Abstract
Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations.
Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻.
So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures.
This paper studies two different aspects of this problem.
First, we give a method to check whether a given action
ρ
:
G
→
Aff
(
H
)
\rho\colon G\to\operatorname{Aff}(H)
is simply transitive by looking only at the induced morphism
φ
:
g
→
aff
(
h
)
\varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h})
between the corresponding Lie algebras.
Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras.
The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras.
As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.
Post-Lie algebra structures on solvable Lie algebra t(2,C)
