arXiv subjects
Publications and source records attributed to Piotr Kosinski.
We reconsider the quasi exactly solvable matrix models constructed recently by R. Zhdanov. The 2$\times$2 matrix operators representing the algebra sl(2) are generalized to matrices of arbitrary dimension and a similar construction is achieved for the algebra sl(n).
By taking a product of two sl(2) representations, we obtain the differential operators preserving some space of polynomials in two variables. This allows us to construct the representations of osp(2,2) in terms of matrix differential operators in two variables. The corresponding operators provide the building blocks for the construction of quasi exactly solvable systems of two and four equations in two variables. Some generalisations are also sketched. The peculiar labelling used for the generators allows us to elaborate a nice deformation of osp(2,2). This gives an appropriate basis for analyzing the quasi exactly solvable systems of finite difference equations.
The bicovariant differential calculus on the three-dimensional Kappa-Poincar'e group and the corresponding Lie-algebra structure are described. The equivalence of this Lie-algebra structure and the three-dimensional $κ$-Poincaré algebra is proved.
The infinitesimal action of Kappa-Poincar'e group on Kappa-Minkowski space is computed both for generators of Kappa-Poincar'e algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out.
The differential calculus on n-dimensional quantum Minkowski space covariant with respect to left action of Kappa-Poincar'e group is constructed and its uniqueness is shown.
The differential calculus on the quantum Heisenberg group is conlinebreak structed. The duality between quantum Heisenberg group and algebra is proved.
The $κ-$Poincare group and its algebra in an arbitrary basis are constructed. The $κ-$de\-formation of the Weyl group and its algebra in any dimensions and in the reference frame in which $g_{00}=0$ are discussed.
The full duality between the $κ$-Poincaré algebra and $κ$-Poincaré group is proved.