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- D. BURDE
- 2005

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)

- DIETRICH BURDE
- 2008

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than… (More)

- DIETRICH BURDE
- 2014

We compare the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals for finite-dimensional Lie algebras. We show that these dimensions coincide for solvable Lie algebras over an algebraically closed field of characteristic zero. We compute this invariant for all complex nilpotent Lie algebras of dimension n ≤ 7. Furthermore… (More)

- Dietrich Burde
- 1999

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)

- DIETRICH BURDE
- 2008

We study symplectic structures on characteristically nilpotent Lie algebras (CN-LAs) by computing the cohomology space H 2 (g, k) for certain Lie algebras g. Among these Lie algebras are filiform CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an… (More)

- Dietrich Burde, Karel Dekimpe, Sandra Deschamps
- 2008

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)

- Dietrich Burde, Karel Dekimpe, Kim Vercammen
- 2008

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)

- Thomas Beně, S And, Dietrich Burde
- 2008

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)

- DIETRICH BURDE
- 2007

We prove an explicit formula for the invariant µ(g) for finite-dimensional semisim-ple, and reductive Lie algebras g over C. Here µ(g) is the minimal dimension of a faithful linear representation of g. The result can be used to study Dynkin's classification of maximal reductive subalgebras of semisimple Lie algebras.