A class of solvable Lie algebras and their Casimir invariants

Abstract

A nilpotent Lie algebra n n,1 with an (n − 1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n n,1 as their nilradical are obtained. Their dimension is at most n + 2. The generalized Casimir invariants of n n,1 and of its solvable extensions are calculated. For n = 4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.

Solvable 3-Lie algebras with a maximal hypo-nilpotent ideal N

“…In this section we describe the one dimensional extension of Lie algebras, first introduced in [6]. As an application of it we get all classes of solvable Lie algebras with the special nilradical given in [19,20]. Then J = F y 0 + I = F y 0 + F e 1 + F e 2 + F e 3 + F e 4 is an ideal of L g .…”

Solvable extensions of a special class of nilpotent Lie algebras

“…In view of this fact Winternitz and colleagues Snobl, Rubin, Karasek, Tremblay, [16,14,22,23,24,25] (see also [26,30]) have established a program whereby they start with a particular nilpotent Lie algebra and try to find all possible solvable extensions of it. In dimension five there are up to isomorphism six nilpotent indecomposable Lie algebras that we denote by N 5,1 , N 5,2 , N 5,3 , N 5,4 , N 5,5 , N 5,6 .…”

“…There also three decomposable Lie algebras R 5 , N 3,1 ⊕ R 2 , N 4,1 ⊕ R. See [12]. The following references [16,14,23,24,25,30] consider sequences of nilpotent Lie algebras that generalize R 5 , N 4,1 ⊕ R, N 5,2 , N 5,4 , N 5,5 , N 5,6 in the sense that the five-dimensional algebras are the first term in the sequence. Then the authors consider all possible extensions to solvable algebras so that the original nilpotent algebra is its nilradical.…”

All solvable extensions of a class of nilpotent Lie algebras of dimensionnand degree of nilpotencyn− 1

“…Earlier articles were devoted to solvable extensions of Heisenberg algebras [45], Abelian Lie algebras [36,37], "triangular" Lie algebras [48,49] and the algebras n n,1 [47].…”

“…The motivation for providing a classification of Lie algebras was discussed in our previous article [47]. Let us mention that string theory and other elementary particle theories require the use of higher dimensional spaces.…”