Rutwig Campoamor-Stursberg

Solvable Lie algebras with naturally graded nilradicals and their invariants

The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analysed, and their generalized Casimir invariants are calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension 2n - 1, indicating that gauge theories (with ghosts) are possible on these subalgebras.

Invariants of solvable rigid Lie algebras up to dimension 8

The invariants of all complex solvable rigid Lie algebras up to dimension 8 are computed. Moreover we show, for rank 1 solvable algebras, some criteria to deduce the non-existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the Casimir invariants of the associated nilradical.

Kinematical superalgebras and Lie algebras of order 3

We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonu–Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.

Higher-order superintegrability of a Holt related potential

In a recent paper, Post and Winternitz (2011 J. Phys. A: Math. Theor. 44 162001) studied the properties of two-dimensional Euclidean potentials that are linear in one of the two Cartesian variables. In particular, they proved the existence of a potential endowed with an integral of third order and an integral of fourth order. In this paper we show that these results can be obtained in a more simple and direct way by noting that this potential is directly related to the Holt potential. It is proved that the existence of a potential with higher-order superintegrability is a direct consequence of the integrability of the family of Holt type potentials.

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

Given a semidirect product g = s ⊎ r of semisimple Lie algebras s and solvable algebras r, we construct polynomial operators in the enveloping algebra U(g) of g that commute with r and transform like the generators of s, up to a functional factor that turns out to be a Casimir operator of r. Such operators are said to generate a virtual copy of s in U(g), and allow to compute the Casimir operators of g in closed form, using the classical formulae for the invariants of 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.

Non-semisimple Lie algebras with Levi factor fraktur so (3), fraktur sfraktur l(2, Bbb R) and their invariants

We analyze the number N of functionally independent generalized Casimir invariants for non-semisimple Lie algebras \frak{s}\overrightarrow{% oplus}_{R}\frak{r} with Levi factors isomorphic to \frak{so}(3) and \frak{sl}(2,R) in dependence of the pair (R,\frak{r}) formed by a representation R of \frak{s} and a solvable Lie algebra \frak{r}. We show that for any dimension n >= 6 there exist Lie algebras \frak{s}\overrightarrow{\oplus}_{R}\frak{r} with non-trivial Levi decomposition such that N(\frak{s}% \overrightarrow{oplus}_{R}\frak{r}) = 0.

Non-solvable contractions of semisimple Lie algebras in low dimension

The problem of non-solvable contractions of Lie algebras is analysed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras of dimension n ≤ 8, and obtain the non-solvable contractions of the latter class of algebras.

Parafermions for higher order extensions of the Poincare algebra and their associated superspace

Parafermions of orders 2 and 3 are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincare algebra. The corresponding superfields are constructed, and some of their main properties are analyzed in detail. In this context, the existence problem of operators acting like covariant derivatives is analyzed, and the associated operators are explicitly constructed.

A new matrix method for the Casimir operators of the Lie algebras and

A method is given to determine the Casimir operators of the perfect Lie algebras and the inhomogeneous Lie algebras in terms of polynomials associated with a parametrized (2N + 1) × (2N + 1)-matrix. For the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation. We further analyse the invariants for the extended Schrodinger algebra in (N + 1) dimensions, which arises naturally as a subalgebra of . The method is extended to other classes of Lie algebras, and some applications to the missing label problem are given.

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.