bialgebra

Definition

A bialgebra is a vector space over a field that has the structure of a (unital) associative algebra and a coalgebra that are compatible.

Formally, a bialgebra is a vector space over a field equipped with the following -linear maps:

• Associative mulitplication: .

• Unit map:

• Coassociative comultiplication:

• Counit map: .

These maps are compatible in the following ways (in commutative diagrams):

Comultiplication and multiplication:

Multiplication and comultiplication with the unit and counit:

Unit and counit: