Title: PD operads and partition Lie algebras.
Partition Lie algebras, introduced by Brantner and Mathew, are homotopy theoretic refinements of Lie algebras that appear in deformation theory in positive characteristic. However, since partition Lie algebras are defined in a purely ∞-categorical way, it is hard to get one’s hands on them. For instance, it is not so clear how to describe them explicitly in terms of point-set models, because there is no operad whose algebras are partition Lie algebras.
In this talk, I will discuss a homotopy-theoretic generalization of the notion of an operad, called a “PD operad”, whose algebras can also carry certain kinds of divided power operations. The homotopy theory of such PD operads and their algebras can be understood quite explicitly in terms of chain complexes, using some homological algebra for representations of the symmetric groups. As a particular example, I will describe the PD operad that controls partition Lie algebras. Based on joint work with Lukas Brantner and Ricardo Campos.