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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 {\it M. Pimsner} and {\it 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 associative algebras 16L60 Quasi-Frobenius rings
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