§ 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 HH over a field KK. together with KK linear maps m:AAAm: A \rightarrow A \otimes A (multiplication), U:AKU: A \rightarrow K (unit), Δ:HHH\Delta: H \rightarrow H \otimes H (comultiplication) S:AAS: A \rightarrow A (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: A=kGA = kG

GG is a group, kGkG is a group algebra. δ(g)gg\delta(g) \equiv g \otimes g, ϵ(g)=1\epsilon(g) = 1, s(g)=g1s(g) = g^{-1}.