×

Extensions of the endomorphism algebra of weak comodule algebras. (English) Zbl 1325.16031

Let \(H\) be a finite-dimensional weak Hopf algebra, \(A\) a weak right \(H\)-comodule algebra and \(B=A^{coH}\). For an \(M\in\mathcal M_A^H\), \(\text{End}_A^H(M)\) is the \(H\)-coinvariant subalgebra of the weak right \(H\)-comodule algebra \(\text{End}_A(M)\) [Y. Wang and L. Zhang, J. Pure Appl. Algebra 215, No. 6, 1133-1145 (2011; Zbl 1255.16037)]. For the induced weak Doi-Hopf module \(M\otimes_BA\), \(\text{End}_A(M\otimes_BA)\) is a weak \(H\)-cleft extension of \(\text{End}_B(M)\) and the smash product \(\text{End}_B(M)*H\).
Theorem 1. Suppose that \(A/B\) is a weak right \(H\)-Galois extension. Then \(\text{End}_A(M\otimes_BA)\cong\text{End}_B(M)*H\) as right \(H\)-comodule algebras.
Let \(l\in H\) be a left (\(r\in H\) right) integral in \(H\) with antipode \(S\). Then \(l\) is said to be \(S\)-fixed if \(S(l)=l\), and \(H\) is unimodular if there exists a non-zero two-sided integral in \(H\). Let \(M,N\in\mathcal M_A^H\) and \(M_A\) and \(N_A\) denote the \(A\)-module structure of \(M\) and \(N\), respectively. If \(M_A\) is a direct summand of some \(A\)-module direct sum of a finite number of copies of \(N_A\), then \(M_A\) is called weakly divided by \(N_A\) denoted by \(M_A\leq N_A\). – The authors show an equivalent condition for a weak right \(H^*\)-Galois extension \(\text{End}_B(M)/\text{End}_A(M)\).
Theorem 2. Suppose that \(A/B\) is a weak right \(H\)-Galois extension, and \(H\) is unimodular with \(S\)-fixed two-sided integral \(t\), and \(T\in H^*\) is its dual integral. Let \(M\in\mathcal M_A^H\). Then (1) \(\text{End}_B(M)/\text{End}_A(M)\) is weak right \(H^*\)-Galois if and only if \(M\otimes_BA\leq M\); (2) \(\text{End}_B(M)\) possesses elements of trace \(1\) if and only if \(M\leq M\otimes_BA\). Moreover, if both (1) and (2) hold, then \((\text{End}_B(M)\otimes_{\text{End}_A(M)}.,.\otimes_{\text{End}_A(M\otimes_BA)}\text{End}_B(M))\) defines a Morita equivalence between \(_{\text{End}_A(M)}\mathcal M\) and \(\mathcal M_{\text{End}_A(M\otimes_BA)}\).

MSC:

16T05 Hopf algebras and their applications
16T15 Coalgebras and comodules; corings
16S50 Endomorphism rings; matrix rings
16S40 Smash products of general Hopf actions

Citations:

Zbl 1255.16037
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Van Oyastaeyen and Y.H. Zhang, “<Emphasis Type=”Italic“>H-module endomorphism rings,” J. Pure Appl. Algebra 102(2), 207-219 (1995). · Zbl 0838.16034
[2] G. Böhm, F. Nill and K. Szlachányi, “Weak Hopf algebras. I. Integral theory and <Emphasis Type=”Italic“>C*-structure,” J. Algebra 221(2), 385-438 (1999). · Zbl 0949.16037
[3] T. Hayashi, “Quantum group symmetry of partition functions of IRF models and its applications to Jones’s index theory,” Comm. Math. Phys. 157(2), 331-345 (1993). · Zbl 0796.17014
[4] Nikshych, D.; Vainerman, L., Finite quantum groupoids and their applications, 211-262 (2002), Cambridge · Zbl 1026.17017
[5] T. Yamanouchi, “Duality for generalized Kac algebras and a characterization of finite groupoid algebras,” J. Algebra 163(1), 9-50 (1994). · Zbl 0830.46047
[6] Y. Wang and L. Y. Zhang, “The structure theorem for weak module coalgebras,” Mat. Zametki 88(1), 3-17 (2010) [Math. Notes 88 (1), 3-15 (2010)]. · Zbl 1231.16029
[7] R. F. Niu, Y. Wang, and L. Y. Zhang, “The structure theorem of endmorphism algebras for weak Doi-Hopf modules,” Acta Math. Hungar. 127(3), 273-290 (2010). · Zbl 1214.16028
[8] Y. Wang and L. Y. Zhang, “The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras,” J. Pure Appl. Algebra 215(6), 1133-1145 (2011). · Zbl 1255.16037
[9] L. Y. Zhang, “Maschke-type theorem and Morita context over weak Hopf algebras,” Sci. China Ser. A 49(5), 587-598 (2006). · Zbl 1121.16036
[10] S. Montgomery, Hopf Algebras and Their Actions on Rings, in CBMS Regional Conf. Ser. in Math. (Amer. Math. Soc., Providence, RI, 1993), Vol. 82. · Zbl 0793.16029
[11] M. E. Sweedler, Hopf Algebras, in Math. Lecture Note Ser. (W. A. Benjamin, New York, 1969).
[12] Caenepeel, S.; Groot, E., Modules over weak entwining structures, 31-54 (2000), Providence, RI · Zbl 0978.16033
[13] G. Böhm, “Doi-Hopf modules over weak Hopf algebras,” Comm. Algebra 28(10), 4687-4698 (2000). · Zbl 0965.16024
[14] A. B. Rodríguez Raposo, “Crossed products for weak Hopf algebras,” Comm. Algebra 37(7), 2274-2289 (2009). · Zbl 1182.18008
[15] X. Zhou and S. H. Wang, “The duality theorem for weak Hopf algebra (co)actions,” Comm. Algebra 38(12), 4613-4632 (2010). · Zbl 1220.16024
[16] D. Nikshych, “On the structure of weak Hopf algebras,” Adv. Math. 170(2), 257-286 (2002). · Zbl 1010.16041
[17] S. Caenepeel and E. De Groot, “Galois theory for weak Hopf algebras,” Rev. Roumaine Math. Pures Appl. 52(2), 151-176 (2007). · Zbl 1146.16018
[18] L. Y. Zhang and Y. C. Li, “Homomorphisms, separable extensions and Morita maps for weak module algebras,” Sibirsk. Mat. Zh. [Siberian Math. J.] 52(1), 210-222 (2011) [Siberian Math. J. 52 (1), 167-177 (2011)]. · Zbl 1225.16016
[19] G. Böhm, “Galois theory for Hopf algebroids,” Ann. Univ. Ferrara Sez. VII (N. S.) 51, 233-262 (2005). · Zbl 1134.16014
[20] X. Y. Zhou, “Homological dimension of weak Hopf-Galois extensions,” Acta Math. Hungar. 138(1-2), 140-146 (2013). · Zbl 1285.16029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.