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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 $\left\{{x}_{i}\right\}$, $\left\{{y}_{i}\right\}$. Let $U={C}_{M}\left(N\right)$. The extension is called symmetric if $E$ commutes with every $u\in U$, and Markov if the extension is strongly separable (i.e. $E\left(1\right)=1$ and ${\sum }_{i}{x}_{i}{y}_{i}={\lambda }^{-1}1$) and there is a (Markov) trace $T:N\to k$ such that $T\left(1\right)={1}_{k}$ and ${T}_{0}=T\circ E:M\to k$ is a trace. The basic construction theorem says that if $N\subseteq M$ is a symmetric Markov extension and ${M}_{1}=M{\otimes }_{N}M=\text{End}\left({M}_{N}\right)$ 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 ${\sum }_{i}{x}_{i}{y}_{i}={\sum }_{i}{y}_{i}{x}_{i}$ then $V={C}_{{M}_{1}}\left(M\right)$ 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 $A={C}_{{M}_{1}}\left(N\right)$ and $B={C}_{{M}_{2}}\left(M\right)$. 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 (assoc. rings and algebras) (MSC2000) 16S40 Smash products of general Hopf actions 16S50 Endomorphism rings: matrix rings 16H05 Separable associative algebras 16L60 Quasi-Frobenius rings