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!

• Hopf algebra and combinatorics
• Hopf algebras and rooted trees

• Hopf algebra 1

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. $\delta(g) \equiv g \otimes g$, $\epsilon(g) = 1$, $s(g) = g^{-1}$.