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Frobenius extensions and weak Hopf algebras. (English) Zbl 1018.16020
Let M/N be a Frobenius extension of k-algebras with Frobenius homomorphism E and dual bases {x i }, {y i }. Let U=C M (N). The extension is called symmetric if E commutes with every uU, and Markov if the extension is strongly separable (i.e. E(1)=1 and i x i y i =λ -1 1) and there is a (Markov) trace T:Nk such that T(1)=1 k and T 0 =TE:Mk is a trace. The basic construction theorem says that if NM is a symmetric Markov extension and M 1 =M N M=End(M N ) then M 1 /M is a symmetric Markov extension; the Frobenius endomorphism E M and the dual bases are described, and the Markov trace is T 0 . If in addition U is Kanzaki separable, T 0 | U is non-degenerate and i x i y i = i y i x i then V=C M 1 (M) is Kansaki separable and the restriction of T 1 =T 0 E M to V is non-degenerate. The construction can be iterated to obtain the Jones tower NMM 1 M 2 . Let A=C M 1 (N) and B=C M 2 (M). Assuming the existence of dual bases for E M resp. E M 1 in A resp. B (depth 2 condition) the authors prove some properties of algebra extensions involving A, B, U and V. Examples of extensions with depth 2 are provided in the final appendix. Then semisimple weak Hopf algebra structures are defined in A and B, and also A resp. B-module algebra structures on M resp. M 1 . Two isomorphisms M 1 M#A and M 2 M 1 #B are also provided. Finally, in the absence of a trace, the basic construction is iterated to the right, extending a result of M. Pimsner and S. Popa [Trans. Am. Math. Soc. 310, No. 1, 127-133 (1988; Zbl 0706.46047)].
16W30Hopf algebras (assoc. rings and algebras) (MSC2000)
16S40Smash products of general Hopf actions
16S50Endomorphism rings: matrix rings
16H05Separable associative algebras
16L60Quasi-Frobenius rings