P_2(G)\). We also prove that the density of the image of \(P_k\) for G implies the same for any connected full rank subgroup.     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Homogeneous Metrics with Nonnegative Curvature

Given compact Lie groups H⊂G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.     

On coset-spaces of compact Lie groups by subgroups of corank 2

Let G be a simple compact connected simply connected Lie group, H its connected Lie subgroup of corank 2 which coincides with the commutator group of the centralizer of a torus, and let Sam(G/H) = 0. We prove that if a compact connected simply connected Lie group G' acts transitively and locally effectively on the manifold G/H, then G' is isomorphic to G. if the root system of G consists of roots of the same length, then the action of G' on G/H is similar to the action of G.     

On a family of lie algebras related to homogeneous surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space ?³ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author’s works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups G ? Aff(3, ?) in the complex tangent space T ^? _p M of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups G on the spaces T _p ^? M for affine homogeneous strictly pseudoconvex surfaces of general position in ?³ that are different from quadrics.     

Nonnegatively curved homogeneous metrics in low dimensions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists. In this case, the homogeneous space G/H is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups (H, K, G) whether non-negative curvature is maintained for small deformations. Building on the work of Schwachhöfer and Tapp (J. Geom. Anal. 19(4):929–943, 2009), we examine all G-invariant fibration metrics on G/H for G a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of non-negative sectional curvature.     

Examples of manifolds which are homogeneous spaces of lie groups of arbitrarily large dimension

We discuss the possibility for a given homogeneous space M=G/H to be diffeomorphic to each term of a sequence (G_n/H_n) of coset spaces, where the G_n 's are connected real Lie groups of arbitrarily large (finite) dimensions. We prove that this possibility does indeed arise when M is the total space of a convenient circle bundle. Typical examples are provided by odd dimensional spheres of dimension at least 3. Our main tool for this result is the theory of connections on principal bundles.     

On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces

Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space ?(Γ,G,H) is a Hausdorff space, where Γ acts on the threadlike homogeneous space G/H as a discontinuous subgroup. Whenever k = rank(Γ) > 3, this space is also shown to be endowed with a smooth manifold structure. But if k = 3, then ?(Γ,G,H) admits a smooth manifold structure as its open dense subset. These phenomena are strongly linked to the features of adjoint orbits of the basis group G on the parameter space ?(Γ,G,H) (which is semi-algebraic in this case) and specifically to their dimensions, as it will be seen throughout the paper. This also allows to provide a proof of the Local Rigidity Conjecture in this setup.     

A class of totally geodesic foliations of Lie groups

This paper is devoted to classifying the foliations ofG with leaves of the formgKh ^?1 whereG is a compact, connected and simply connected Lie group andK is a connected closed subgroup ofG such thatG/K is a rank-1 Riemannian symmetric space. In the case whenG/K=S ⁿ, the homotopy type of space of such foliations is also given.     

Homogeneous spaces with invariant projectively flat affine connections

We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.

     

Stability of Discontinuous Groups Acting on Homogeneous Spaces

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ,G,H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ,H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ,G) and study the case of Heisenberg groups.     

On permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωⁿ⁺¹G→Ωⁿ⁺¹(G/H)→ΩⁿH for n=1,2, and the Eilenberg-Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H_∗(Ωⁿ(G/H);F₂). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by $\text{D}$(G) the space of C^∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U$\text{G}$_$\text{c}$ of the complexification of the Lie algebra of G for which the linear form $\text{? ψ(π(u 1 ?))}$ on $\text{D}$(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U$\text{G}$_$\text{c}$.     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.