Formal deformations, contractions and moduli spaces of Lie algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions.Comment: 27 page

Leibniz algebra deformations of a Lie algebra

In this note we compute Leibniz algebra deformations of the 3-dimensional nilpotent Lie algebra $\mathfrak{n}_3$ and compare it with its Lie deformations. It turns out that there are 3 extra Leibniz deformations. We also describe the versal Leibniz deformation of $\mathfrak{n}_3$ with the versal base.Comment: 15 page

Versal deformation of the Lie algebra L2L_2

We investigate deformations of the infinite dimensional vector field Lie algebra spanned by the fields $e_i = z^{i+1}d/dz$, where $i \ge 2$. The goal is to describe the base of a ``versal'' deformation; such a versal deformation induces all the other nonequivalent deformations and solves the deformation problem completely. \u

Stratification of moduli spaces of Lie algebras, similar matrices and bilinear forms

In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of nxn matrices under scalar similarity and bilinear forms under the cogredient action. For similar matrices, we give a complete description of a stratification of the space by some very simple projective orbifolds of the form P^n/G, where G is a subgroup of the symmetric group sigma_{n+1} acting on P^n by permuting the projective coordinates. For bilinear forms, we give a similar stratification up to dimension 4

Extensions of associative algebras

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras. </jats:p

Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac-Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.Comment: 17 page

A characterization of nilpotent Leibniz algebras

W. A. Moens proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper we show that with the definition of Leibniz-derivation from W. A. Moens the similar result for non Lie Leibniz algebras is not true. Namely, we give an example of non nilpotent Leibniz algebra which admits an invertible Leibniz-derivation. In order to extend the results of paper W. A. Moens for Leibniz algebras we introduce a definition of Leibniz-derivation of Leibniz algebras which agrees with Leibniz-derivation of Lie algebras case. Further we prove that a Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. Moreover, the result that solvable radical of a Lie algebra is invariant with respect to a Leibniz-derivation was extended to the case of Leibniz algebras.Comment: arXiv admin note: text overlap with arXiv:1103.472

Deformations of modules of differential forms

We study non-trivial deformations of the natural action of the Lie algebra $\mathrm{Vect}({\mathbb R}^n)$ on the space of differential forms on ${\mathbb R}^n$. We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of \cite{ff}.Comment: Published by JNMP at http://www.sm.luth.se/math/JNM

On Deformations of n-Lie algebras

The aim of this paper is to review the deformation theory of $n$-Lie algebras. We summarize the 1-parameter formal deformation theory and provide a generalized approach using any unital commutative associative algebra as a deformation base. Moreover, we discuss degenerations and quantization of $n$-Lie algebras.Comment: Proceeding of the conference Dakar's Workshop in honor of Pr Amin Kaidi. arXiv admin note: text overlap with arXiv:hep-th/9602016 by other author