About Leibniz cohomology and deformations of Lie algebras
Alice Fialowski, Louis Magnin, Ashis Mandal
TL;DR
This paper compares Leibniz and Lie algebra cohomology to identify conditions for additional Leibniz deformations and provides explicit descriptions of versal deformations for specific low-dimensional Lie algebras.
Contribution
It introduces an elementary method to compare Leibniz and Lie cohomology, enabling easy verification of extra Leibniz deformations, and explicitly describes versal deformations of certain low-dimensional Lie algebras.
Findings
Conditions for extra Leibniz deformations identified
Complete description of deformations for 4-dimensional diamond Lie algebra
Analysis of 5-dimensional analogue deformations
Abstract
We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding whether it has more Leibniz deformations than just the Lie ones. We also give the complete description of a Leibniz (and Lie) versal deformation of the 4-dimensional diamond Lie algebra, and study the case of its 5-dimensional analogue.
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