bialgebras

A bialgebra is an algebraic structure that is simultaneously an algebra and a coalgebra on the same underlying vector space, defined over a field k. Let A be a vector space over k with multiplication m: A ⊗ A → A and unit η: k → A making A an algebra, and comultiplication Δ: A → A ⊗ A and counit ε: A → k making A a coalgebra. The two structures are compatible in the sense that Δ and ε are algebra homomorphisms, equivalently that m and η are coalgebra homomorphisms. Concretely, this means Δ(ab) = Δ(a)Δ(b) and ε(ab) = ε(a)ε(b) for all a,b ∈ A, where A ⊗ A is given its natural algebra structure, and that the unit and counit satisfy the usual unital and counital relations.

The standard axioms require associativity of m and unitality with η, coassociativity of Δ and counitality with ε:

- m ∘ (m ⊗ id) = m ∘ (id ⊗ m), and m ∘ (η ⊗ id) = id = m ∘ (id ⊗ η);

- (Δ ⊗ id) ∘ Δ = (id ⊗ Δ) ∘ Δ, and (ε ⊗ id) ∘ Δ = id = (id ⊗ ε) ∘ Δ.

Examples include the group algebra k[G], with Δ(g) = g ⊗ g and ε(g) = 1 for all g ∈ G, and the universal enveloping algebra U(g) of a Lie algebra g, with its standard coproduct extended from Δ(x) = x ⊗ 1 + 1 ⊗ x for x ∈ g. The polynomial algebra k[x] with Δ(x) = x ⊗ 1 + 1 ⊗ x is a cocommutative bialgebra. In finite dimensions, commutative bialgebras relate to coordinate rings of affine group schemes.

Hopf algebras are bialgebras equipped with an antipode S, providing an algebraic inverse. Bialgebras play key roles in representation theory, combinatorics, and quantum groups.