arXiv research

Survey on metrics on compact Lie groups.

Study describes isometry groups of specific Lie groups.

Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.

Investigate local Lie group structure of bisections over compact manifolds

Characterizes isolated compact subgroups in Lie groups.

The reduction of biharmonic maps equation in terms of the Maurer-Cartan form for all smooth map of any compact Riemannian manifolds into a compact Lie group with bi-invariant Riemannian metric is obtained. By this formula, all the biharmonic curves into a compact Lie group and all biharmonic maps from a 2-dimensional o…

No exact G₂-structures on compact Lie group quotients.

Investigates solving curvature equations on special Lie groups.

Study classifies Lie group representations with non-empty boundary orbit space.

The paper proves convexity results for a specific type of Lie groups.

We prove that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the grou…

The 2-rank of a compact Lie group $G$ is the maximal possible rank of the elementary 2-subgroup ${\mathbb Z}_{2}\times... {\mathbb Z}_{2}$ of $G$. The study of 2-ranks (and $p$-rank for any prime $p$) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups h…

Study convolution of invariant valuations on Lie groups.

Let $G$ be a compact Lie group. (Compact) topological $G$-manifolds have the $G$-homotopy type of (finite-dimensional) countable $G$-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear $G$-manifolds [Elf96], wherein the Lie group $G$ is linear (such as compact).

Proves cohomology of elliptic structures on Lie groups can be algebraic.

We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.

Study on simplicity of Lie skew braces, proving new results for compact cases.

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then apply our method to construct many examples of compact minimal submanifolds of the special u…

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine the group structures of the fixed points subgroup. They correspond to the classif…

We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.

Bi-invariant metrics on Lie groups and homogeneous spaces are extremal and rigid.

New complex structures found on tangent bundles of Lie groups.

Explores new perspectives in transverse index theory for Lie group actions.

This paper classifies geodesic orbit metrics on compact Lie group $G_2$.

Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.

Geometrically revisits and models homogeneous spaces of compact Lie group $G_2$.

In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e. a gauge groupoid, …

Develops Lie algebraic approach for compact complex homogeneous manifolds.

Constructs explicit p-harmonic functions on specific Lie groups.

Balanced metrics found on Lie groups and their quotients.

We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative de Rham currents and the noncommutative Chern characters, and their computation f…

This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gawȩdzki-Reis [J. Geom. Phys. 50(1):28-55, 2004], we exhibit basic equivariant bundle gerbes over non-simply connected c…

Solves classification of compact Clifford-Klein forms for specific Lie groups.

Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.

Compact Lie groups can be realized as automorphism groups of Riemannian manifolds.

Researchers classify geodesic orbit spaces for compact Lie groups of rank two.

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the cor…

In this paper we construct infinitely many examples of a Riemannian submersion from a simple, compact Lie group $G$ with bi-invariant metric onto a smooth manifold that cannot be a quotient of $G$ by a group action. This partially addresses a question of K. Grove's about Riemannian submersions from Lie groups.

In this survey, we discuss a series of linearization problems--for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and alm…

New p-harmonic and harmonic morphisms found on Lie groups.

Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.

The main result of this paper is the classification of the real irreducible representations of compact Lie groups with vanishing homogeneity rank.

Explicitly describes pluriclosed metrics on compact Lie groups.

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for $n \ge 6$, by using a method of Riemannian submersions. In the pres…

Compact Lie group actions with a free point are determined by two vector fields.

Vector fields invariant under Lie group action are finitely generated by polynomial fields.

We discuss a Moser type argument to show when a deformation of a Lie group homomorphism and of a Lie subgroup is trivial. For compact groups we obtain stability results.