A subalgebra of a finite dimensional Lie algebra is said to be a -subalgebra if there is a chief series of such that for every , we have or . This is analogous to the concept of -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.
A pair of sequences of nilpotent Lie algebras denoted by and are introduced. Here denotes the dimension of the algebras that are defined for ; the first term in the sequences are denoted by 6.11 and 6.19, respectively, in the standard list of six-dimensional Lie algebras. For each of and all possible solvable extensions are constructed so that and serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating...
The purpose of this paper is to study some properties of Filiform Lie Algebras (FLA) and to prove the following theorem: a FLA, of dimension n, is either derived from a Solvable Lie Algebra (SLA) of dimension n+1 or not derived from any LA.
A new concept of spectrum for a solvable Lie algebra of operators is introduced, extending the Taylor spectrum for commuting tuples. This spectrum has the projection property on any Lie subalgebra and, for algebras of compact operators, it may be computed by means of a variant of the classical Ringrose theorem.
Soit une algèbre de Lie complètement résoluble sur un corps de caractéristique zéro. Soit un idéal -invariant de l’algèbre symétrique de . L’application de Dixmier pour associe à un idéal premier de l’algèbre enveloppante de . Soit l’algèbre des opérateurs différentiels à coefficients séries formelles. Dans l’algèbre des opérateurs différentiels à coefficients polynomiaux, il y a un idéal à gauche qui contient et les champs de vecteurs adjoints. Il y a un plongement canonique...