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Morita contexts and partial group Galois extensions for Hopf group coalgebras. (English) Zbl 1326.16029

Let \(k\) be a field, \(\pi\) a discrete group, \(C=\{C_\alpha\}_{\alpha\in\pi}\) a \(\pi\)-coalgebra and \(H=(\{H_\alpha\}_{\alpha\in\pi},\Delta,\varepsilon)\) a \(T\)-Hopf \(\pi\)-coalgebra with comultiplication \(\Delta=\{\Delta_{\alpha,\beta\in\pi}\colon C_{\alpha\beta}\to C_\alpha\otimes C_\beta\}_{\alpha,\beta\in\pi}\) and counit \(\varepsilon\colon C_1\to k\), and antipode \(S:\{S_\alpha\colon H_\alpha\to H_{\alpha^{-1}}\}_{\alpha\in\pi}\), \(\varphi=\{\varphi_\beta\colon H\alpha\to H_{\beta\alpha\beta^{-1}}\}_{\alpha,\beta\in\pi}\) a family of algebra isomorphisms, and \(\pi\)-integral \(\lambda=\{\lambda_\alpha\}_{\alpha\in\pi}\in\prod_{\alpha\in\pi}H_\alpha^*\) [A. Virelizier, J. Pure Appl. Algebra 171, No. 1, 75-122 (2002; Zbl 1011.16023)]. A Hopf \(\pi\)-coalgebra \(H\) is called co-Frobenius if \(H\) has non-zero space of left (respectively, right) \(\pi\)-integrals. Let \(A=\bigoplus_{\alpha\in\pi}A_\alpha\) be a \(\pi\)-graded algebra. Then \(A\) is said to be a partial left \(H\)-module \(\pi\)-graded algebra if there exists a family of \(k\)-linear maps \(\nearrow:=\{\nearrow H_\alpha\otimes A_\alpha\to A_\alpha\}_{\alpha\in\pi}\) such that
(1) for all \(h\in H_{\alpha\beta}\), \(a\in A_\alpha\) and \(b\in A_\beta\), \(h\nearrow(ab)=(h_{(1,\alpha)})\nearrow a)(h_{(2,\beta)}\nearrow b)\);
(2) for any \(\alpha\in\pi\) and \(1_\alpha\in H_\alpha\), \(a\in A\), \(1_\alpha\nearrow a=a\); and
(3) for \(h,g\in H_\alpha\), \(a\in A_\alpha\), \(1_A\in A_1\), \(h\nearrow(g\nearrow a)=(h_{(1,1)}\nearrow 1_A)(h_{(2,\alpha)}g\nearrow a)\).
Similarly, a partial right \(H\)-module \(\pi\)-graded algebra is defined. \(A\) is called a partial \(H\)-bimodule \(\pi\)-graded algebra if
(1) \(A\) is a partial \(H\)-bimodule with the partial left \(H\)-module structure map \(\nearrow\) and the partial right \(H\)-module structure map \(\swarrow \) such that \((h\nearrow a)\swarrow g=h\nearrow(a\swarrow g)\) for all \(a\in A_\alpha\), \(h,g\in H_\alpha\) and
(2) \(A\) is not only a partial left \(H\)-module \(\pi\)-graded algebra with \(\nearrow\), but also a partial right \(H\)-module \(\pi\)-graded algebra with \(\swarrow\). Then a partial \(\pi\)-twisted smash product is defined for a \(T\)-coalgebra \(H\) and a partial \(H\)-bimodule \(\pi\)-graded algebra \(A\). Moreover, let \(A\) be a partial right \(\pi\)-\(H\)-comodule algebra and \(H\) a co-Frobenius Hopf-\(\pi\)-coalgebra, a right generalized partial smash product \(\pi\)-graded algebra \(A\odot H^*\) is also defined and a Morita context \((A\odot H^*, A^{coH},\tau,\mu)\) is constructed for some \(\tau\colon A\otimes_{A^{coH}}A\to A\odot H^*\) and \(\mu\colon A\otimes_{A\odot H^*}A\to A^{coH}\). Equivalent conditions are then given for the Morita context to be strict. It is also shown that any partial group Galois extension induces a unique partial group entwining map compatible with the right partial coaction.

MSC:

16T15 Coalgebras and comodules; corings
16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions

Citations:

Zbl 1011.16023
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References:

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