= Hopf algebra
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A <Hopf algebra> over a <field> $k$ is a <bialgebra> $H$ with an <antipode> $S$ satisfying $m(S\otimes1)\Delta=u\varepsilon=m(1\otimes S)\Delta$, where $m,u,\Delta,\varepsilon$ denote multiplication, unit, coproduct, and counit. These identities generalize taking an inverse.
For a <group algebra> $k[G]$, define $\Delta(g)=g\otimes g$, $\varepsilon(g)=1$, 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.
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