Engineering Mathematics II pp 1-12 | Cite as

# Classification of Low Dimensional 3-Lie Superalgebras

## Abstract

A notion of *n*-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with *n*-ary multiplication law. A notion of Lie algebra can be extended to \(\mathbb Z_2\)-graded structures giving a notion of Lie superalgebra. Analogously a notion of *n*-Lie algebra can be extended to \(\mathbb Z_2\)-graded structures by means of a graded Filippov identity giving a notion of *n*-Lie superalgebra. We propose a classification of low dimensional 3-Lie superalgebras. We show that given an *n*-Lie superalgebra equipped with a supertrace one can construct the \((n+1)\)-Lie superalgebra which is referred to as the induced \((n+1)\)-Lie superalgebra. A Clifford algebra endowed with a \(\mathbb Z_2\)-graded structure and a graded commutator can be viewed as the Lie superalgebra. It is well known that this Lie superalgebra has a matrix representation which allows to introduce a supertrace. We apply the method of induced Lie superalgebras to a Clifford algebra to construct the 3-Lie superalgebras and give their explicit description by ternary commutators.

## Keywords

n-Lie algebras n-Lie superalgebras Clifford algebras Induced n-Lie superalgebras## Notes

### Acknowledgement

The authors is gratefully acknowledge the Estonian Science Foundation for financial support of this work under the Research Grant No. ETF9328. This research was also supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The authors are also grateful for partial support from Linda Peetres Foundation for cooperation between Sweden and Estonia provided by Swedish Mathematical Society.

## References

- 1.Abramov, V.: Super 3-Lie algebras induced by super Lie algebras. (to appear in Advances in Applied Clifford Algebras)Google Scholar
- 2.Abramov, V., Kerner, R., Le Roy, B.: Hypersymmetry: A \(\mathbb{Z}_3\)-graded generalization of supersymmetry. J. Math. Phys.
**38**, 1650–1669 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S.: Structure and cohomology of 3-Lie algebras induced by Lie algebras. In: Makhlouf, A., Paal, E., Silvestrov, S.D., Stolin, A. (eds.) Algebra, Geometry and Mathematical Physics, pp. 123–144. Springer Proceedings in Mathematics & Statistics, Mulhouse, France (2014)Google Scholar
- 4.Daletskii, Yu. L., Kushnirevitch,V. A.: Formal differential geometry and Nambu-Takhtajan algebra. In: Budzynski, R., Pusz, W., Zakrzewski, S. (eds.) Quantum Groups and Quantum Spaces,
**40**, pp. 293–302. Banach Center Publications (1997)Google Scholar - 5.
- 6.Mathai, V., Quillen, D.: Superconnections, Thom classes, and equivariant differential forms. Topology
**25**(1), 85–110 (1986)MathSciNetCrossRefzbMATHGoogle Scholar