Non-commutative integral forms and twisted multi-derivations

Abstract

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.