On a pair of difference equations for the 4F3_4F_3 type orthogonal polynomials and related exactly-solvable quantum systems

We introduce a pair of novel difference equations, whose solutions are expressed in terms of Racah or Wilson polynomials depending on the nature of the finite-difference step. A number of special cases and limit relations are also examined, which allow to introduce similar difference equations for the orthogonal polynomials of the $_3 F_2$ and $_2 F_1$ types. It is shown that the introduced equations allow to construct new models of exactly-solvable quantum dynamical systems, such as spin chains with a nearest-neighbour interaction and fermionic quantum oscillator models.Comment: 8 pages, to be published in Springer Proceedings in Mathematics & Statistic

Wigner quantum systems (Lie superalgebraic approach)

We present three groups of examples of Wigner Quantum Systems related to the Lie superalgebras $osp(1/6n)$, $sl(1/3n)$ and $sl(n/3)$ and discuss shortly their physical features. In the case of $sl(1/3n)$ we indicate that the underlying geometry is noncommutative.Comment: 8 pages, Latex. to be published in Rep. Math. Phy

Parafermionic algebras, their modules and cohomologies

We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded 2-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant's theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincar\'e characteristics of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.Comment: 10 pages, talk presented at the International Workshop "Lie theory and its applications in Physics" (17-23 June 2013, Varna, Bulgaria

On the n-particle Wigner quantum oscillator: non-commutative coordinates and particle localisation

After a general introduction to Wigner Quantum Systems, we define the three-dimensional n-particle Wigner Quantum Oscillator, and its relation to the Lie superalgebra sl(1|3n). In this framework, coordinates, momentum and the angular momentum of the particles are defined and investigated. This investigation is done in state spaces, using representation theory of sl(1|3n). For the case of n=1, the main properties are listed, with an emphasis on the non-commutative coordinates of the particle and its unconventional consequences. For general n, we study energy spectra, angular momentum, and in particular the particle configuration and its interpretation. Throughout, a comparison with the canonical oscillator solution is given

Wigner quantum oscillators

This work was supported by the European Community, contract n. ERB-CIPA-CT92-2011 (Cooperation in Science and Technology with Central and Eastern European Countries) and Grand PHY-215 of the Bulgarian Foundation for Scientific ResearchConsiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Unitarizable represtantions of the deformed para-Bose superalgebra (U_q osp(1/2) at roots of 1

The work was supported by the Grant PHY-416 of the Bulgarian Foundation for Scientific ResearchConsiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Many-body Wigner quantum systems

Consiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Wigner quantum oscillators: osp(3/2) oscillators

This work was supported by the Grant "PHY"-215 of the Bulgarian Foundation for Scientific ResearchConsiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Highest weight representations of the quantum algebra U_h(gl#infinity#)

Consiglio Nazionale delle Ricerche - Biblioteca Centrale -. P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal