Marcelo Aguiar

A unified approach to Hopf algebras.

Abstract:

Hopf algebras are prominent in three areas of mathematics. In algebraic topology, where the notion originated, they arise from topological groups or H-spaces. In combinatorics, as pointed out by Rota, they encode natural operations of assembly and disassembly one may perform on a given type of discrete structure. In representation theory, they arise as deformations of simple Lie algebras, the "quantum groups" of Drinfeld. Each of these sources of Hopf algebras appears to be fairly distant from the others, at first.

This talk will describe a theory being developed jointly with Mahajan that allows us to view the above constructions in a unified manner. In particular, the profusion of "combinatorial Hopf
algebras" witnessed in the recent literature, receives in this work a conceptual organization and a substantial generalization.

The theory involves notions such as a bilax monoidal functors and Joyal's species. In spite of the language and tools used, this theory is very keen on combinatorics, both through concrete examples and the structures underlying the abstract constructions.