# Frobenius extensions and weak Hopf algebras

### Abstract

• We study a symmetric Markov extension of k-algebras N \into M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N \into M \into M_1 \into M_2 can be obtained by taking relative tensor products with centralizers A = C_{M_1}(N) and B = C_{M_2}(M). Under this condition, we prove that N \into M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N = M^A. The endomorphism algebra M_1 = \End_N M is shown to be isomorphic to the smash product algebra M # A. We also extend results of Szymanski, Vainerman and the second author, and the authors.
• ### Authors

• Nikshych, Dmitri

### Publication Date

• October 1, 2001

### Published In

• Journal of Algebra  Journal
• ### Keywords

• Frobenius extension
• Jones tower
• actions
• basic construction
• conditional expectation
• endomorphism ring
• symmetric Markov extension
• trace
• weak Hopf algebra

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