Braided module and comodule algebras, Galois extensions and elements of trace 1.
Let \((H,c)\) be a finite-dimensional braided bialgebra over a field \(k\). \((H,c)\) is called rigid if a certain map from \(H^*\otimes H\) to \(H\otimes H^*\), using \(c^{-1}\), evaluation and coevaluation maps, is bijective. \textit{M. Takeuchi} studied duality results for rigid Hopf algebras (in a braided category), for example a duality between right \(H\)-comodule algebras and left \(H^*\)-module algebras [J. Pure Appl. Algebra 138, No. 1, 59-82 (1999; Zbl 0936.16038)]. The paper under review extends these results to the setting of \(H\)-braided spaces and algebras introduced by the last two authors [J. Algebra 261, No. 1, 54-101 (2003; Zbl 1017.16032)]. A left \(H\)-braided space \((V,s)\) is a \(k\)-vector space \(V\) and a bijective map \(s\) from \(H\otimes V\) to \(V\otimes H\) which is compatible with the operations of \(H\) and with the braid \(c\). Similar definitions were given for \((V,s)\) to be a left \(H\)-module or a right \(H\)-comodule. Replacing \(V\) by an algebra \(A\), a definition of \(H\)-braided algebra was given. Let \(H\) be rigid. In the paper under review, the authors show that \((V,s)\) is a left \(H\)-braided space if and only if \((V,t)\) is a left \(H^*\)-braided space, where \(t\) is a certain map from \(H^*\otimes V\) to \(V\otimes H^*\) involving \(s^{-1}\). Similarly, \((A,s)\) is a left \(H\)-module algebra if and only if \((A,t)\) is a left \(H^*\)-module algebra. To discuss left module and right comodule duality in their setting, a variant \(H^!\) of \(H^*\) is introduced. Then \((V,s)\) is a right \(H\)-comodule if and only if \((V,t)\) is a left \(H^!\)-module, and a similiar result holds for right \(H\)-comodule algebras \((A,s)\) and left \(H^!\)-module algebras \((A,t)\). Takeuchi [op. cit.] had noted that this does not work if \(H^*\) is used instead of \(H^!\). There are other results involving smash products, invariants and Galois extensions, which lead to generalizations of results of \textit{M. Cohen} and \textit{D. Fischman}, [J. Algebra 149, No. 2, 419-437 (1992; Zbl 0788.16029)], and of \textit{M. Cohen, D. Fischman} and \textit{S. Montgomery}, [J. Algebra 133, No. 2, 351-372 (1990; Zbl 0706.16023)].
scientific article; zbMATH DE number 5126077
scientific article