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Concept
Bialgebra
Summary
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.
As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is , 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:
B is a vector space over K;
there are K-linear maps (multiplication) ∇: B ⊗ B → B (equivalent to K-multilinear map ∇: B × B → B) and (unit) η: K → B, such that (B, ∇, η) is a unital associative algebra;
there are K-linear maps (comultiplication) Δ: B → B ⊗ B and (counit) ε: B → K, such that (B, Δ, ε) is a (counital coassociative) coalgebra;
compatibility conditions expressed by the following commutative diagrams:
Multiplication ∇ and comultiplication Δ
where τ: B ⊗ B → B ⊗ B is the linear map defined by τ(x ⊗ y) = y ⊗ x for all x and y in B,
Multiplication ∇ and counit ε
Comultiplication Δ and unit η
Unit η and counit ε
The K-linear map Δ: B → B ⊗ B is coassociative if .
The K-linear map ε: B → K is a counit if .
Coassociativity and counit are expressed by the commutativity of the following two diagrams (they are the duals of the diagrams expressing associativity and unit of an algebra):
The four commutative diagrams can be read either as "comultiplication and counit are homomorphisms of algebras" or, equivalently, "multiplication and unit are homomorphisms of coalgebras".
Official source
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