An irreducible component of the variety of Leibniz algebras having trivial intersection with the variety of Lie algebras

Abstract

We show the rigidity of a parameterized family of solvable Leibniz non-Lie algebras in arbitrary dimension, obtaining an irreducible component in the variety that does not intersect the variety of Lie algebras non-trivially. Moreover it is shown that for any the Abelian Lie algebra appears as the algebra of derivations of a solvable Leibniz algebra.

Keywords:

• Leibniz algebra
• rigidity
• irreducible component
• derivations

AMS Subject Classifications:

• 17A32

Acknowledgements

The authors acknowledge valuable comments by the referee that improved the presentation of this work. The second author (RCS) would also like to acknowledge the financial support of the MICINN research project MTM2010-18556 during the preparation of this work.

Notes

1. Technically, much of the results can be applied to algebraically closed fields of characteristic zero.