"Geometry and Analysis on Groups" Research Seminar
Post-Lie algebra structures and Poisson structures
Christof Ender (Wien)
Post-Lie algebra structures (PLA-structures) arise in many different contexts, e.g., in connection with homology of partition sets, operad theory, geometric structures on Lie groups, crystallographic groups, Yang-Baxter equations, Rota-Baxter operators and other topics.
In the first part of the talk, we want to discuss the notion of post-Lie algebra structures in the context of affine and nil-affine crystallographic actions of groups. J. Milnor conjectured that every virtually polycyclic group admits an affine crystallographic action.
This conjecture was open for a long time; finally, it turned out to be false.
However, it was proven by K. Dekimpe that every virtually polycyclic group admits a so-called nil-affine crystallographic action. These actions correspond on a Lie algebra level to PLA-structures.
In the second part, we want to discuss new results on post-Lie algebra structures.
Studying PLA-structures on nilpotent Lie algebras, we find that certain classes of PLA-structures can be described by the well-studied notion of Poisson algebras and other classes of PLA-structures correspond to associative algebras and Poisson-admissible algebras.
For certain families of nilpotent Lie algebras, this allows us to conclude that all commutative PLA-structures are associative algebras or Poisson algebras.
This is joint work with D. Burde.