6533b7d6fe1ef96bd12666fe
RESEARCH PRODUCT
Two-step nilpotent Leibniz algebras
Manuel ManciniGianmarco La Rosasubject
Numerical AnalysisAlgebra and Number Theory17A32 22A30 20M99Mathematics::History and OverviewMathematics::Rings and AlgebrasMathematics - Rings and AlgebrasSettore MAT/02 - AlgebraRings and Algebras (math.RA)Coquegigrue problemFOS: MathematicsDiscrete Mathematics and CombinatoricsNilpotent Leibniz algebrasGeometry and TopologySettore MAT/03 - GeometriaLeibniz algebrasLie racksdescription
In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. In particular, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical $(2n+1)-$dimensional Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. We also show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2022-03-01 |