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Weak Hopf Algebras: I. Integral Theory and C-Structure

✍ Scribed by Gabriella Böhm; Florian Nill; Kornél Szlachányi

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 324 KB
Volume: 221
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1999.7984

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✦ Synopsis

We give an introduction to the theory of weak Hopf algebras proposed as a coassociati¨e 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 g A and of a *

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