Simply transitive NIL-affine actions of solvable Lie groups

Abstract: Although not every $1$-connected solvable Lie group $G$ admits a simply transitive action via affine maps on $\mathbb{R}^n$, it is known that such an action exists if one replaces $\mathbb{R}^n$ by a suitable nilpotent Lie group $N$, depending on $G$. However, not much is known about which pairs of Lie groups $(G,N)$ admit such an action, where ideally you only need information about the Lie algebras corresponding to $G$ and $N$. The most-studied case is when $G$ is assumed to be nilpotent, then the existence of a simply transitive action is related to the notion of complete pre-Lie algebra structures.

In recent work with Marcos Origlia, we showed how this problem is related to the semisimple splitting of the Lie algebra corresponding to $G$. Our characterization not only allows us to check whether a given action is simply transitive, but also whether a simply transitive action exists given the Lie groups $G$ and $N$. As a consequence, we list the possibilities for such actions up to dimension $4$.

differential geometry

Audience: researchers in the topic

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Virtual seminar on geometry with symmetries

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