Cocommutativity

Cocommutativity is a structural property of coalgebras and Hopf algebras that expresses symmetry of the comultiplication. Let C be a coalgebra over a field k with comultiplication Δ: C → C ⊗ C and counit ε: C → k. The flip map τ: C ⊗ C → C ⊗ C is defined by τ(x ⊗ y) = y ⊗ x. C is cocommutative if Δ = τ ∘ Δ, meaning swapping the two tensor factors leaves the comultiplication unchanged.

In a Hopf algebra H, equipped with a compatible multiplication and unit, cocommutativity of the coalgebra structure Δ is a key simplifying condition. If H is cocommutative, many structural results become more tractable and the representation theory reflects the underlying symmetry of the comultiplication.

Examples include:

- The group algebra k[G] of a group G, with the standard comultiplication Δ(g) = g ⊗ g for all g ∈ G. This map satisfies Δ = τ ∘ Δ, so k[G] is cocommutative.

- The universal enveloping algebra U(g) of a Lie algebra g, where primitive elements satisfy Δ(x) = x ⊗ 1 + 1 ⊗ x for x ∈ g; such Δ is cocommutative.

Dual perspective and related facts:

- If C is finite-dimensional and cocommutative, its linear dual C* is a commutative algebra. Conversely, the finite-dimensional dual of a finite-dimensional commutative algebra is a cocommutative coalgebra.

- In topology, many natural coalgebras, such as those arising from singular homology with the Alexander–Whitney diagonal, are cocommutative.

Cocommutativity thus captures a fundamental symmetry in the coalgebraic framework, paralleling the notion of commutativity in algebras.