Galois and Cleft Monoidal Cowreaths. Applications.

Format:

Book

Author/Creator:

Bulacu, D.

Contributor:

Torrecillas, B.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.270

Language:

English

Subjects (All):

Galois theory.

Hopf algebras.

Physical Description:

1 online resource (145 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"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"-- Provided by publisher.

Contents:

Cover

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.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470465322

1470465329

OCLC:

1259589112