Invertible weak entwining structures and weak \(C\)-cleft extensions. (English) Zbl 1116.18003
A (right) weak entwining structure is a map \(\psi:C\otimes A\to A\otimes C\) where \(A\) is an algebra and \(C\) is a coalgebra that interacts appropriately with the algebra and coalgebra structure. There is also a left version of this structure. An invertible weak entwining structure is an appropriately interacting left and right weak entwining maps with the same algebra and coalgebra. This paper studies the relationship between the left and right structures and shows that one of the conditions in the original definition of invertible weak entwining structure is redundant. It also studies \(C\)-cleft versions that utilize equilizers and coequilizers.
Reviewer: David Auckly (Manhattan)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 16T05 | Hopf algebras and their applications |
Keywords:
entwining structureReferences:
| [1] | Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R., Rodríguez Raposo, A.B.: Weak C-cleft extensions, weak entwining structures and weak Hopf algebras. J. Algebra 284, 679–704 (2005) · Zbl 1085.16027 · doi:10.1016/j.jalgebra.2004.07.043 |
| [2] | Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R., Rodríguez Raposo, A.B.: Weak C-cleft extensions and weak Galois extensions. J. Algebra 299, 276–293 (2006) · Zbl 1117.16024 · doi:10.1016/j.jalgebra.2005.09.012 |
| [3] | Böhm, G.: Doi–Hopf modules over weak Hopf algebras. Comm. Algebra 28, 4687–4698 (2000) · Zbl 0965.16024 · doi:10.1080/00927870008827113 |
| [4] | Brzeziński, T.: On modules associated to coalgebra-Galois extensions. J. Algebra 215, 290–317 (1999) · Zbl 0936.16030 · doi:10.1006/jabr.1998.7738 |
| [5] | Brzeziński, T.: The structure of corings. Induction functors, Maschke-type theorems, and Frobenius and Galois-type properties. Algebr. Represent. Theory 5, 389–410 (2002) · Zbl 1025.16017 · doi:10.1023/A:1020139620841 |
| [6] | Brzeziński, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191, 467–492 (1998) · Zbl 0899.55016 · doi:10.1007/s002200050274 |
| [7] | Brzeziński, T., Turner, R.B., Wrightson, A.P.: The structure of weak coalgebra Galois extensions. Comm. Algebra 34, 1489–1519 (2006) · Zbl 1097.16013 · doi:10.1080/00927870500455056 |
| [8] | Caenepeel, S., De Groot, E.: Modules over weak entwining structures. Contemp. Math. 267, 31–54 (2000) · Zbl 0978.16033 |
| [9] | Caenepeel, S., Militaru, G., Zhu, S.: Frobenius and separable functors for generalized module categories and nonlinear equations. In: Lecture Notes in Mathematics, vol. 1787. Springer, Berlin Heidelberg New York (2002) · Zbl 1008.16036 |
| [10] | Schauenburg, P., Schneider, H.-J.: On generalized Hopf–Galois extensions. J. Pure Appl. Algebra 202, 168–194 (2005) · Zbl 1081.16045 · doi:10.1016/j.jpaa.2005.01.005 |
| [11] | Sweedler, M.: The predual theorem to the Jacobson–Bourbaki theorem. Trans. Amer. Math. Soc. 213, 391–406 (1975) · Zbl 0317.16007 · doi:10.1090/S0002-9947-1975-0387345-9 |
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