Frobenius Extensions and Weak Hopf Algebras

We study a symmetric Markov extension of k-algebras N 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 M M1 M2 can be obtained by taking relative tensor products with centralizers A = CM1(N) and B = CM2(M). Under this condition, we prove that N M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N = MA. The endomorphism algebra M1 = EndNM is shown to be isomorphic to the smash product algebra M # A. We also extend results of W. Szyma ski (1994, Proc. Amer. Math. Soc. 120, 519–528), D. Nikshych and L. Vainerman (2000, Funct. Anal.171, 278–307), and L. Kadison and D. Nikschych (Comm. Algebra, 29(9) 2001).