Affine actions on Lie groups and post-Lie algebra structures

By Dietrich Burde, K. Dekimpe, Kim Vercammen. 2011

64 citations recorded by the source corpus.

We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space $V$. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras $\Lg$ and $\Ln$. One result is, for example, that if there exists a post-Lie algebra structure on $(\Lg,\Ln)$, where $\Lg$ is nilpotent, then $\Ln$ must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/20d0bbd060832b54c2b18b60aa26a9451b3848e5

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In Algebra and Number Theory

1. Valeriy G. Bardakov
2. Xiaomin Tang
3. Vsevolod Gubarev

In Geometry and Topology

1. Valeriy G. Bardakov
2. Xiaomin Tang
3. Vsevolod Gubarev
4. S. Stević

In Mathematical Physics

1. Valeriy G. Bardakov
2. Xiaomin Tang
3. Vsevolod Gubarev

Citations in the corpus

• Rota—Baxter groups, skew left braces, and the Yang—Baxter equation (2021, background): “A space A with two bilinear operations [, ] and · is called a post-Lie algebra [8, 37] if [, ] is a Lie bracket and the next two identities hold”
• Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras (2017, background): “Recently, many authors study some post-Lie algebras and post-Lie algebra structures [8, 9, 25, 26, 30].”