# Poincaré and sl(2) algebras of order 3

@article{Goze2007PoincarAS, title={Poincar{\'e} and sl(2) algebras of order 3}, author={Michel Goze and Michel Rausch de Traubenberg and Adrian Tanasa}, journal={Journal of Mathematical Physics}, year={2007}, volume={48}, pages={093507-093507} }

In this paper, we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on sl(2,C) and iso(1, 3) the Poincare algebra in four dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3.

## 22 Citations

### About Filiform Lie Algebras of Order 3

- Mathematics
- 2015

The aim of this work is to review recent advances in generalizing filiform Lie (super)algebras into the theory of Lie algebras of order F. Recall that the latter type of algebras constitutes the…

### Kinematical superalgebras and Lie algebras of order 3

- Mathematics
- 2008

We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonu–Wigner contractions…

### HIGHER-ORDER EXTENSIONS OF THE POINCARÉ ALGEBRA

- Mathematics
- 2012

Cubic extensions of the Poincare algebra can be constructed in consistency with physical assumptions, and provide new insights for the description of phenomena. In this paper we review some recent…

### Filiform Lie algebras of order 3

- Mathematics
- 2014

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of…

### Parafermions for higher order extensions of the Poincaré algebra and their associated superspace

- Mathematics
- 2009

Parafermions of orders 2 and 3 are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincaré algebra. The corresponding superfields are…

### Hopf algebras for ternary algebras

- Mathematics
- 2008

We construct a universal enveloping algebra associated with the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincare–Birkhoff–Witt theorem is proven is this context.…

### Some Results on Cubic and Higher Order Extensions of the Poincare Algebra

- Mathematics
- 2008

In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincare algebra. We first give some general properties of Lie superalgebras with some emphasis on…

### Cubic extentions of the Poincaré algebra

- Mathematics
- 2006

A systematic study of nontrivial cubic extensions of the Poincare algebra in four dimensions is undertaken. Explicit examples are given with various techniques (Young tableau, characters, etc.).

## References

SHOWING 1-10 OF 44 REFERENCES

### Finite dimensional Lie algebras of order F

- Mathematics
- 2002

F-Lie algebras are natural generalizations of Lie algebras (F=1) and Lie superalgebras (F=2). When F>2 not many finite-dimensional examples are known. In this article we construct finite-dimensional…

### Lie subalgebras of the Weyl algebra. Lie algebras of order 3 and their application to cubic supersymmetry

- Mathematics
- 2005

In the first part we present the Weyl algebra and our results concerning its finite-dimensional Lie subalgebras. The second part is devoted to a more exotic algebraic structure, the Lie algebra of…

### Fractional superLie algebras and groups

- Mathematics
- 2000

The nth root of a Lie algebra and its dual (that is the fractional supergroup) based on the permutation group Sn invariant forms is formulated in the Hopf algebra formalism. Detailed discussion of…

### Valued Deformations of Algebras

- Mathematics
- 2002

We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as…

### Nontrivial Extensions of the 3D-Poincaré Algebra and Fractional Supersymmetry for Anyons

- Mathematics
- 1997

Nontrivial extensions of three-dimensional Poincare algebra, beyond the supersymmetric one, are explicitly constructed. These algebraic structures are the natural three-dimensional generalizations of…

### 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…

### On the Classification of Rigid Lie Algebras

- Mathematics
- 2001

After having given the classification of solvable rigid Lie algebras of low dimensions, we study the general case concerning rigid Lie algebras whose nilradical is filiform and present their…

### Non-trivial extension of the Poincar\'e algebra for antisymmetric gauge fields

- Mathematics, Physics
- 2004

We investigate a non-trivial extension of the $D-$dimensional Poincar\'e algebra. Matrix representations are obtained. The bosonic multiplets contain antisymmetric tensor fields. It turns out that…

### Z3 ‐graded algebras and the cubic root of the supersymmetry translations

- Mathematics
- 1992

A generalization of supersymmetry is proposed based on Z3 ‐graded algebras. Introducing the objects whose ternary commutation relations contain the cubic roots of unity, e2πi/3, e4πi/3 and 1, the…

### Fractional supersymmetry and Fth-roots of representations

- Mathematics
- 1999

A generalization of super-Lie algebras is presented. It is then shown that all known examples of fractional supersymmetry can be understood in this formulation. However, the incorporation of…