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 ${\Omega }^1(A)$) that vanishes on the dual basis of ${\Omega }^1(A)$. To any flat hom-connection $\nabla$ 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 $\nabla$-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 ${\mathcal{O}}_q(\mathrm{SL}(2))$ 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 ${\mathcal{O}}_q(\mathrm{SL}(2))$ is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the $\nabla$-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.