Bialgebras, Convolution, and Antipode Identities — Examples

Compare group algebra and function algebra for a finite nonabelian group; compute convolution of linear maps; show why treating S as an ordinary inverse of id is meaningless. Polynomial examples work in every characteristic, with characteristic-dependent primitive elements.

These are planned worked examples and counterexamples for bialgebras-convolution-and-antipode-identities, with empty item lists pending normal authoring and proof review. Each example must calculate from the stated hypotheses, and each counterexample must verify both its premises and the failed conclusion. This companion is a dependency leaf and supplies no theorem to another page.