New York Journal of Mathematics
Volume 22 (2016) 1085-1109


Ludwik Dąbrowski, Tom Hadfield, Piotr M. Hajac, and Elmar Wagner

Braided join comodule algebras of bi-Galois objects

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Published: September 21, 2016
Keywords: Hopf algebra, principal coaction, (anti-)Drinfeld double
Subject: 46L85, 58B32

A bi-Galois object A is a bicomodule algebra for Hopf-Galois coactions with trivial invariants. In the spirit of Milnor's construction, we define the join of noncommutative bi-Galois objects (quantum torsors). To ensure that the diagonal coaction on the join algebra of the right-coacting Hopf algebra is an algebra homomorphism, we braid the tensor product A\ot A with the help of the left-coacting Hopf algebra. Our main result is that the diagonal coaction is principal. Then we show that an anti-Drinfeld double is a symmetric bi-Galois object with the Drinfeld-double Hopf algebra coacting on both left and right. In this setting, we consider a finite quantum covering as an example. Finally, we take the noncommutative torus with the natural free action of the classical torus as an example of a symmetric bi-Galois object equipped with a *-structure. It yields a noncommutative deformation of a nontrivial torus bundle.


Ludwik Dąbrowski was partially supported by the PRIN 2010-11 grant "Operator Algebras, Noncommutative Geometry and Applications" and WCMCS (Warsaw). He also gratefully acknowledges the hospitality of ESI (Vienna), IHES (Bures-sur-Yvette) and IMPAN (Warsaw). Tom Hadfield was financed via the EU Transfer-of-Knowledge contract MKTD-CT-2004-509794. Piotr M. Hajac was partially supported by NCN grant 2011/01/B/ST1/06474. Elmar Wagner was partially sponsored by WCMCS, IMPAN (Warsaw) and CIC-UMSNH (Morelia).

Author information

Ludwik Dąbrowski:
SISSA (Scuola Internazionale Superiore di Studi Avanzati), Via Bonomea 265, 34136 Trieste, Italy

Tom Hadfield:
G-Research, Whittington House, 19-30 Alfred Place, London WC1E 7EA, United Kingdom

Piotr M. Hajac:
Instytut Matematyczny, Polska Akademia Nauk, ul. Sniadeckich 8, Warszawa, 00-656 Poland

Elmar Wagner:
Instituto de Fisica y Matemáticas, Universidad Michoacana, Edificio C-3, Ciudad Universitaria, Morelia, C.P. 58040, Mexico