- Mateusz Aniserowicz
- Institute of Mathematics University of Bialystok Poland
- Anatolij Antonevich
- Institute of Mathematics University of Bialystok Poland
Weighted Composition Opretators: Application and sperctal propertiesA bounded linear operator $ B $, acting on a Banach space $ F(X) $ of functions on an arbitrary set $X$, is called a
*weighted shift operator*or*weighted composition operator*if it can be represented in the form $ \label{WSO1} Bu(x)=a_0(x)u(\alpha(x)), \ \ x \in X, $ where $\alpha : X \to X$ is a map, $a_0(x)$ is a given function on $ X $. Operator of the form $Bu(x)=\sum_k a_k(x) u(\alpha^k(x))$ is called*functional operator*. Such operators, operator algebras generated by them and functional equations related to such operators have been studied in application to theory of dynamical systems, integro-functional, differential-functional, functional and difference equations, automorphisms and endomorphisms of Banach algebras, nonlocal boundary value problems, nonclassical boundary value problems for equation of string vibration, the general theory of operators and operator algebras. The main problem is to clarify the relationship between the spectral properties of the weighted shift operators and the dynamical properties of the map $ \alpha$, i.e. the behavior of the trajectories $$\{\alpha^n(x), n \in \mathbb{N}\},$$ where $\alpha^n(x)= \alpha(\alpha^{n-1}(x)).$ At the talk will be given a short survey of investigations in this area. - Agata Bezubik
- Institute of Mathematics, University of Bialystok, Poland
Verma bases for representations of Lie algebrasWell known Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. However, for practical computations, the sum over the Weyl group and the need to divide two elements from weight lattice make this formula rather difficult to use. To obtain the character of an irreducible highest weight module, Freudenthal’s recursive formula is more practical. On the other hand, tracing of the identity element, Weyl’s character formula turns to the Weyl dimension formula, which provides an efficient algorithm for obtaining a polynomial $p$ of $l$ variables such that $p(n_1 , ..., n_l)$ is the dimension of the highest-weight module with highest weight $n_1 λ_1 + ... + n_lλ_l$, where $λ_j$ are fundamental weights. We give some of these polynomials explicitly. Connection with Verma basis of the space of given irreducible representation is discussed.
*Joint work with Severin Pošta.* - Mark Bodner
- MIND Research Institute, Irvine, California
Breaking of Icosahedral Symmetry and the geometric construction of FullerenesThe icosahedral symmetry group $H_3$ of order 120 and its dihedral subgroup $H_2$ of order 10 are used for geometric construction of polytopes that are known to exist in nature. Specifically described is the existence and structure of large fullerenes in terms of breaking of the icosahedral symmetry of the $C_{60}$ buckyball by the insertion into its middle additional $H_2$ decagons. The branching rule for the $H_3$ orbit of the fullerene $C_{60}$ yields a union of 8 orbits of $H_2$, 4 of them are regular pentagons and 4 are regular decagons. By inserting into the branching rule one, two, three, or n additional decagonal orbits of $H_2$, one builds the polytopes $C_{70}$, $C_{80}$, $C_{90}$, or nanotubes in general. A minute difference should be taken into account when even or odd numbers of $H_2$ decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group. Also defined are twisted fullerenes whose surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted Fullerenes, all possessing precise icosahedral symmetry
- Goce Chadzitaskos
- Czech Technical University in Prague
Quantum mechanics on discrete chainsThe quantum mechanics has passed more than eighty years of successful development. The quantum mechanics on different configuration spaces, discrete ones among them, have been developed during last thirty years. We present methods how to construct quantum mechanics on configuration spaces done by discrete set of points with a symmetry group action on it. We discuss traditional quantization, deformation quantization and coherent states quantization.
- Patrick Da Silva
- Berlin Mathematical School
- Pierre-Philippe Dechant
- Durham University
What Clifford algebra can do for Coxeter groups and root systemsClifford algebra often provides a natural geometric interpretation for algebraic results. We discuss this framework in the context of Coxeter groups and root systems. For instance, we show that it provides geometric - different - complex structures for the eigenvalues of the Coxeter element. A construction in terms of Clifford spinors furthermore yields a 4d root system for any 3d root system, which includes all exceptional 4d groups, and as a side effect induces 4d polytopes and their symmetries from the Platonic solids. We explore the connections with Arnold's trinities and the McKay correspondence. We further discuss some recent work on affine extensions of non-crystallographic root systems.
- Pavel Exner
- Nuclear Physics Institute, Czech Republic
- Zofia Grabowiecka
- University of Bialystok
Universal $C^*$-algebra and crossed product of $C^*$-algebra by an endomorphism.The purpose of the presentation is to introduce universal $C^*$-algebra $C^*(\mathcal{G} , \mathcal{R} )$ and crossed product of a $C^*$-algebra by an endomorphism. Moreover some properties of Cuntz-Krieger $C^*$-algebra will be discussed and it will be shown that this algebra is both universal $C^*$-algebra and crossed product of $C^*$-algebra by an endomorphism and transfer operator.
- Lenka Háková
- Czech Technical University in Prague
Interpolation of digital data using orbit functions of $B_3$ and $C_3$In this talk we consider new families of orbit functions, so called $S^s-$ and $S^l-$ functions, in three-dimensional case. These functions are defined for the Weyl groups with two lenghts of root, namely $B_3$ and $C_3$. We show the discrete orthogonality relations and the formulas for corresponding Fourier trasform. We conclude with several examples.
- Jiří Hrivnák
- Czech Technical University in Prague
Ten types of Weyl group orbit functionsThe notions for introduction of special functions of Weyl groups are reviewed. The affine Weyl groups corresponding to the root systems of simple Lie algebras are recalled and their fundamental domains described. Sign homomorphisms which allow general explicit description of the functions are defined. The resulting three types of orbit functions and ten types of orbit functions for the root systems with two different lengths of roots are obtained and their properties discussed.Fourier transforms of E-functions of O(5) and G(2)We discuss the properties of six types of special functions related to the Weyl groups of O(5) and G(2). These functions of two real variables, known as
*E-*functions, are generalization of the common exponential functions for each group. The Fourier transforms of these function are described. We focus on discrete transforms on lattices - the symmetries of these lattices are inherited from the groups O(5) and G(2). Examples of application to interpolation are presented. - Franz Hinterleitner
- Institute of Theoretical Physics and Astrophysics, Masaryk University, Brno, Czech Republic
Discretization of Geometry in Loop Qauntum GeometryLoop quantum grvity attempts to unite discreteness and uncertainties of quantum theory with the main achievment of general relativity, namely the mutation of space and time from a rigid stage for physical dynamics to a dynamical actor. It is based on a 3+1 split of four-dimensional space-time and the introduction of triad fields and connections as canonical field variables on three-dimensional hypersurfaces, which are to become the basic quantum operators, acting on state functionals of the connection (connection representation). The necessary invariance of quantum states under triad gauge rotations leads in a natural way to the introduction of loop holonomies, i.e. parallel transport along closed curves, so that states becom functionals of holonomies, rather than of the connection directly. At the present stage of the theory loops are combined into so-called spin-networks, which give rise to a well-defined Hilbert space. From the triad variables one constructs quantum operators of metric quantities, like length, area, volume, which turn out to have discret spectra. In this way, in contrast to the usual approach to general relativity, metric quantities are not introduced from the beginning, but come out as (discrete) expectation values in quantum states of the gravitational field, the dynamics of which is governed by quantum Einstein equations. The result is a 'granular"" geometry with ""atoms of space""; time evolution depends on observers.
- Irena Hinterleitner
- Brno University of Technology
Geodesic mappingsIn our talk we present geodesic lines and afterwards we introduce geodesic mappings, conditions for their existence and some properties.
- Ondřej Kajínek
- Czech Technical University in Prague
Orbit convolution in image processingConvolution is one of the basic tools in the image processing. In practice, convolution is computed using a goniometric transform, i.e., Fourier transform with much less effort. Convolution theorem describes how to simplify convolution to goniometric transform. This theorem is generalised to $C$-orbit transform to provide similar simplification to so called orbit convolution. Simple image processing is used as a demonstration of how the convolution operates on the fundamental region of $A_{2}$ Weyl group orbit.
- Dalibor Karásek
- Czech Technical University in Prague
Solvable extensions of nilpotent Lie algebras from the cohomological viewpointA brief review of the significance of third cohomology group for the extendability of Lie algebras will be presented. Afterwards, known facts about vanishing cohomology groups of nilpotent Lie algebras will be used to shed light on some observed properties of solvable Lie algebras with a given nilradical.
- Arthemy Kiselev
- Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen
The calculus of multivectors on noncommutative jet spacesWe state and prove the main properties of Schouten bracket on jet spaces in a class of noncommutative geometries. Our reasoning confirms that traditional differential calculus does not appeal to a graded commutativity assumption in the setup.
- Stepan Manko
- Czech Technical University in Prague
Inverse Scattering for Energy-Dependent Schrödinger EquationsWe develop the direct and inverse scattering theory for one-dimensional energy-dependent Schrödinger equations \[ -y''+q(x)y+2kp(x)y=k^2y \hskip{5cm} (\star) \] on the half-line with highly singular potentials $q$. We study scattering problem for the equation ($\star$) along with the energy-dependent boundary condition \[ \cos\alpha\, y^{[1]}(0,k)+k\sin\alpha\, y(0,k)=0, \hskip{4cm} (\star\star) \] where the quasiderivative $y^{[1]}$ is defined as follows: \[ y^{[1]}(x,k):=y'(x,k)-u(x)y(x,k). \] Schrödinger equations of the form ($\star$) arise in various models of quantum and classical mechanics; for instance, the Klein-Gordon equation modeling interactions between colliding relativistic spinless particles (anti-particles) with zero mass is a special case of ($\star$) when $q=-p^2$. The
*Jost solution*$f$ is a solution of ($\star$) obeying the asymptotics \[ f(x,k)=e^{ikx}\big(1+o(1)\big),\qquad x\to\infty. \] Next we introduce the*scattering function*$S$ by requiring that the solution \[ g(x,k):=\overline{f(x,k)}+S(k)f(x,k) \] of ($\star$) should satisfy the boundary condition ($\star\star$); $g$ is called the*scattering solution*of ($\star$). The*direct scattering problem*is to construct the scattering function $S$ given $(u,p,\alpha)$, while the*inverse scattering problem*is to recover the triple $(u,p,\alpha)$ from the scattering function $S$. More generally, we study properties of the direct and inverse scattering maps $(u,p,\alpha)\mapsto S$ and $S\mapsto(u,p,\alpha)$ respectively, show that they are continuous and also obtain an explicit reconstruction formula for $(u,p,\alpha)$ in terms of $S$. As a motivation for our study we may use the fact that inverse scattering theory for energy-dependent Schrödinger equations ($\star$) can be applied effectively to a class of inverse problems to determine coefficients in hyperbolic or elliptic partial differential equations in mathematical physics. Inverse scattering for energy-dependent Schrödinger equations has also appeared in the literature in the context of the (sound, electromagnetic, or elastic) waves propagation in nondispersive media where the wave speed depends on position. - Alexander Minakov
- Czech Technical University in Prague
Parametrices for the modified Korteweg — de Vries equation in a modulated elliptic wave regionWe consider the Cauchy problem for the modified Korteweg — de Vries equation on the line. The initial function is step-like, that is it tends to some constants when $x\to \pm\infty$. We study the asymptotical behavior of the solution as $t\to\infty.$ Earlier in [1] we got the formula for the main term of the asymptotics in the most interesting domain $(-6c^2+\varepsilon)t<x<(4c^2-\varepsilon)t$. It was obtained from the model Riemann — Hilbert problem, which was explicitly solvable in terms of the elliptic functions. Here we justify the transition from the original Riemann — Hilbert problem to the model one. It is done due to the analysis of the so-called parametrices, and in our case they are constructed in terms of the Airy function and its derivative.
[1] Kotlyarov V.P., Minakov A.A. RiemannHilbert problem to the modified Kortevegde Vries equation: Long-time dynamics of the steplike initial data, J. Math. Phys. 51, 093506 (2010)

- Josef Mikeš
- Palacky University Olomouc
- Lenka Motlochová
- Université de Montréal
Cubature rules arising from hybrid characters of compact simple Lie groupsAny compact simple Lie group $G$ with two root lengths (types $B_n$, $C_n$, $F_4$, and $G_2$) allows one to define two types of Weyl group orbit functions related to "mixed sign" homomorphisms on the corresponding Weyl group of $G$. This gives rise to so-called hybrid characters which are connected to certain families of orthogonal polynomials and which have a few remarkable properties leading to new cubature formulas. There are Gaussian cubatures in the short root case and less efficient Radau cubatures in long root case. We show basic ideas of the connection between cubatures and root systems, and present explicit cubature formulas for Lie group $G_2$.
- Anatoly Nikitin
- Institute of Mathematics of Nat. Acad. Sci. of Ukraine
Supersymmetric and superintegrable models of quantum mechanicsThere are two properties of quantum mechanical systems which can make them exactly solvable: supersymmetry and superintegrability. Being formally independent, both of them are guide signs in searches for exactly solvable problems. In this presentation a classification of supersymmetric and superintegrable QM models is proposed. This classification is restricted to systems of coupled Schroedinger equations with matrix potentials being 2x2 dimensional matrices. In addition, systems with arbitrary spin which admit generalized Laplace-Runge vectors are discussed. It is shown that the known relations between supersymmetry and superintegrability valid for scalar systems cannot be extended to the case of matrix potentials.
- Petr Novotný
- Czech Technical University in Prague
Graded Contractions of Representations of Lie Algebra sl(3,C)We present the concept of graded contractions for representations of Lie algebras, which allows one to construct representations of some solvable Lie algebras from known irreducible representations of simple Lie algebras. We focus on the construction of faithful indecomposable representations and mutually nonequivalent representations. As an example we contract one class of irreducible representations of simple Lie algebra sl(3,C) to the representations of seven dimensional solvable Lie algebra.
- Satoshi Ohya
- Czech Technical University in Prague
Parasupersymmetry in quantum graphsParasupersymmetry is a generalization of supersymmetry to parastatistics: it is a symmetry between bosons and parafermions. Though originally found in the context of parastatistics, parasupersymmetry can appear even in purely bosonic quantum mechanical systems. In this talk we study hidden parasupersymmetry structures in quantum mechanics for a single free spinless particle on compact equilateral graphs with $N$ edges. We show that, if the system is invariant under $\mathbb{Z}_{N}$-cyclic rotations of graph edges, parasupersymmetry appears in the spectrum with some particular class of $\mathbb{Z}_{N}$-invariant boundary conditions at the graph vertices. We would also like to discuss a novel supersymmetry – cyclic parasupersymmetry – realized in equilateral quantum graphs.
- Jiří Patera
- Université de Montréal
- Severin Pošta
- Czech Technical University in Prague
- Marzena Szajewska
- Institute of Mathematics, University of Bialystok, Poland
Icosahedral symmetry breaking: $C_{60}$ to $C_{78}$, $C_{96}$ and to related nanotubesExact icosahedral symmetry of $C_{60}$ is viewed as union of twelve orbits of the symmetric subgroup of order 6, denote by $A_2$, of the icosahedral group of order 120. Eight of the orbits are hexagons and four are triangles. The orbits form a stack of parallel layers centred on the axis of $C_{60}$ passing through the centres of two opposite hexagons on the surface of $C_{60}$. By inserting into the middle of the stack three $A_2$ orbits of 6 points each, one can match the structure of $C_{78}$. Repeating the insertion, one gets $C_{96}$; multiple such insertions generate nanotubes of any desirable length.
- Miloš Tater
- Nuclear Physics Institute, Czech Republic
Polynomial solutions of the Heun equationThe classical Heun equation has the form $$ Q(z) S''(z)+P(z) S'(z)+V(z) S(z)=0, $$ where $Q$ is a cubic complex polynomial, $P$ is a quadratic polynomial and $V$ is at most linear. We are interesred in polynomial solutions of this equation,
*i.e.*the set of all $V$'s for which the above equation has a polynomial solution $S$ of a given degree $n$. The main goal is to study the union of the roots of the latter set of $V$'s when n goes to infinity. We prove that the union of the roots tends to a certain compact connecting the three roots of $Q$ which is given by a condition that a certain abelian integral is real-valued. We further prove that roots of $S$ tend to singular trajectories of a quadratic differential as $n\rightarrow\infty$. We terminate the talk by some natural generalizations. - Keith Taylor
- Dalhousie University
A Fourier Transform and the C*-algebra of Crystal GroupsThe objects of abstract harmonic analysis associated with non-commutative discrete groups are usually intractable in their complexity. However, the symmetry group of a crystal always has an abelian subgroup of finite index. This enables a complete description of their representation theory that has been understood for more than a century. Nevertheless, some other standard objects such as the group C*-algebra had not been well understood. We will present an elementary approach to defining a concrete Fourier transform for functions on a crystal symmetry group that yields an explicit description of the group C*-algebra and promises to be a useful tool in developing the theory of the recently defined wavelets with crystal symmetry shift groups.
- Agnieszka Tereszkiewicz
- Institute of Mathematics, University of Bialystok, Poland
Orbit functions for non-crystallographic groupsDihedral groups $I_2(m), m = 5, 8, 10, 12$ are interesting from the physical point of view because all of them and $H_3$ describe well known quasicrystals. These dihedral groups are presented in details. Orbit functions for them are defined, by analogy to well know orbit functions of Weyl groups. Properties of orbit functions are presented. The similarities and differences to the crystallographic case are discussed.
- Irina Yehorchenko
- Institute of Mathematics of National Academy of Sciences of Ukraine
Differential Invariants of Transitively Differential AlgebrasWe consider transitively differential algebras of degree 3 and construct functional basis of second order differential invariants for some subalgebras of this algebra.
- Jean-Bernard Zuber
- LPTHE University Paris 6
Sum rules on tensor multiplicitiesI discuss identities recently found in multiplicities of irreps in tensor products of simple Lie algebras or fusion products of affine algebras. In contrast, these identities under complex conjugation of representations fail generically for finite groups