Co-Frobenius Hopf algebras and the coradical filtration

Details Co-Frobenius Hopf algebras and the coradical filtration Co-Frobenius Hopf algebras and the coradical filtration

2001-02-03

arxiv.org/abs/math/0102028v1

Math. Z. 243 (2003), no. 1, 145--154.

Mathematics

Quantum Algebra

13 pages

Scientific paper

We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius, i. e. there is a non-zero integral on it. As a consequence, we show that algebras of functions on quantum groups at roots of one are co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. This characterization is in the framework of the lifting method due to H.-J. Schneider and the first-named author. Here is our main result. Let H be a Hopf algebra whose coradical is a Hopf subalgebra. Let gr H be the associated graded Hopf algebra and let R be the diagram of H. Then the following are equivalent: (1) H is co-Frobenius, (2) gr H is co-Frobenius, (3) R is finite dimensional, (4) the coradical filtration of H is finite. This Theorem allows to construct many new examples of co-Frobenius Hopf algebras and opens the way to the classification of ample classes of such Hopf algebras.

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