Two-step nilpotent Leibniz algebras (Q2070845)

A (left) Leibniz algebra \(L\) has a (not necessarily antisymmetric) bracket satisfying the axiom \[ [x,[y,z]]=[[x,y],z]+[y,[x,z]]. \] A notion of nilpotent Leibniz algebra is introduced, by similarity with Lie algebras. In particular, a Leibniz algebra is 2-steps nilpotent if \([L,[L,L]]=(0)\). A description of \(2\)-steps nilpotent Leibniz algebra is done, in terms of Kronecker modules and of bilinear forms. A description of these objects is done in the real and complex case, leading to three families: Heisenberg, Kronecker and Dieudonné. The complete description of the Heisenberg case is lead in the real case. The problem of integrations of Leibniz algebras into Lie racks (coquecigrue problem) is then studied, using Covez's local integration. It is shown that in the real case, any nilpotent Leibniz algebra has a global integration. As examples, Leibniz algebras \(L\) such that \([L,L]\) is one-dimensional are integrated. It is finally shown that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the underlying Lie algebra of this group.