We study a certain class of non-maximal rank contractions of the nilpotent Lie algebra $\frak{g}_{m}$ and show that these contractions are completable Lie algebras. As a consequence a family of solvable complete Lie algebras of non-maximal rank is given in arbitrary dimension.

Rutwig Campoamor-Stursberg

We introduce the concept of weight graph for the weight system $P\frak{g}(T)$ of a finite dimensional nilpotent Lie algebra $\frak{g}$ and analyze the necessary conditions for a $(p,q)$-graph to be a weight graph for some $\frak{g}$.

Otto Rutwig Campoamor Jose Maria Ancochea

We prove that if $\frak{g}^{\prime}$ is a contraction of a Lie algebra $\frak{g}$ then the number of functionally independent invariants of $\frak{g}^{\prime}$ is at least that of $\frak{g}$. This allows to determine explicitly the number of invariants of Lie algebras carrying a supplementary structure, such as linear contact or linear forms whose differential is symplectic.

Rutwig Campoamor-Stursberg

We describe the invariants for the coadjoint representation of all real Lie algebras with nontrivial Levi decomposition up to dimension eight.

Rutwig Campoamor-Stursberg

It is shown that for N=2 supersymmetry a hidden symmetry arises from the hybrid structure of a quartic algebra. The implications for invariant Lagrangians and multiplets are explored.

Rutwig Campoamor-Stursberg Michel Rausch de Traubenberg

The generalized Casimir invariants of real indecomposable Lie algebras admitting a nontrivial Levi decomposition are determined.

R. Campoamor-Stursberg

We construct large families of characteristically nilpotent Lie algebras by considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and which arises as a central extension fo the filiform Lie algebra L_{n}. By studying the graded cohomology spaces we obtain that the sill algebras are isomorphic to the nilradicals of solvable, complete Lie algebra laws. For extremal cocycles these laws are also rigid. Considering supplementary cocycles we construcy, for dimensions n>8, nonfiliform characteristically nilpotent Lie algebras and show that for certain deformations these are compatible with central extensions.

Jose Maria Ancochea-Bermudez Otto Rutwig Campoamor-Stursberg

We classify the (n-5)-filiform Lie algebras which have the additional property of a non-abelian derived subalgebra. We show that this property is strongly related with the structure of the Lie algebra of derivations; explicitely we show that if a (n-5)-filiform algebra is characteristically nilpotent, then it must be 2-abelian. We also give applications of k-abelian Lie algebras to the construction of solvable rigis algebras, as well as to the theory of nilalgebras of parabolic subalgebras in the example of the exceptional simple model E_{6}.

Otto Rutwig Campoamor

In this work large families of naturally graded nilpotent Lie algebras in arbitrary dimension and characteristic sequence (n,q,1), with n odd, satisfying the centralizer property, are given. This condtion constitutes a generalization, for a nilpotent Lie agebra, of the structural properties charactrizing the Lie algebra $Q_{n}$. By considering certain cohomological classes of the space $H^{2}(\frak{g},\mathbb{C})$, it is shown that, with few exceptions, the isomorphism classses of these algebras are given by central extensions of $Q_{n}$ by $\mathbb{C}^{p}$ which preserve the nilindex and the natural graduation.

Jose Maria Ancochea Otto Rutwig Campoamor

By using the concept of weight graph associated to certain nilpotent Lie algebras $\frak{g}$, we find necessary and sufficient conditions for a semidirect product $\frak{g}\oplus T_{i}$, where $T_{i}<T$ is a subalgebra of a maximal torus of derivations $T$ of $\frak{g}$ which induces a decomposition of $\frak{g}$ into one dimensional weight spaces, to be 2-step solvable. In particular we show that the semidirect product of such a Lie algebra with its torus of derivations cannot be itself 2-step solvable.

Jose Maria Ancochea Otto Rutwig Campoamor