Refinement of Ado's Theorem in Low Dimensions and Application in Affine Geometr

TL;DR

This paper constructs minimal faithful representations for complex Lie algebras of dimension up to 4, leading to simple affine structures with applications in affine geometry and algebra.

Contribution

It provides explicit minimal faithful representations for low-dimensional complex Lie algebras and introduces affine representations with desirable geometric properties.

Findings

Constructed minimal faithful representations for all complex Lie algebras of dimension ≤ 4.

Established affine representations inducing compatible left-symmetric algebras.

Showed these affine representations lack nontrivial one-parameter translation subgroups.

Abstract

In this paper, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension $\leq 4$. In particular, in our construction, in the case that the faithful representation has the same dimension of the Lie algebra, it can induce an \'etale affine representation with base zero which has a natural and simple form and gives a compatible left-symmetric algebra on the Lie algebra. Such affine representations do not contain any nontrivial one-parameter subgroups of translation.