(H, C)-groups with positive line bundles

Let Ĝ be a connected complex Lie group and X = Ĝ/Ĥ a complex manifold homogeneous under a holomorphic Ĝ-action. In order to understand X, e.g. how it fits into a fine classification, details of its function theory, etc., one should use as much Lie theoretic information about Ĝ as is possible. In particular it is often useful to study the orbit structure of real subgroups of Ĝ. Such orbits are usually not complex sub-manifolds of X.

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup such that $X := G/H$ is Kaehler and the codimension of the top non-vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $X$ is biholomorphic to a complex homogeneous manifold constructed using well-known basic building blocks, i.e., $\mathbb C, \mathbb C^*$, Cousin groups, and flag manifolds.

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa. The Lie algebra cohomology a la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e . in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories. Cohomology of Lie algebras: general definitions We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing. The extension of core information theoretic inequalities defined in the setting of Euclidean space to this broad class of Lie groups is potentially relevant to a number of problems relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular structure determination, and quantum information theory. In this paper, several definitions are extended from the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained by pooling different sensory inputs.

Let $G$ be a Lie group, and let $D$ be a discrete subgroup of $G$ such that the right coset space $D\backslash G$ has finite right-invariant volume. We will exhibit an injection of left-invariant means on ${l^\infty }(D)$ into left-invariant means on the left uniformly continuous bounded functions of $G$. When $G$ is an abelian Lie group with finitely many connected components, we also show surjectivity, and when $G$ is the additive group ${{\mathbf {R}}^n}$ and $D$ is ${{\mathbf {Z}}^n}$, the bijection will explicitly take the form of an integral over the unit cube ${[0,1]^n}$.

This paper classifies all left-invariant Hermitian and hyper-Hermitian structures on Lie groups whose commutator groups are one-dimensional. Furthermore, for each natural number n = 2k and n = 4k (k > 1 for the case of hyper-Hermitian structures), we construct an n-dimensional Lie group with a one-dimensional commutator group that possesses left-invariant Hermitian and hyper-Hermitian structures, respectively.

Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains

The stratified spaces of a symplectic lie group action

Abstract. For a topological group the intersection of all kernels of ordinary representations is studied. We show that is contained in the center of if is a connected pro-Lie group. The class is determined explicitly if is the class of connected Lie groups or the class of almost-connected Lie groups: in both cases, it consists of all compactly-generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as for some connected pro-Lie group . However, the dimension of is bounded by the cardinality of the continuum if is locally compact and connected. Examples are given to show that becomes complicated if contains groups with infinitely many connected components.

We shall study the properties of the Chern classes of complex line bundles on toroidal groups by the theory of de Rham cohomology and describe the conditions for a toroidal group to be a quasi-abelian variety using a Z-valued skew-symmetric form. In case of complex tori, these are called Riemann conditions.

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers M N of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns “random” sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {S N j} of H 0(M, L N ), we show that for almost every sequence {S N j}, the associated sequence of zero currents &1/N Z S N j ; tends to the curvature form ω of L. Thus, the zeros of a sequence of sections s N ∈H 0(M, L N ) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S N j } of H 0(M, L N ) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

Let L → X be a positive line bundle on a compact complex manifold X. For compact submanifolds Y, S of X and a holomorphic submersion Y → S with compact fibre, we study curvature of a natural connection on certain line bundles on S.

One of the fundamental problems of gravity to be an infinite range interaction is whether it has a gauge group via the internal structure, so that Einstein type theories couldn’t explain it yet. Gauge groups in Yang Mill type theories, i.e. weak [11] and strong [3, 8] interactions, are generally non Abelian Lie group via the Hermitian structure induced by complex valued smooth vector fields. A group labeled by U(1)⊗Ũ(d) obtained as the reduction of unitary Lie groups is presented as candidate a gauge group for each behavior at large scale and at small scale of a gauge field, in this context, integrated Einstein Yang Mills formalism on a unique geometrical frame is given. Also as seen that the (anti) self dual gauge theory of the group G = U(1) ⊗ Ũ(4) is indeed a G = U(1) theory in real 4 dimension, i.e the Maxwell theory for the electromagnetism to be an infinite range interaction. Mathematics Subject Classification: 22E15, 53C07, 53C80, 70S05, 70S15, 83C05

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing.