Weak Hopf Algebras I: Integral Theory and C^*-structure

TL;DR

This paper introduces the theory of weak Hopf algebras, focusing on their structure, integrals, and C^*-algebra properties, establishing foundational results like the existence of Haar measures and canonical elements.

Contribution

It develops an axiomatic framework for weak Hopf algebras, analyzing their integral theory and C^*-structure, including existence and uniqueness results for Haar measures.

Findings

Existence of a unique Haar measure in C^*-weak Hopf algebras

Canonical grouplike element implementing the antipode square

Relationship between algebraic properties and integrals

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.