Nontrivial Levi Decomposition Research Articles

Representations of Ten-Dimensional Levi Decomposition Lie Algebras

The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the present paper, we obtain a matrix representation for each of these Lie algebras. We are able to find such representations by exploiting properties of the radical, principally, when it has a trivial center, in which case we can obtain such a representation by restricting the adjoint representation. Another important subclass of algebras is where the radical has a codimension one abelian nilradical and for which a representation can readily be found. In general, finding matrix representations for abstract Lie algebras is difficult and there is no algorithmic process, nor is it at all easy to program by computer, even for algebras of low dimension. The present paper represents another step in our efforts to find linear representations for all the low dimensional abstract Lie algebras.

Representations of nine-dimensional Levi decomposition Lie algebras

We obtain a matrix representation for each of the indecomposable 9-dimensional real Lie algebras that have a non-trivial Levi decomposition.

Closed $${\text {G}}_{2}$$ G 2 -structures on non-solvable Lie groups

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$ and the radical is unimodular and centerless. Moreover, we classify unimodular Lie algebras with non-trivial Levi decomposition admitting closed G$_2$-structures.

Minimal representations of Lie algebras with non-trivial Levi decomposition

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.

Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition

The problem of group classification of quasilinear elliptic-type equations in the two-dimensional space is considered. We obtain the lists of all equations of this class that admit the semisimple Lie algebras of symmetry operators and the Lie algebras of symmetry operators with nontrivial Levi decomposition.

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.

A note on the classification of nine dimensional Lie algebras with nontrivial Levi decomposition

We point out an omission in the classification of nine dimensional real indecomposable Lie algebras having a nontrivial Levi decomposition.

Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical

Let g = s n r be an indecomposable Lie algebra with nontrivial semisimple Levi subalgebra s and nontrivial solvable radical r. In this note it is proved that r cannot be isomorphic to a filiform nilpotent Lie algebra. The proof uses the fact that any Lie algebra g = snr with filiform radical would degenerate (even contract) to the Lie algebra snfn, where fn is the standard graded filiform Lie algebra of dimension n = dim r. This leads to a contradiction, since no such indecomposable algebra snr with r = fn exists

Structure of real Lie algebras

Restrictions are presented on the radical structure of an indecomposable Lie algebra. These restrictions are consequences of both the Levi theorem and the representation theory of semisimple Lie algebras. As an application, all real Lie algebras of dimension 9 that admit a nontrivial Levi decomposition are found. These algebras are expressed in terms of known solvable and simple algebras of smaller dimensions and in terms of representations of simple algebras.

Low-dimensional real Lie algebras

All real Lie algebras of dimension up to 8 that admit a nontrivial Levi decomposition are found.

A multidimensional extension of the Bianchi type-IX cosmology

Abstract A higher-dimensional homogeneous spacetime is investigated satisfying the vacuum Einstein equations. It is assumed that the algebra of Killing vectors L admits a non-trivial Levi decomposition L = N ⨴so(3), i.e. that the subalgebras N and so(3) do not commute. It is found that the model behaves in a non-chaotic way and cosmological dimensional reduction inevitably occurs. This model completes all the possible types within the class of higher-dimensional extensions of Bianchi type-IX cosmology.

Classification of multidimensional spacetimes

We classify all eleven-dimensional spatially homogeneous spacetimes, which contain a seven-dimensional compact subspace and admit a simple-transitive group of isometry, by enumerating the real ten-dimensional Lie algebras which contain an n-dimensional (n⩾7) compact Lie subalgebra. The main result of this paper consists in giving a complete list of the distinct real ten-dimensional algebras which admit a non-trivial Levi decomposition. It is hoped that this investigation may be of some help in studying eleven-dimensional and ten-dimensional cosmologies which are considered in the context of Kaluza-Klein, supergravity and superstring theories.