Bialgebra

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.) 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 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 ( i d B ⊗ Δ ) ∘ Δ = ( Δ ⊗ i d B ) ∘ Δ {displaystyle (mathrm {id} _{B}otimes Delta )circ Delta =(Delta otimes mathrm {id} _{B})circ Delta } . The K-linear map ε: B → K is a counit if ( i d B ⊗ ϵ ) ∘ Δ = i d B = ( ϵ ⊗ i d B ) ∘ Δ {displaystyle (mathrm {id} _{B}otimes epsilon )circ Delta =mathrm {id} _{B}=(epsilon otimes mathrm {id} _{B})circ Delta } . 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'. These statements are meaningful once we explain the natural structures of algebra and coalgebra in all the vector spaces involved besides B: (K, ∇0, η0) is a unital associative algebra in an obvious way and (B ⊗ B, ∇2, η2) is a unital associative algebra with unit and multiplication