Section
Coalgebras
Coalgebras, bialgebras, and Hopf algebras.
Core idea
This section collects definitions of algebraic structures with a comultiplication.
Let be a commutative ring. A -coalgebra is a -module equipped with -linear maps
called the comultiplication and counit, such that
and
after the canonical identifications . Thus a coalgebra is a comonoid in the monoidal category of -modules.
Let be a commutative ring. A -bialgebra is a unital associative -algebra and a -coalgebra on the same module such that and are unital algebra homomorphisms. The tensor product algebra structure used here is
Equivalently, and are coalgebra homomorphisms.
Let be a commutative ring. A Hopf algebra over is a bialgebra together with a -linear map , called the antipode, such that
Equivalently, is the two-sided inverse of for the convolution product