Related papers: An overview on complex Kleinian groups

This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of…

Recently, it has been established that the discrete star Laplace and the discrete Dirac operator, i.e. the discrete versions of their continuous counterparts when working on the standard grid, are rotation-invariant. This was done starting…

This survey gives an overview of three central algebraic themes related to the study of splines: duality, group actions, and homology. Splines are piecewise polynomial functions of a prescribed order of smoothness on some subdivided domain…

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. But the spin group is not the only subgroup of the Clifford algebra. An algebraist's perspective on…

We give an easy diagrammatical description of the parabolic Kazhdan-Lusztig polynomials for the Weyl group $W_n$ of type $D_n$ with parabolic subgroup of type $A_n$ and consequently an explicit counting formula for the dimension of the…

A discrete DJS-hypergroup is constructed in connection with the linearization formula for the product of two spherical elements for a quantum Gelfand pair of two compact quantum groups. A similar construction is discussed for the case of a…

We study the dynamics of planons, particles whose mobility is restricted to a plane, through the classification of coadjoint orbits and unitary irreducible representations of the centrally extended planon group. Planons are closely related…

It is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$,…

Klein foams are analogues of Riemann and Klein surfaces with one-dimensional singularities. We prove that the field of dianalytic functions on a Klein foam $\Omega$ coincides with the field of dianalytic functions on a Klein surface…

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature.…

Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of…

It is shown that the horizontal holonomy group of a K-contact sub-Riemannian manifold either coincides with the holonomy group of a Riemannian manifold, or it is a codimension-one normal subgroup of the later group. The question of…

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space $G/H$ if…

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley-Klein framework. This approach affords a unified and global study of the conformal structure of the…

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…

The class of locally compact near abelian groups is introduced and investigated as a class of metabelian groups formalizing and applying the concept of scalar multiplication. The structure of locally compact near abelian groups and its…

The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop…

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media…

There are contextual sets of multiple qubits whose commutation is parametrized thanks to the coset geometry $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=\left\langle x,y\right\rangle$. One defines geometric…

We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and…