J. Tolar

On fine gradings and their symmetries

In this paper fine gradings ofgl(n, C) associated with the Pauli matrices inn dimensions are studied with the subsequent graded contractions ofsl(n, C) in view. It is shown that, ifn≥3 is a prime, the discrete symmetries of the gradings involve the specialn-dimensional representations ofSL(2,Fn), whereFn is the finite field of ordern. These symmetries may be used to simplify the system of contraction equations.

Graded contractions of the Pauli graded sl(3, C)

The Lie algebra sl(3, C) is considered in the basis of generalized Pauli matrices. Corresponding grading is the Pauli grading here. It is one of the four gradings of the algebra which cannot be further refined. The set S of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of sl(3, C), which is essentially the finite group SL(2, Z3), and the induced symmetry of the system S. A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classes are non-parametric and 13 solutions depend on one or two continuous parameters, providing a continuum of equivalence classes and subsequently continuum of non-isomorphic Lie algebras. Solutions of the contraction equations of Pauli graded sl(3, C) are identified here as specific solvable Lie algebras of dimensions up to 8. Earlier algorithms for identification of Lie algebras, given by their structure constants, had to be made more efficient in order to distinguish non-isomorphic Lie algebras encountered here. Resulting Lie algebras are summarized in tabular form. There are 88 indecomposable solvable Lie algebras of dimension 8, 77 of them being nilpotent. There are 11 infinite sets of parametric Lie algebra which still deserve further study.

Transformation matrix for the isotropic harmonic oscillator eigenvectors in {n1n2n3} and {nlm} representations

In this paper the properties of the elements (n1n2n3¦nlm〉 of the transformation matrix connecting the eigenvectors ¦n1n2n3) and ¦nlm) of the isotropic harmonic oscillator are investigated. An explicit expression and recursion relations for these elements are derived.

Symmetries of finite Heisenberg groups for multipartite systems

A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Z_{n_i}, i = 1, ... , k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces of dimensions n_1, ... , n_k. The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.

Telescopic system with a rotating objective element

The angular resolution is the ability of a telescope to render detail: the higher the resolution the finer is the detail. It is, together with the aperture, the most important characteristic of telescopes. We propose a new construction of telescopes with improved ratio of angular resolution and area of the primary optical element (mirror or lense). For this purpose we use the rotation of the primary optical element with one dominating dimension. The length of the dominating dimension of the primary optical element determines the angular resolution. During the rotation a sequence of images is stored in a computer and the images of observed objects can be reconstructed using a relatively simple software. The angular resolution is determined by the maximal length of the primary optical element of the system. This construction of telescopic systems allows to construct telescopes of high resolution with lower weight and fraction of usual costs.

On Representations of sl(n, C) Compatible with a Z2-grading

This paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is applied to finite-dimensional representations of sl(n,C) in relation to its Z 2 -gradings. For representation theory of sl(n,C) the Gel’fand-Tseitlin method turned out very efficient. We show that it is not generally true that every irreducible representation can be compatibly graded.

Finite-dimensional *-product and matrix algebras

Abstract*-product formulation of finite-dimensional quantum mechanics introduces an other multiplication law between square real matrices (which play a role of classical observables on discrete phase space). Then the analogy of classical mechanics on discrete space is done by multiplication law where the element of resulting matrix is the product of corresponding elements. The possible deformation is studied.

Quantization, manifolds, topology

In this report certain topological global aspects of quantum theory are reviewed. We consider systems admitting localization on configuration spaces which are differentiable manifolds. Our approach is based on a generalization of systems of imprimitivity — the quantum Borel kinematics which are shown to te classified in terms of global characteristics of topological origin.