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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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