# Non-commutative integral forms and twisted multi-derivations

### Tomasz Brzeziński

Swansea University, UK### Laiachi El Kaoutit

Universidad de Granada, Spain### Christian Lomp

Universidade do Porto, Portugal

## 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 $Ω_{1}(A)$) that vanishes on the dual basis of $Ω_{1}(A)$. 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 $O_{q}(SL(2))$ 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 $O_{q}(SL(2))$ 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.

## Cite this article

Tomasz Brzeziński, Laiachi El Kaoutit, Christian Lomp, Non-commutative integral forms and twisted multi-derivations. J. Noncommut. Geom. 4 (2010), no. 2, pp. 281–312

DOI 10.4171/JNCG/56