Related papers: Remarks on quantization of classical r-matrices

We suggest a somewhat non-standard view on a set of curious, paradoxical from the standpoint of simple classical physics and everyday experience phenomena. There are the quantisation (discrete set of values) of the observables (e.g.,…

This review article is devoted to analyze the main properties characterizing the cosmological singularity associated to the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the…

Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability…

First, the properties of a classical model of spontaneous symmetry breakdown are analyzed. Then, the pros and cons of some pedagogical non-relativistic quantum-mechanical models, also used to illustrate spontaneous symmetry breakdown, are…

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M,…

Complete eigenstructure, e.g., eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes…

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the…

We compare classical and quantum dynamics of a particle in the de Sitter spacetimes with different topologies to show that the result of quantization strongly depends on global properties of a classical system. We present essentially…

We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and…

First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…

We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as…

The construction of quantum knot invariants from solutions of the Yang--Baxter equation (R-matrices) is reviewed with the emphasis on a class of R-matrices admitting an interpretation in intrinsically three-dimensional terms.

We study an elementary inequality supporting the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such new Schatten p-norm estimates and new majorization

We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same…

The fact that not all measurements can be carried out simultaneously is a peculiar feature of quantum mechanics and responsible for many key phenomena in the theory, such as complementarity or uncertainty relations. For the special case of…

We develop several methods, based on the geometric relationship between the eigenspaces of a matrix and its adjoint, for determining whether a square matrix having distinct eigenvalues is unitarily equivalent to a complex symmetric matrix.…

Scattering properties and time delays for general (non-symmetric) potentials in terms of the respective S-matrices are discussed paradigmatically in one dimension and in comparison to symmetric potentials. Only for the latter the Wigner and…

We derive computable formulas for the structured backward errors of a complex number $\lambda$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric,…

A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map is introduced…

An $n\times n$ real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI_{n}$ for some positive real number $q$. If $M$ is a principal sub-matrix of a quasi-orthogonal matrix $Q$, we say that $Q$ is a quasi-orthogonal extension of $M$. In a…