Hopf monoids relative to a hyperplane arrangement

Marcelo Aguiar
Cornell University
Thursday, October 17, 2019 - 3:35pm to 4:35pm
Vincent Hall 16

The talk is based on recent and ongoing work with Swapneel
Mahajan. We will introduce a notion of Hopf monoid relative to a real
hyperplane arrangement. When the latter is the braid arrangement, the
notion is closely related to that of a Hopf monoid in Joyal's category
of species, and to the classical notion of connected graded Hopf
algebra. We are able to extend many concepts and results from the
classical theory of connected Hopf algebras to this level. The
extended theory connects to the representation theory of a certain
finite dimensional algebra, the Tits algebra of the arrangement. This
perspective on Hopf theory is novel even when applied to the classical
case. We will outline our approach to generalizations of the classical
Leray-Samelson, Borel-Hopf, and Cartier-Milnor-Moore theorems to this
setting. Background on hyperplane arrangements will be reviewed.