Lie algebras with associative structures. Applications to the study of 2-step nilpotent Lie algebras

Abstract

We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.

Cartan class of Invariant forms on Lie groups

“…Its bracket µ is a deformation of the bracket µ 0 of the Heisenberg algebra h 2p+1 . But, from [18], µ 0 is rigid in the variety 2N ilp(2p + 1) of (2p + 1)-dimensional 2-step nilpotent Lie algebras. This means that any deformation of µ 0 in a 2-step nilpotent Lie algebra is isomorphic to µ 0 .…”