Formal Deformations, Contractions and Moduli Spaces of Lie Algebras

Abstract

Abstract. 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 the 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.

The Moduli Space of 1|1-Dimensional Complex Associative Algebras

“…In [3], the theory of extensions of an algebra W by an algebra M is described in the language of codifferentials. Consider the diagram 0 → M → V → W → 0 of associative K-algebras, so that V = M ⊕ W as a K-vector space, M is an ideal in the algebra V , and W = V /M is the quotient algebra.…”

The Moduli Space of 3-Dimensional Associative Algebras

“…A construction of a versal deformation for Lie algebras was given in [4], which carries over without any difficulties for associative algebras. A generalization of this construction to the case of infinity algebras appeared in [5]. A versal deformation of an associative algebra given by the codifferential d is a formal deforma- …”

Infinitesimal deformations of naturally graded filiform Leibniz algebras

“…In particular, each point of our moduli space corresponds to one geometric object (class of isomorphism). The theory of deformations is one of the most effective approaches in the investigation of solvable and nilpotent Lie algebras (see for example, [3][4][5][6]). …”