A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite-dimensional Hopf algebra, this dimension is exactly one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite-dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one. © 2003 Published by Elsevier Inc.
Bulacu, D., & Caenepeel, S. (2003). Integrals for (dual) quasi-Hopf algebras. Applications. Journal of Algebra, 266(2), 552–583. https://doi.org/10.1016/S0021-8693(03)00175-3