Bialgebra

Summary

A bialgebra is a vector space that is both an associative algebra and a coalgebra in a compatible way. The compatibility is that the comultiplication and the counit are algebra homomorphisms. Group algebras, monoid algebras, polynomial rings and universal enveloping algebras of Lie algebras are bialgebras. The compatibility lets one form tensor products of modules, which is the key feature of Hopf algebras and quantum groups.