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. 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.

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.

(B, ∇, η, Δ, ε) is a bialgebra over K if it has the following properties:

The K-linear map Δ: B → B ⊗ B is coassociative if $(\mathrm {id} _{B}\otimes \Delta )\circ \Delta =(\Delta \otimes \mathrm {id} _{B})\circ \Delta$ $({\mathrm {id}}_{B}\otimes \Delta )\circ \Delta =(\Delta \otimes {\mathrm {id}}_{B})\circ \Delta$ .

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