As is well known, using the Hilbert Nullstellensatz (and a more recent result of Cartier) one can show that commutative finitely generated Hopf algebras over $\mathbb{C}$ are equivalent to algebraic groups (cf the accepted [answer][1] of David Speyer). What happens when one considers finitely generated Hopf $*$-algebras? Does there exist an analogous equivalence with the star coming (perhaps) from matrix adjoints?





  [1]: https://mathoverflow.net/questions/9046/hopf-algebras-arising-as-group-algebras