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Let g be a Lie algebra of dimension n over a field K . Then g determines a multiplication table relative to each basis {e1, . . . , en}. If [ei, ej] = ∑n k=1 γ k i,jek , then (γ k i,j) ∈ K n is called a structure for g and the γ i,j the structure constants of g. The elements of Ln(K) are exactly the Lie algebra structures. They form an affine algebraic… (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)

- DIETRICH BURDE, 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
- 2005

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry and 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… (More)

- DIETRICH BURDE
- 2008

We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space H(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
- 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
- 2006

We classify the cohomology spaces H(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H(g,K). We prove the general result that the existence of an affine cohomology class implies an affine structure of… (More)

- DIETRICH BURDE
- 2008

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of… (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, 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)