Antipode

Summary

The antipode of a bialgebra is a linear map $S$ that is a two-sided inverse of the identity map for the convolution product. A bialgebra with an antipode is a Hopf algebra. The antipode generalizes the inverse map $g \mapsto g^{-1}$ of a group, and it is used to define dual representations. When it exists it is unique, and it reverses the order of products and of coproducts.