# Non-solvable contractions of semisimple Lie algebras in low dimension

@article{CampoamorStursberg2007NonsolvableCO, title={Non-solvable contractions of semisimple Lie algebras in low dimension}, author={Rutwig Campoamor-Stursberg}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2007}, volume={40}, pages={5355 - 5372} }

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.

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## References

SHOWING 1-10 OF 59 REFERENCES

### Contractions of the Low‐Dimensional Real Lie Algebras

- Mathematics
- 1972

A complete and detailed classification and analysis of all the Inonu‐Wigner contractions of all the real Lie algebras of dimension 1, 2, and 3 is presented. Starting with a more natural…

### Central extensions of the families of quasi-unitary Lie algebras

- Mathematics
- 1998

The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras of the Cartan series and the…

### A new matrix method for the Casimir operators of the Lie algebras and

- Mathematics
- 2005

A method is given to determine the Casimir operators of the perfect Lie algebras and the inhomogeneous Lie algebras in terms of polynomials associated with a parametrized (2N + 1) × (2N + 1)-matrix.…

### Deformation and Contraction of Lie Algebras

- Mathematics
- 1967

This paper deals with the theory of deformation of Lie algebras. A connection is established with the usual contraction theory, which leads to some ``more singular'' contractions. As a consequence it…

### Contractions of low-dimensional Lie algebras

- Mathematics
- 2006

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and…

### Group theory in physics

- Mathematics
- 1984

OF VOLUME 2: The Role of Lie Algebras. Relationships between Lie Groups and Lie Algebras. The Three-Dimensional Rotation Groups. The Structure of Semi-Simple Lie Algebras. Semi-Simple Real Lie…

### The three‐dimensional real Lie algebras and their contractions

- Mathematics
- 1991

In this paper, the general contractions of the three‐dimensional real Lie algebras are determined. A short summary will be given of the history, definitions, properties, examples, and applications of…

### Contractions: Nijenhuis and Saletan tensors for general algebraic structures

- Mathematics
- 2001

We study generalizations in many directions of the contraction procedure for Lie algebras introduced by Saletan. We consider products of an arbitrary nature, not necessarily Lie brackets, and we…

### COHOMOLOGY AND DEFORMATIONS IN GRADED LIE ALGEBRAS

- Mathematics
- 1966

Abstract : The theories of deformations of associative algebras, Lie algebras, and of representations and homomorphisms of these all show a striking similarity to the theory of deformations of…

### Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics: A first look at cohomology of groups and related topics

- Mathematics
- 1995

Preface 1. Lie groups, fibre bundles and Cartan calculus 2. Connections and characteristic classes 3. A first look at cohomology of groups and related topics 4. An introduction to abstract group…