{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T23:06:20Z","timestamp":1772751980166,"version":"3.50.1"},"reference-count":0,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Noncommut. Geom."],"published-print":{"date-parts":[[2010,3,2]]},"abstract":"<jats:p>\n                    Non-commutative connections of the second type or\n                    <jats:italic>hom-connections<\/jats:italic>\n                    and associated integral forms are studied as generalisations of\n                    <jats:italic>right connections<\/jats:italic>\n                    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\n                    <jats:italic>twisted multi-derivations<\/jats:italic>\n                    . The notion of a\n                    <jats:italic>free<\/jats:italic>\n                    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\n                    <jats:inline-formula>\n                      <jats:tex-math>A<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    induces a unique hom-connection on\n                    <jats:inline-formula>\n                      <jats:tex-math>A<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    (with respect to the induced differential calculus\n                    <jats:inline-formula>\n                      <jats:tex-math>\u03a9^1(A)<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    ) that vanishes on the dual basis of\n                    <jats:inline-formula>\n                      <jats:tex-math>\u03a9^1(A)<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . To any flat hom-connection\n                    <jats:inline-formula>\n                      <jats:tex-math>\u2207<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    on\n                    <jats:inline-formula>\n                      <jats:tex-math>A<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    one associates a chain complex, termed a\n                    <jats:italic>complex of integral forms<\/jats:italic>\n                    on\n                    <jats:inline-formula>\n                      <jats:tex-math>A<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . The canonical cokernel morphism to the zeroth homology space is called a\n                    <jats:inline-formula>\n                      <jats:tex-math>\u2207<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -\n                    <jats:italic>integral<\/jats:italic>\n                    . 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\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathcal{O}_q(\\mathrm{SL}(2))<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf\u2013Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podle\u2019s sphere\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathcal{O}_q(\\mathrm{SL}(2))<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the\n                    <jats:inline-formula>\n                      <jats:tex-math>\u2207<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.\n                  <\/jats:p>","DOI":"10.4171\/jncg\/56","type":"journal-article","created":{"date-parts":[[2010,3,2]],"date-time":"2010-03-02T17:47:56Z","timestamp":1267552076000},"page":"281-312","source":"Crossref","is-referenced-by-count":10,"title":["Non-commutative integral forms and twisted multi-derivations"],"prefix":"10.4171","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6270-3439","authenticated-orcid":false,"given":"Tomasz","family":"Brzezi\u0144ski","sequence":"first","affiliation":[{"name":"Swansea University, UK"}]},{"given":"Laiachi","family":"El Kaoutit","sequence":"additional","affiliation":[{"name":"Universidad de Granada, Spain"}]},{"given":"Christian","family":"Lomp","sequence":"additional","affiliation":[{"name":"Universidade do Porto, Portugal"}]}],"member":"2673","container-title":["Journal of Noncommutative Geometry"],"original-title":[],"link":[{"URL":"http:\/\/www.ems-ph.org\/fulltext\/10.4171\/JNCG\/56","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T13:04:13Z","timestamp":1762952653000},"score":1,"resource":{"primary":{"URL":"https:\/\/ems.press\/doi\/10.4171\/jncg\/56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,2]]},"references-count":0,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.4171\/jncg\/56","relation":{},"ISSN":["1661-6952","1661-6960"],"issn-type":[{"value":"1661-6952","type":"print"},{"value":"1661-6960","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,3,2]]}}}