bialgebra
Sign in to saveIn mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.)
~8 min read
Article
9 sectionsContents
- Formal definition
- Coassociativity and counit
- Compatibility conditions
- Examples
- Group bialgebra
- Other examples
- See also
- Notes
- References
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.)
Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism.