Dietrich Burde

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We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of(More)
In this paper we study degenerations of nilpotent Lie algebras. If λ, µ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to µ , λ → deg µ , if µ lies in the Zariski closure of the orbit of λ. It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We(More)
We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension µ(g) of a faithful g-module for some nilpotent Lie algebras g. In particular, we describe an infinite family of filiform nilpotent Lie(More)
To any connected and simply connected nilpotent Lie group N , one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N , via such affine transformations. We succeed in translating the existence question of such a simply transitive affine(More)
We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras(More)
We consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL n (K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie algebra degenerations. We give fundamental results on such degenerations, including invariants and necessary(More)