§ Hopf Algebras and combinatorics
Started from algebraic topology in the 40s. In late 70s, Rota figured out that many combinatorial objects have the structure of a Hopf algebra.
A hopf algebra is a vector space over a field . together with linear
(co-inverse/antipode). Best explained by examples!
The idea is that groups act by symmetries. Hopf algebras also act, we can think of as providing quantum symmetries.
§ Eg 1: Group algebra:
is a group, is a group algebra. ,