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A class of solvable Lie algebras and their Casimir invariants. (English) Zbl 1063.22023

Summary: A nilpotent Lie algebra \({\mathfrak n}_{n,1}\) with an \((n- 1)\)-dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with \({\mathfrak n}_{n,1}\) as their nilradical are obtained. Their dimension is at most \(n+ 2\). The generalized Casimir invariants of \({\mathfrak n}_{n,1}\) 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.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B70 Graded Lie (super)algebras
22E60 Lie algebras of Lie groups

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