Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

Abstract

Article:Faria Martins, J orcid.org/0000-0001- 8113-3646 (2016) ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Abstract After a thorough treatment of all algebraic structures involved, we address two dimensional holonomy operators with values in crossed modules of Hopf algebras and in crossed modules of associative algebras (called here crossed modules of bare algebras.) In particular, we will consider two general formulations of the two-dimensional holonomy of a (fully primitive) Hopf 2-connection (exact and blur), the first being multiplicative the second being additive, proving that they coincide in a certain natural quotient (defining what we called the fuzzy holonomy of a fully primitive Hopf 2-connection).

Some Remarks on MCI Crossed Modules

“…Remark 4.1. However, there exist some other crossed module structures which are MCI crossed modules, such as crossed modules of racks [7], of Hopf algebras [10], of polygroups [1]. Therefore, such structures can not be included in the above theorem.…”

Graphical calculus of Hopf crossed modules

“…The non-diagrammatic version of the following notions appear in [11,15]. Therefore we have the category of Hopf crossed modules.…”

A semi-abelian extension of a theorem by Takeuchi

“…In the context of Hopf algebras, the notion of Hopf crossed module was defined independently by Fernández Vilaboa, López López, and Villanueva Novoa [18] (see also [30,19]). We will show here that in the cocommutative case, the category HXMod K,coc of Hopf crossed modules is equivalent to the category of internal groupoids in Hopf K,coc and is consequently also equivalent to the category of internal crossed modules in Hopf K,coc .…”