Hopf algebra

A Hopf algebra over a field $k$ is a bialgebra $H$ with an antipode $S$ satisfying $m(S\otimes 1)\Delta =u\epsilon =m(1\otimes S)\Delta$, where $m,u,\Delta ,\epsilon$ denote multiplication, unit, coproduct, and counit. These identities generalize taking an inverse.

For a group algebra $k[G]$, define $\Delta (g)=g\otimes g$, $\epsilon (g)=1$, and $S(g)=g^{-1}$, then extend linearly. The coproduct makes a group element act simultaneously on both factors of a tensor product of representations. A Hopf algebra therefore packages not only multiplication but also how symmetries act on products and how they are reversed.