Frobenius and separable functors for the category of entwined modules over cowreaths. II: applications
Throughout the review, \(k\) denotes a field, \(H\) denotes a quasi-Hopf \(k\)-algebra, \(A\) denotes a right \(H\)-comodule \(k\)-algebra with \(\rho\colon A\rightarrow A\otimes H\), \(\rho (a)=\sum a_{\langle 0\rangle}\otimes a_{\langle 1\rangle}\), \(C\) denotes a right \(H\)-module \(k\)-coalgebra, \(\mathcal{M}_k\) denotes the category of \(k\)-vector spaces, \(\mathcal{M}_A\) denotes the category of right \(A\)-modules, \(\mathcal{M}(H)^C_A\) denotes the category of \((H,A,C)\)-Doi-Hopf modules. The paper under review is intended as the study of the Frobenius and separable properties for the forgetful functor \(F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A\). For \(\psi\colon C\otimes A\rightarrow A\otimes C\), \(\psi (c\otimes a)=\sum a_{\langle 0\rangle}\otimes c\cdot a_{\langle 1\rangle}\), \((C,\psi)\) is a coalgebra in \(\mathcal{T}(\mathcal{M}_k)^\#_A\), \((A,C,\psi)\) is a cowreath in \(\mathcal{M}_k\), \(\mathcal{M}(H)^C_A\) is just the category \(\mathcal{M}_k(\psi)^C_A\) of right entwined modules over \((A,C,\psi)\) in \(\mathcal{M}_k\). This allowed the authors to apply the results from their previous paper [Part I, Algebr. Represent. Theory 23, No. 3, 1119--1157 (2020; Zbl 1447.16031)]. In the present paper, the authors prove that: (1) If \(C\) is a Frobenius coalgebra in \(\mathcal{M}_H\) then \(F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A\) is a Frobenius functor (Proposition 2.3); (2) If \(A=H\) then \(F\colon\mathcal{M}(H)^C_H\rightarrow\mathcal{M}_H\) is a Frobenius functor if and only if \(C\) is a Frobenius coalgebra in \(\mathcal{M}_H\) (Theorem 2.4); (3) If \(C\) is a coseparable coalgebra in \(\mathcal{M}_H\) then \(F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A\) is a separable functor (Proposition 2.6); (4) If \(H\) has a bijective antipode then there is a bijective correspondence between the set of normalized Casimir morphisms for \(C\) in \(\mathcal{M}_H\) and the set of normalized Casimir morphisms for \((C,\psi)\) in \(\mathcal{T}(\mathcal{M}_k)^\#_H\) satisfying some additional condition (Theorem 2.7). Consequently, if \(F\colon\mathcal{M}(H)^C_H\rightarrow\mathcal{M}_H\) is a separable functor then this does not imply that \(C\) is a coseparable coalgebra in \(\mathcal{M}_H\). For a quasi-Hopf \(k\)-algebra \(H\) with bijective antipode, a right \(H\)-comodule \(k\)-algebra \(A\), an \(H\)-bimodule \(k\)-coalgebra \(C\), let \(_H\mathcal{M}_A\) denote the category of \((H,A)\)-bimodules, let \(_H\mathcal{M}^C_A\) denote the category of two-sided \((H,A)\)-bimodules over \(C\). In this case, the authors prove that: (1) \(A\otimes H^\mathrm{op}\) is a right \(H\otimes H^\mathrm{op}\)-comodule \(k\)-algebra, \(C\) is a a right \(H\otimes H^\mathrm{op}\)-module \(k\)-coalgebra, \(_H\mathcal{M}^C_A\) is isomorphic to the category \(\mathcal{M}(H\otimes H^\mathrm{op})^C_{A\otimes H^\mathrm{op}}\) of \((H\otimes H^\mathrm{op},A\otimes H^\mathrm{op},C)\)-Doi-Hopf modules (Proposition 3.2); (2) If \(C\) is a Frobenius coalgebra in \(_H\mathcal{M}_H\) then \(F\colon {_H\mathcal{M}^C_A}\rightarrow {_H\mathcal{M}_A}\) is a Frobenius functor; (3) If \(A=H\) then \(F\colon {_H\mathcal{M}^C_H}\rightarrow {_H\mathcal{M}_H}\) is a Frobenius functor if and only if \(C\) is a Frobenius coalgebra in \(_H\mathcal{M}_H\); (4) If \(A=C=H\) then \(F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}\) is a Frobenius functor if and only if \(H\) is finite dimensional and unimodular (Theorem 3.4). In the case when \(H\) is finite dimensional, the authors prove that: (1) \(H\) is coseparable as a coalgebra in \(_H\mathcal{M}_H\) if and only if \(H\) is unimodular and cosemisimple (Proposition 3.6); (2) \(F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}\) is a separable functor if and only if \(H\) is unimodular (Theorem 3.7). Consequently, in the case when \(H\) is finite dimensional: (1) The forgetful functor \(F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}\) is a separable if and only if it is Frobenius; (2) If \(F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}\) is a separable functor then this does not imply that \(H\) is coseparable as a coalgebra in \(_H\mathcal{M}_H\). For a quasi-Hopf \(k\)-algebra \(H\), an \(H\)-bicomodule \(k\)-algebra \(A\), an \(H\)-bimodule \(k\)-coalgebra \(C\), let \(\mathcal{YD}(H)^C_A\) denote the category of right \((H,A,C)\)-Yetter-Drinfeld modules. In this case, the authors prove that: (1) If \(C\) is a Frobenius coalgebra in \(_H\mathcal{M}_H\) then \(F\colon\mathcal{YD}(H)^C_A\rightarrow\mathcal{M}_A\) is a Frobenius functor (Proposition 4.2); (2) If \(A=C=H\) then \(F\colon\mathcal{YD}(H)^H_H\rightarrow\mathcal{M}_H\) is a Frobenius functor if and only if \(H\) is finite dimensional and Frobenius as a coalgebra in \(_H\mathcal{M}_H\) if and only if \(H\) is finite dimensional and unimodular (Theorem 4.5); (3) If \(A=C=H\) is finite dimensional then \(F\colon\mathcal{YD}(H)^H_H\rightarrow\mathcal{M}_H\) is a separable functor if and only if \(H\) is coseparable as a coalgebra in \(_H\mathcal{M}_H\) if and only if \(H\) is unimodular and cosemisimple (Theorem 4.9).
scientific article; zbMATH DE number 6954166
scientific article