The moduli space of three-dimensional Lie algebras

Abstract

Abstract. In this paper, we consider the versal deformations of three dimensional Lie algebras. We classify Lie algebras and study their deformations by using linear algebra techniques to study the cohomology. We will focus on how the deformations fasten the space of all such structures together. This space is known as the moduli space. We will give a geometric description of this space, derived from deformation theory, in order to illustrate general features of Lie algebras' moduli spaces.

Formal Deformations, Contractions and Moduli Spaces of Lie Algebras

“…These moduli spaces were studied in detail in [9,10,14]. When constructing moduli spaces of Lie and L ∞ algebras, we had noticed that jump deformations were always transitive.…”

“…The set of equivalence classes of Lie algebras over a fixed vector space is called the moduli space of Lie algebras on that vector space. In [9,14], the moduli space of Lie algebras of dimension 3 was carefully analyzed, and in [10], a construction of the moduli space of 4-dimensional Lie algebras was given. The main idea which we used in our analysis was the computation of the miniversal deformation, which allows one to determine all possible deformations of the Lie algebra (see [3,5]).…”

“…We also give a description of the moduli space of real 3-dimensional Lie algebras. In [9, 14,10], only complex Lie algebras were studied. We also use a miniversal deformation approach to give a complete description of the contractions of all 4-dimensional complex Lie algebras.…”

Extensions of associative algebras

“…Second, when studying the 3 | 0-dimensional complex L ∞ algebras, which are just the ordinary 3-dimensional complex Lie algebras, we noticed that the moduli space had a stratification by complex orbifolds [7,22], and we verified that the same type of stratification occurs for 4-dimensional complex Lie algebras in [8]. This led us to a conjecture that finite-dimensional complex Lie algebras have a certain natural stratification by complex orbifolds, which we have since been exploring.…”

The Bianchi variety

“…For instance, in [4] this is done on the basis of the invariants of Lie structures, in [2] the co-differential graded calculus is used, in [2] the outer derivations, etc. A shortcoming of the original Bianchi method, as well as of the above-cited works, is that they do not allow a satisfactory description of deformations of 3-dimensional Lie algebras (see [2,3,8] and references therein). It should be especially stressed the recently emerged important role of Poisson geometry in various questions related with Lie algebras and, first of all, classification, representation, deformations, etc.…”