arXiv subjects

Irina Markina

Publications and source records attributed to Irina Markina.

At least 37 records · Page 2Linked to original sources

arXiv2015

Complete classification of $H$-type algebras: I

Let $\mathscr N$ be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle.\,,.\rangle$ and let $\mathscr N=V\oplus_{\perp}Z$, where $Z$ is the centre of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space $V$ arises as a representation space of a Clifford algebra $\Cl(\mathbb R^{r,s})$ and the representation map $J\colon \Cl(\mathbb R^{r,s})\to(V)$ is related to the Lie algebra structure by $\langle J_zv,w\rangle=\langle z,[v,w]\rangle$ for all $z\in \mathbb R^{r,s}$ and $v,w\in V$. The classification is based on the range of parameters $r$ and $s$ and is completed for the Clifford modules $V$, having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters $0\leq r,s<\infty$. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.

math.RT↗

arXiv2015

Pseudo-metric 2-step nilpotent Lie algebras

The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that any 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with Lie brackets. This choice of the standard pseudo-metric form allows to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo $H$-type algebras have bases with rational structural constants, which implies that the corresponding pseudo $H$-type groups admit lattices.

math.RT↗

arXiv2015

On the Hodge-type decomposition and cohomolgy groups of $k$-Cauchy-Fueter complexes over domains in the quaternionic space

The $k$-Cauchy-Fueter operator $ D_0^{(k) } $ on one dimensional quaternionic space $\mathbb{H}$ is the Euclidean version of helicity $\frac k 2$ massless field operator on the Minkowski space in physics. The $k$-Cauchy-Fueter equation for $k\geq 2$ is overdetermined and its compatibility condition is given by the $k$-Cauchy-Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we show the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous $k$-Cauchy-Fueter equation $ D_0^{(k) } u=f$ on a smooth domain $Ω$ in $\mathbb{H}$ is solvable if and only if $f$ satisfies the compatibility condition and is orthogonal to the set $\mathscr H^1_{ (k) }(Ω)$ of Hodge-type elements. This set is isomorphic to the first cohomology group of the $k$-Cauchy-Fueter complex over $Ω$, which is finite dimensional, while the second cohomology group is always trivial.

math.CV↗

arXiv2015

Free nilpotent and $H$-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices.

math.DG↗

arXiv2014

Rolling against a sphere: The non transitive case

We study the control system of a Riemannian manifold $M$ of dimension $n$ rolling on the sphere $S^n$. The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone $C(M)$ of $M$. Using Berger's list, we reduce the possible holonomies to a few families. In particular, we focus on the cases where the holonomy is the unitary and the symplectic group. In the first case, using the rolling formalism, we construct explicitly a Sasakian structure on $M$; and in the second case, we construct a 3-Sasakian structure on $M$.

math.DG↗

arXiv2014

Classification of pseudo $H$-type algebras

We present a partial classification of the pseudo $H$-type algebras with minimal admissible Clifford modules. Furthermore, we prove that the subspace $\mathfrak{v}_{r,s}$ of $\mathfrak{n}_{r,s}$ is strongly bracket generating if and only if $r=0$ or $s=0$. Additionally, we discuss the classification of pseudo $H$-type algebras related to non-equivalent irreducible Clifford modules.

math.RT↗

arXiv2014

Extremal functions for modules of systems of measures

We study Fuglede's $p$-module of systems of measures in condensers in Euclidean spaces and on polarizable Carnot groups. We apply and generalize a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in $\mathbb R^2$, to a variety of settings. In the planar case, we apply Rodin's method to obtain estimates for the conformal module of a parallelogram and of a ring domain using directional dilatations. In $\mathbb R^n,$ we identify the extremal function and compute the $p$-module of images of families of connecting curves and of separating sets with respect to the plates of a condenser under homeomorphisms of certain regularity. Then we calculate the module and find the extremal measures for the spherical ring domain on polarizable Carnot groups and extend Rodin's theorem to the spherical ring domain on the Heisenberg group.

math.CA↗

arXiv2013

Existence of the lattice on general $H$-type groups

Let $\mathscr N$ be a two step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle\cdot\,,\cdot\rangle$ and let $\mathscr N=V\oplus_{\perp}Z$, where $Z$ is the center of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We prove that if $(V,\langle\cdot\,,\cdot\rangle_V)$ is the Clifford module for the Clifford algebra $\Cl(Z,\langle\cdot\,,\cdot\rangle_Z)$ such that the homomorphism $J\colon \Cl(Z,\langle\cdot\,,\cdot\rangle_Z)\to\End(V)$ is skew symmetric with respect to the scalar product $\langle\cdot\,,\cdot\rangle_V$, or in other words the Lie algebra $\mathscr N$ satisfies conditions of general $H$-type Lie algebras ~\cite{Ciatti, GKM}, then there is a basis with respect to which the structural constants of the Lie algebra $\mathscr N$ are all $\pm 1$ or 0.

math.RT↗

arXiv2013

Sub-Riemannian geometry of Stiefel manifolds

In the paper we consider the Stiefel manifold $V_{n;k}$ as a principal $U(k)$- bundle over the Grassmann manifold and study the cut locus from the unit element. We gave the complete description of this cut locus on $V_{n;1}$ and presented the sufficient condition on the general case. At the end, we study the complement to the cut locus of $V_{2k;k}$.

math.OC↗

arXiv2013

On the Alexandrov Topology of sub-Lorentzian Manifolds

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which is linked to the property of strong causality of a space-time. We studied three possible ways to define the Alexandrov topology on sub-Lorentzian manifolds, which usually give different topologies, but agree in the Lorentzian case. We investigated their relationships to each other and the manifold's original topology and their link to causality.

math.DG↗

arXiv2012

An intrinsic formulation for rolling pseudo-Riemannian manifolds

In the present work we define the rolling of one pseudo-Riemannian manifold over another without slipping and twisting. We compare the definition of the rolling without slipping and twisting of two manifolds isometrically embedded into a pseudo-Euclidean space with the rolling defined only by the intrinsic data, namely by the metric tensors on manifolds. The smooth distribution on the configuration space, encoding the no-slipping and no-twisting kinematic conditions is constructed. Some results concerning the causal character of the rolling curves are also included. Several examples are presented along the paper to illustrate concepts and help to understand the theoretical results.

math.DG↗

arXiv2012

On Cauchy-Szegö kernel for quaternionic Siegel upper half space

The work is dedicated to the construction of the Cauchy-Szegö kernel for the Cauchy-Szegö projection integral operator from the space of $L^2$-integrable functions defined on the boundary of the quaternionic Siegel upper half space to the space of boundary values of the quaternionic regular functions of the Hardy space over the quaternionic Siegel upper half space.

math.FA↗

arXiv2012

Controllability on infinite-dimensional manifolds

Following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable.

math.DG↗

arXiv2012

Sub-semi-Riemannian geometry of general $H$-type groups

We introduce a special class of nilpotent Lie groups of step 2, that generalizes the so called $H$(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general $H$-type groups. We also present some results on sectional curvature and the Ricci tensor of general $H$-type groups.

math.DG↗

arXiv2012

Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmueller space and curve

We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmüller curve associated to the space of normalized univalent functions. We find formulas for the normal geodesics with respect to the pullback of the invariant Kählerian metrics, namely, the Velling-Kirillov metric on the class of normalized univalent functions and the Weil-Petersson metric on the universal Teichmüller space. The geodesic equations are sub-Riemannian analogues of the Euler-Arnold equation and lead to the CLM, KdV, and other known non-linear PDE.

math.DG↗

arXiv2012

Sub-Riemannian geometry on infinite-dimensional manifolds

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $M$, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle $TM$, called the horizontal distribution. Similarly to the finite-dimensional case, we are able to split possible candidates for minimizing curves into two categories: semi-rigid curves that depend only on $\calH$, and normal geodesics that depend both on $\calH$ itself and on the metric on $\calH$. In this sense, semi-rigid curves in the infinite-dimensional case generalize the notion of singular curves for finite dimensions. In particular, we study the case of regular Lie groups. As examples, we consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and the Virasoro-Bott group with their respective horizontal distributions chosen to be the Ehresmann connections with respect to a projection to the space of normalized univalent functions. In these cases we prove controllability and find formulas for the normal geodesics with respect to the pullback of the invariant Kählerian metric on the class of normalized univalent functions. The geodesic equations are analogues to the Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDE.

math.DG↗

arXiv2011

Evolution of smooth shapes and integrable systems

We consider a homotopic evolution in the space of smooth shapes starting from the unit circle. Based on the Loewner-Kufarev equation we give a Hamiltonian formulation of this evolution and provide conservation laws. The symmetries of the evolution are given by the Virasoro algebra. The `positive' Virasoro generators span the holomorphic part of the complexified vector bundle over the space of conformal embeddings of the unit disk into the complex plane and smooth on the boundary. In the covariant formulation they are conserved along the Hamiltonian flow. The `negative' Virasoro generators can be recovered by an iterative method making use of the canonical Poisson structure. We study an embedding of the Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian, construct the tau-function, the Baker-Akhiezer function, and finally, give a class of solutions to the KP equation.

math-ph↗

arXiv2010

Hopf fibration: geodesics and distances

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also present some motivations related to quantum mechanics.

math.DG↗