Weak Hopf Algebras I: Integral Theory and C^*-structure
G. Bohm, F. Nill, K. Szlachanyi
TL;DR
This work develops a cohesive axiomatic framework for weak Hopf algebras (WHAs), emphasizing coassociativity with weakened unit/counit axioms and the pivotal canonical subalgebras $A^L$ and $A^R$. It builds an integral theory linking integrals to Frobenius structure, semisimplicity, and Haar measures, and shows how WHAs naturally generalize Hopf algebras while supporting rich representation-theoretic phenomena via weak Hopf modules. The text then introduces a finite-dimensional $C^*$-structure, establishing the existence and properties of Haar measures, the selfduality of the dual WHA, and the canonical grouplike element $g$ that implements $S^2$ and modular automorphisms. Together these results lay the groundwork for a robust theory of noncommutative symmetry with potential applications to operator algebras and quantum field theory, with Part II promising further development of representation categories and dimensions.
Abstract
We give an introduction to the theory of weak Hopf algebras proposed recently as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the "classical" theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square of the antipode, are related to the existence of non-degenerate, normalized, or Haar integrals. In case of C^*-weak Hopf algebras we prove the existence of a unique Haar measure h in A and of a canonical grouplike element g in A implementing the square of the antipode and factorizing into left and right algebra elements. Further discussion of the C^*-case will be presented in Part II.