Journal of Noncommutative Geometry
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Published online: 2010-03-02
Non-commutative integral forms and twisted multi-derivations
Tomasz Brzeziński[1], Laiachi El Kaoutit[2] and Christian Lomp[3] (1) Swansea University, UK(2) Universidad de Granada, Spain
(3) Universidade do Porto, Portugal
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 Ω1(A)) that vanishes on the dual basis of Ω1(A). To any flat hom-connection ∇ 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 ∇-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 $\mathcal{O}$q(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 Podle’s sphere $\mathcal{O}$q(S2) is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the ∇-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.
Keywords: Hom-connection, integral form, twisted multi-derivation, covariant differential calculus, quantum group
Brzeziński Tomasz, El Kaoutit Laiachi, Lomp Christian: Non-commutative integral forms and twisted multi-derivations. J. Noncommut. Geom. 4 (2010), 281-312. doi: 10.4171/JNCG/56