Том 216, № 1 (2016)

In the paper, we discuss some problems concerning the structural properties of crossed products. While expansions of C*-algebras under group actions have been studied rather extensively, known difficulties in the transition to irreversible dynamical systems require the development of new methods. The first step in this direction is to study actions of inverse semigroups, whose properties are closest to those of groups. The main tool to achieve the goal is the concept of grading. The detection of the grading structure in the corresponding constructions seems to be very promising.

The goal of our article is a study of related mathematical and physical objects: orthogonal pairs in sl(n) and mutually unbiased bases in ℂⁿ. An orthogonal pair in a simple Lie algebra is a pair of Cartan subalgebras that are orthogonal with respect to the Killing form. The description of orthogonal pairs in a given Lie algebra is an important step in the classification of orthogonal decompositions, i.e., decompositions of the Lie algebra into a direct sum of Cartan subalgebras pairwise orthogonal with respect to the Killing form. One of the important notions of quantum mechanics, quantum information theory, and quantum teleportation is the notion of mutually unbiased bases in the Hilbert space ℂⁿ. Two orthonormal bases {e_i}_i = 1ⁿ, {f_j}_j = 1ⁿare mutually unbiased if and only if\( {\left|\left\langle {e}_i\left|{f}_j\right.\right\rangle \right|}^2=\frac{1}{n} \)for any i, j = 1,…, n. The notions of mutually unbiased bases in ℂⁿand orthogonal pairs in sl(n) are closely related. The problem of classification of orthogonal pairs in sl(n) and the closely related problem of classification of mutually unbiased bases in ℂⁿare still open even for the case n = 6. In this article, we give a sketch of our proof that there is a complex four-dimensional family of orthogonal pairs in sl(6). This proof requires a lot of algebraic geometry and representation theory. Further, we give an application of the result on the algebraic geometric family to the study of mutually unbiased bases. We show the existence of a real four-dimensional family of mutually unbiased bases in ℂ⁶, thus solving a long-standing problem. Bibliography: 24 titles.

In this paper, we investigate the asymptotics of the normalized dimensions of strict Young diagrams. We describe the results of corresponding computer experiments. The strict Young diagrams parametrize the projective representations of the symmetric group Sn. So, the asymptotics of the normalized dimensions of diagrams gives us the asymptotics of the dimensions of projective representations. Sequences of strict diagrams of high dimension consisting of up to one million boxes were generated. It was proved by an exhaustive search that the first 250 diagrams of all these sequences have the maximum possible dimensions. Presumably, these sequences contain infinitely many diagrams of maximum dimension, and thus give the correct asymptotics of their growth. Also, we investigate the behavior of the normalized dimensions of typical diagrams with respect to the Plancherel measure on the Schur graph. The calculations strongly agree with A. M. Vershik’s conjecture on the convergence of the normalized dimensions of the maximum and Plancherel typical diagrams not only for the standard Young graph, but also for the Schur graph.

The paper provides a short overview of a series of articles devoted to C*-algebras generated by a self-mapping on a countable set. Such an algebra can be seen as a representation of the universal C*-algebra generated by a family of partial isometries satisfying a set of conditions. These conditions are determined by the initial mapping.

We study the Cauchy problem for the nonlinear Fokker–Planck–Kolmogorov equations for probability measures on a Hilbert space that corresponds to stochastic partial differential equations. Sufficient conditions for the uniqueness of probability solutions for a cylindrical diffusion operator and for a possibly degenerate diffusion operator are given. A new general existence result is established without explicit growth restrictions on the coefficients.