Galois and cleft monoidal cowreaths. Applications / D. Bulacu and B. Torrecillas.

Format:

Book

Author/Creator:

Bulacu, Daniel, 1973- author.

Torrecillas, B. (Blas), 1958- author.

Series:

Memoirs of the American Mathematical Society ; no. 1322.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1322

Language:

English

Subjects (All):

Galois theory.

Hopf algebras.

Physical Description:

v, 133 pages : illustrations ; 26 cm.

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 ν-Doi-Hopf structures, respectively. In particular, we obtain that some constructions of Brzeziński and Schauenburg produce examples of cleft wreaths, and therefore of cleft cowreaths, too.

Contents:

Introduction and preliminaries

Schneider type theorem for monoidal cowreaths

Cleft cowreaths and wreath algebras

Applications.

Notes:

"March 2021, volume 270, number 1322 (sixth of 7 numbers)."

Includes bibliographical references.

ISBN:

9781470447526

1470447525

OCLC:

1243743173