LEADER 04020nam 22005293 450 001 9910970824903321 005 20231110230402.0 010 $a9781470465322 010 $a1470465329 035 $a(CKB)4100000011975390 035 $a(MiAaPQ)EBC6661103 035 $a(Au-PeEL)EBL6661103 035 $a(OCoLC)1259589112 035 $a(RPAM)22488161 035 $a(EXLCZ)994100000011975390 100 $a20210901d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGalois and Cleft Monoidal Cowreaths. Applications 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2021. 215 $a1 online resource (145 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.270 311 08$a9781470447526 311 08$a1470447525 320 $aIncludes bibliographical references. 327 $aCover -- Title page -- Part 1. Introduction and Preliminaries -- Chapter 1. Introduction -- Chapter 2. Preliminaries -- 2.1. Coflat and robust objects -- 2.2. Cowreaths in monoidal categories and entwined module categories associated to them -- Part 2. A Schneider type theorem for monoidal cowreaths -- Chapter 3. A pair of adjoint functors -- Chapter 4. The fundamental theorem for entwined modules over monoidal cowreaths -- Chapter 5. Total integrals and the structure theorem -- Part 3. Cleft cowreaths and wreath algebras -- Chapter 6. Cleft cowreaths -- Chapter 7. Cleft cowreaths versus wreath algebras -- Part 4. Applications -- Chapter 8. Cleft (co)wreaths arising from Doi-Hopf modules over (quasi-)bialgebras -- Chapter 9. Cleft (co)wreaths arising from crossed products by a coalgebra -- Chapter 10. Cleft (co)wreaths arising from -Doi-Hopf cowreaths -- Chapter 11. Cleft (co)wreaths arising from generalized crossed products -- Chapter 12. Cleft (co)wreaths arising from quasi-Hopf bimodules -- Bibliography -- Back Cover. 330 $a"We introduce (pre-)Galois and cleft monoidal cowreaths. Generalizing a result of Schneider, to any pre-Galois cowreath we associate a pair of adjoint functors L R and give necessary and sufficient conditions for the adjunction to be an equivalence of categories. Inspired by the work of Doi we also give sufficient conditions for L R to be an equivalence, and consequently conditions under which a fundamental structure theorem for entwined modules over monoidal cowreaths holds. We show that a cowreath is cleft if and only if it is Galois and has the normal basis property; this generalizes a result concerning Hopf cleft extensions due to Doi and Takeuchi. Furthermore, we show that the cleft cowreaths are in a one to one correspondence with what we call cleft wreaths. The latter are wreaths in the sense of Lack and Street, equipped with two additional morphisms satisfying some compatibility relations. Note that, in general, the algebras defined by cleft wreaths cannot be identified to (generalized) crossed product algebras, as they were defined by Doi and Takeuchi, and Blattner, Cohen and Montgomery. This becomes more transparent when we apply our theory to cowreaths defined by actions and coactions of a quasi-Hopf algebra, monoidal entwining structures and Doi-Hopf structures, respectively. In particular, we obtain that some constructions of Brzezinski and Schauenburg produce examples of cleft wreaths, and therefore of cleft cowreaths, too"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aGalois theory 606 $aHopf algebras 615 0$aGalois theory. 615 0$aHopf algebras. 676 $a512/.32 700 $aBulacu$b D$01801285 701 $aTorrecillas$b B$01801286 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910970824903321 996 $aGalois and Cleft Monoidal Cowreaths. Applications$94346424 997 $aUNINA