Braids and Hopf algebras
Craig Westerland (University of Minnesota, Twin Cities)
The Milnor–Moore theorem identifies a large class of Hopf algebras as enveloping algebras of the Lie algebras of their primitives. If we broaden our definition of a Hopf algebra to that of a braided Hopf algebra, much of this structure theory falls apart. The most obvious reason is that the primitives in a braided Hopf algebra no longer form a Lie algebra. In this talk, we will discuss recent work to understand what precisely is the algebraic structure of the primitives in a braided Hopf algebra in order to “repair” the Milnor–Moore theorem in this setting. It turns out that this structure is closely related to the dualizing module for the braid groups, which implements dualities in the (co)homology of the braid groups.