Glasnik Matematicki, Vol. 54, No. 2 (2019), 477-499.

BICOVARIANT DIFFERENTIAL CALCULI FOR FINITE GLOBAL QUOTIENTS

David N. Pham

Department of Mathematics & Computer Science, Queensborough C. College, City University of New York, Bayside, NY 11364, USA
e-mail: dpham90@gmail.com

Abstract.   Let (M,G) be a finite global quotient, that is, a finite set M with an action by a finite group G. In this note, we classify all bicovariant first order differential calculi (FODCs) over the weak Hopf algebra 𝕜 (G ⋉ M) ≃ 𝕜 [G ⋉ M]*, where G ⋉ M is the action groupoid associated to (M,G), and 𝕜[G ⋉ M] is the groupoid algebra of G ⋉ M. Specifically, we prove a necessary and sufficient condition for a FODC over 𝕜(G ⋉ M) to be bicovariant and then show that the isomorphism classes of bicovariant FODCs over 𝕜(G ⋉ M) are in one-to-one correspondence with subsets of a certain quotient space.

2010 Mathematics Subject Classification.   58B32, 16T05, 18B40

Key words and phrases.   Global quotients, noncommutative differential geometry, first order differential calculi, weak Hopf algebras

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.2.10

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