Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

Chirvasitu, Alexandru

doi:10.2140/ant.2014.8.1179

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Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

Alexandru Chirvasitu

Vol. 8 (2014), No. 5, 1179–1199

DOI: 10.2140/ant.2014.8.1179

Abstract

The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$ ) is commutative, cocommutative, or pointed, or when $K$ contains the coradical of $H$ . We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting “exact sequence” $A \to H \to C$ is always a cosemisimple coalgebra, and that the expectation $H \to A$ is positive when $H$ is a CQG algebra.

Keywords

cosemisimple Hopf algebra, CQG algebra, faithfully flat, right coideal subalgebra, quotient left module coalgebra, expectation

Mathematical Subject Classification 2010

Primary: 16T20

Secondary: 16T15, 16T05, 20G42

Milestones

Received: 11 August 2013

Revised: 6 March 2014

Accepted: 21 April 2014

Published: 16 September 2014

Authors

Alexandru Chirvasitu
Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350 United States

Vol. 8, No. 5, 2014