The Quantum Deformed Dirac Equation from the k-Poincare` Algebra

In this letter we derive a deformed Dirac equation invariant under the k-Poincare` quantum algebra. A peculiar feature is that the square of the k-Dirac operator is related to the second Casimir (the k-deformed squared Pauli-Lubanski vector). The ``spinorial'' realization of the k-Poincare` is obtained by a contraction of the coproduct of the real form of SO_q(3,2) using the 4-dimensional representation which results to be, up some scalar factors, the same of the undeformed algebra in terms of the usual gamma matrices.Comment: 6 pages, Late

Coherent Orthogonal Polynomials

We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put thus --in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions-- Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis ${|x>}$, for an alternative countable basis ${|n>}$. The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine a unitary representation of a non-compact Lie algebra, whose second order Casimir ${\cal C}$ gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl-Heisenberg algebra $h(1)$ with ${\cal C}=0$ for Hermite polynomials and $su(1,1)$ with ${\cal C}=-1/4$ for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the ${\cal L}^2$ functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space ${\cal L}^2$ and, in particular, generalized coherent polynomials are thus obtained.Comment: 11 page

Quantum κ\kappa-Poincare in Any Dimensions

The $\kappa$-deformation of the D-dimensional Poincar\'e algebra $(D\geq 2)$ with any signature is given. Further the quadratic Poisson brackets, determined by the classical $r$-matrix are calculated, and the quantum Poincar\'e group "with noncommuting parameters" is obtained.Comment: (PLAIN TeX, 10 pp.