## § 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 $H$ over a field $K$. together with $K$ linear
maps $m: A \rightarrow A \otimes A$ (multiplication),
$U: A \rightarrow K$ (unit), $\Delta: H \rightarrow H \otimes H$ (comultiplication)
$S: A \rightarrow A$ (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: $A = kG$

$G$ is a group, $kG$ is a group algebra. $\delta(g) \equiv g \otimes g$,
$\epsilon(g) = 1$, $s(g) = g^{-1}$.