Internal labelling operators and contractions of Lie algebras

We analyze under which conditions the missing label problem associated to a reduction chain $\frak{s}^{\prime}\subset \frak{s}$ of (simple) Lie algebras can be completely solved by means of an In\"on\"u-Wigner contraction $\frak{g}$ naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labeling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions of semisimple algebras.Comment: 20 pages, 2 table

A comment concerning cohomology and invariants of Lie algebras with respect to contractions and deformations

Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint module. A criterion to decide whether a given deformation is invertible or not is given in dependence of the Poincar\'e polynomial.Comment: 13 pages, 1 tabl

Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras

Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie algebras $I\frak{u}(p,q)$. This procedure is extended to contractions of $I\frak{u}(p,q)$ isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Lie algebras $I\frak{su}(p-1,q)$, providing an additional analytical method to obtain their invariants. Further, matrix formulae for the invariants of other inhomogeneous Lie algebras are presented.Comment: Final ammended versio

Invariant Tensors Formulae via Chord Diagrams

We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras $so(n)$ and $sp(n)$.Comment: 18 pages, 8 figures. To appear in a special issue of Journal of Mathematical Science

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invariants of $\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.Comment: 20 pages, 2 Appendice

Unitary representations of three dimensional Lie groups revisited: An approach via harmonic functions

Harmonic functions of the three dimensional Lie groups defined on certain manifolds related to the Lie groups themselves and carrying all their unitary representations are explicitly constructed. The realisations of these Lie groups are shown to be related with each other by either natural operations as real forms or In\"on\"u-Wigner contractions.Comment: The title was changed; More details are given for the constuction of harmonic functions 19 page