Non-commutative integral forms and twisted multi-derivations

• Type: Article in International Scientific Journal
• Date: 2010

Authors

brzezinski, tomasz (Author) Other; el kaoutit, laiachi (Author) Other; lomp, christian (Author) FCUP

Journal

Title: Journal of Noncommutative Geometry

Vol. 4

Pages: 281-312

ISSN: 1661-6952

Publisher: European Mathematical Society Publishing House

Indexing

ISI Web of Knowledge - 11 Citations

Scopus - 13 Citations

Other information

Authenticus ID: P-003-BQA

DOI: 10.4171/JNCG/56

Abstract (EN): Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every injective module admits a hom-connection with respect to any differential graded algebra. The bulk of the article is devoted to describing a method of constructing hom-connections from twisted multi-derivations. The notion of a free twisted multi-derivation is introduced and the induced first order differential calculus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A (with respect to the induced differential calculus (Omega(1)(A)) that vanishes on the dual basis of Omega(1) (A). To any flat hom-connection del on A one associates a chain complex, termed a complex of integral forms on A. The canonical cokernel morphism to the zeroth homology space is called a del-integral. Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras (quantum groups). The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group O(q)F(SL)2)) is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf-Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podles sphere O(q)(S(2)) is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the del-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.

Language: English

Type (Professor's evaluation): Scientific