Related papers: Polyadic systems, representations and quantum grou…
We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent…
Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous correpresentations of $C^*$-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is…
The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures. We restrict…
The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a…
General properties of ternary semigroups and groups are considered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built, connection with the standard theory is considered and…
We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic…
In this text different theoretical approaches towards multi-quark states are introduced, namely hadrocharmonia, compact tetraquarks and hadronic molecules. The predictions derived from either of them are contrasted with current and possible…
The concept of quantum commutativity with respect to an action or coaction of a given Hopf algebra is used for the algebraic description of a system of particles and their interaction with certain quantum field. Graded commutativity and…
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.
Some introductory concepts and basic definitions of the Lie superalgebras and their quantum deformations are exposed. Especially the induced representation methods in both cases are described. Based on the Kac representation theory we have…
The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier…
We study finite dimensional algebras that appear as fibers of quantum orders over a given point of variety of center. We present the formula for the number of irreducible representations and check it for it for the algebra of twisted…
En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los…
The $q$--deformation $U_q (h_4)$ of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra.Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the…
Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group…
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of…
We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum…
We introduce the notion of partial representation of a weak Hopf algebra. We present the universal algebra $H_{par}^w$, which factorizes these partial representations by algebra morphisms. Also, it is shown that $\Hp$ is isomorphic to a…
These are lecture notes prepared for a minicourse given at the Cimpa Research School "Algebraic and geometric aspects of representation theory", held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction…
Coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model: (i) classical realizations which are the starting point for geometric quantization; (ii) induced…