arXiv research

Computes the component group of arbitrary real algebraic groups.

Builds geometric structures for algebraic groups over real closed fields.

The paper studies spaces of non-compact real algebraic curves and their uniformisation.

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…

Computes the component group of real reductive groups.

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …

The paper explores alternative definitions of complex Lie groups using real numbers.

The paper classifies orbit closures of symplectic Lie algebras.

Constructs a topological cover of real line's multiplicative group.

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

Study algebraic K-theory for specific groups of non-orientable surfaces.

We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, real algebraic groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surfa…

The paper explores different realizations of complex Lie groups using various number fields.

The paper studies invariant measures for specific actions in algebraic groups.

We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the h…

Criterion for nilpotent Lie groups to have nilsolitons.

Study extends JB-algebra structure group results to infinite dimensions.

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic …

Analytic curves linked to algebraic ones via Schottky groups.

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend o…

This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…

Maximal Laplacian algebras applied to invariant theory solved inverse problems.

Generic Hitchin representations avoid hyperplanes in Lie algebras.

Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace …

The main goal is to classify 4-dimensional real Lie algebras $\g$ which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the …

The paper classifies orbits of semisimple elements in real semisimple Lie algebras.

In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.

Let $\mathcal{J}^1$ be the real form of complex simple Jordan algebra with the automorphism group $G$ of type $F_{4(-20)}$. Explicitly, we give the orbit decomposition of $\mathcal{J}^1$ under the action of $G$ and determine the Lie group structure of stabilizer for each $G$-orbit on $\mathcal{J}^1$.

The paper studies algebraic integer relations and sequences converging to 4.

We give necessary conditions for certain real analytic tube generic submanifolds in C^n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C^n. During the proof, we show that the local CR automorphism group of a minimal, finitely n…

We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.

A study is made of real Lie algebras admitting a hypersymplectic structure, and we provide a method to construct such hypersymplectic Lie algebras. We use this method in order to obtain the classification of all hypersymplectic structures on four-dimensional Lie algebras, and we describe the associated metrics on the c…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…

Study complex deformations of the circle using group cohomology and Virasoro algebra.

Survey recent constructions of cyclic cocycles for Lie groups.

A special symplectic Lie group is a triple $(G,ω,\nabla)$ such that $G$ is a finite-dimensional real Lie group and $ω$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $\nabla$. In this paper starting from a special symplectic Lie group we show how to ``defo…

Investigates properties of moment maps and stratifications on Lie groups.

Constructs real algebraic functions with specified preimages.

Infinite volume requires no atoms at the bottom of the spectrum for certain groups.

Study real forms and GIT quotients in algebraic varieties.

Study on 4D Lie groups and related almost hypercomplex manifolds.

We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …

The study examines the asymptotic behavior of cohomology groups of algebraic group subgroups.

We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.