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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.[1]: 46 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.[1]: 46 (These statements are equivalent since they are expressed by the same commutative diagrams.)[1]: 46
Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism.[2]: 45
As reflected in the symmetry of the commutative 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.
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- ^ a b c Kassel 2012, p. 46.
- ^ Kassel 2012, p. 45.