A quasi-hopf algebra is an algebraic structure that generalizes the concept of a Hopf algebra by relaxing some of its axioms. In a quasi-hopf algebra, the coassociativity of the comultiplication is modified by a twist element, allowing for a broader class of algebras that still possess many characteristics of Hopf algebras. This structure retains a duality between algebra and coalgebra but introduces flexibility through the use of a 2-cocycle in its definitions.
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