Category:Bialgebras

Bialgebra

In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra, such that these structures are compatible.

Compatibility means that the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms: these statements are equivalent in that they are expressed by the same diagrams. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism.

As reflected in the symmetry of the diagrams, the definition of bialgebra is self-dual, so if one can define a dual of B (which is always possible if B is finite-dimensional), then it is automatically a bialgebra.