Normal Hopf subalgebras, depth two and Galois extensions

2004/11/06 by Lars Kadison, Kadison, Lars
Mathematics · #16S40 #16W30 (13B05 #20L05 #81R50) #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.QA #math.RA #msc:16S40 #msc:16W30 #msc:20L05

paper · pdf · doi:10.48550/arxiv.math/0411129

superseded by my more recent preprints math.QA/0502188 and math.QA/0503194

arxiv created 2005/04/06 · arxiv updated 2009/12/01

Abstract

Let S be the left R-bialgebroid of a depth two extension with centralizer R as defined in math.QA/0108067. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left S-Galois extension of A^{\rm op}. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We find a class of examples of the alternative Hopf algebroids in math.QA/0302325. We also characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

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