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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 $$u\in U$$, and Markov if the extension is strongly separable (i.e. $$E(1)=1$$ and $$\sum_ix_iy_i=\lambda^{-1}1$$) and there is a (Markov) trace $$T\colon N\to k$$ such that $$T(1)=1_k$$ and $$T_0=T\circ E\colon M\to k$$ is a trace. The basic construction theorem says that if $$N\subseteq M$$ is a symmetric Markov extension and $$M_1=M\otimes_NM=\text{End}(M_N)$$ 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 $$\sum_ix_iy_i=\sum_iy_ix_i$$ then $$V=C_{M_1}(M)$$ is Kansaki separable and the restriction of $$T_1=T_0\circ E_M$$ to $$V$$ is non-degenerate. The construction can be iterated to obtain the Jones tower $$N\subseteq M\subseteq M_1\subseteq M_2$$. 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 $$M_1\simeq M\#A$$ and $$M_2\simeq M_1\#B$$ 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)].

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16S40 Smash products of general Hopf actions 16S50 Endomorphism rings; matrix rings 16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) 16L60 Quasi-Frobenius rings
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