Related papers: An Integrability Condition for Simple Lie Groups I…

It is shown that for any compact Lie group $G$ (odd or even dimensional), the tangent bundle $TG$ admits a left-invariant integrable almost complex structure, where the Lie group structure on $TG$ is the natural one induced from $G$. The…

Let $G$ be a connected complex Lie group. A real form of $G$ is a closed subgroup $H\subset G$ whose Lie algebra $\mathfrak{h}$ is a real form of the Lie algebra $\mathfrak{g}$ of $G$. A pair $(G,H)$ of this type is reductive, and the…

In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for…

The coupling of the tangent bundle $TM$ with the Lie algebra bundle $L$ (K.Mackenzie,2005, Definition 7.2.2) plays the crucial role in the classification of the transitive Lie algebroids for Lie algebra bundle $L$ with fixed finite…

Let $H$ be a closed normal subgroup of a compact Lie group $G$ such that $G/H$ is connected. This paper provides a necessary and sufficient condition for every complex representation of $H$ to be extendible to $G$, and also for every…

We find new conditions that the existence of nullity of the curvature tensor of an irreducible homogeneous space $M=G/H$ imposes on the Lie algebra $\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ of the full isometry group…

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian…

Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r…

We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie…

Let G be a Lie group over a local field of positive characteristic which admits a contractive automorphism f (i.e., the forward iterates f^n(x) of each group element x converge to the neutral element 1). We show that then G is a torsion…

Let $G$ be a Lie group, and let $(g,J)$ be a left invariant almost pseudo-Hermitian structure on $G$. It is shown that if $(g,J)$ is also nearly pseudo-K\"{a}hler, then the tangent bundle $TG$ (with its natural Lie group structure induced…

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let Aut$(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of GL$(\frak G)$, where $\frak G$…

Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action…

We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness,…

For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a…

Let G be a connected reductive algebraic group over an algebraically closed field k. In a recent paper, Bate, Martin, R\"ohrle and Tange show that every (smooth) subgroup of G is separable provided that the characteristic of k is very good…

We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading.…

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth…

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta…

We show that locally every beta-integrable (2,n)-Segre structure can be reduced to a torsion-free S^1*GL(n,R)-structure. This is done by observing that such reductions correspond to sections with holomorphic image of a certain `twistor…