quasiassociativity

Quasiassociativity is a weakening of the associativity law in algebraic structures, where the strict equality (ab)c = a(bc) is replaced by a controlled form of associativity described by an associator. In many frameworks, the product or composition is not strictly associative, but coherence data ties different ways of parenthesizing products together via a specified isomorphism.

In categorical terms, a monoidal category provides a setting for quasiassociativity: the tensor product is associative only up to a natural isomorphism called the associator, typically denoted α. The associator must satisfy coherence conditions, most notably the pentagon axiom, which ensures consistency of rebracketing multiple factors.

In algebra, quasi-Hopf algebras (introduced by Drinfeld) are Hopf algebras equipped with an invertible element φ in H ⊗ H ⊗ H, called the associator, that encodes a nontrivial failure of coassociativity of the coproduct. The relation (Δ ⊗ id)Δ(h) = φ[(id ⊗ Δ)Δ(h)]φ⁻¹ holds for elements h in H, with φ satisfying a pentagon-like coherence condition. This framework generalizes strict Hopf algebras and yields rich representation theory and applications in quantum groups.

Another common setting is twisting group algebras by a 3-cocycle ω in group cohomology. Such a twist makes the product associative only up to ω, producing a quasiassociative algebra where ω measures the deviation from strict associativity.

Quasiassociativity thus arises in contexts where symmetry is implemented through higher coherence data rather than strict equations, including deformation theory, quantum groups, and topological field theories. It reduces to ordinary associativity when the associator is trivial. It is related to, but distinct from, higher homotopy associativity found in A∞-algebras.