Abstract: There are well-known close links between the free Lie algebra, the symmetric group, and the braid arrangement of hyperplanes. ln joint work in progress with Swapneel Mahajan, we propose an extension of this theory to hyperplane arrangements (real, central, and finite).

A central role is played by a generalized notion of Hopf monoid. We develop a Lie theory for these objects which includes analogs of the Eulerian and Dynkin idempotents, the theorems of Cartier-Milnor-Moore and of Poincare-Birkhoff-Witt, among others. In the new setting, these classical notions and results acquire a new geometric meaning. The classical results are obtained by specializing to braid arrangements.

This talk will be accessible to graduate students.