{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T21:37:27Z","timestamp":1773265047608,"version":"3.50.1"},"reference-count":0,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Eur. Math. Soc."],"accepted":{"date-parts":[[2013,8,5]]},"published-print":{"date-parts":[[2016,2,16]]},"abstract":"<jats:p>\n                    In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum\n                    <jats:inline-formula>\n                      <jats:tex-math>(k)_F<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum\n                    <jats:inline-formula>\n                      <jats:tex-math>(k)_F<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor\n                    <jats:inline-formula>\n                      <jats:tex-math>\\overline{HP_\\ast}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues\n                    <jats:inline-formula>\n                      <jats:tex-math>C_{NC}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>D_{NC}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    of Grothendieck's standard conjectures\n                    <jats:inline-formula>\n                      <jats:tex-math>C<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>D<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Assuming\n                    <jats:inline-formula>\n                      <jats:tex-math>C_{NC}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , we prove that NNum\n                    <jats:inline-formula>\n                      <jats:tex-math>(k)_F<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    can be made into a Tannakian category NNum\n                    <jats:inline-formula>\n                      <jats:tex-math>^\\dagger(k)_F<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    by modifying its symmetry isomorphism constraints. By further assuming\n                    <jats:inline-formula>\n                      <jats:tex-math>D_{NC}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , we neutralize the Tannakian category Num\n                    <jats:inline-formula>\n                      <jats:tex-math>^\\dagger(k)_F<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    using\n                    <jats:inline-formula>\n                      <jats:tex-math>\\overline{HP_\\ast}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Via the (super-)Tannakian formalism, we then obtain well-defined\n                    <jats:italic>noncommutative motivic Galois (super-)groups<\/jats:italic>\n                    . Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.\n                  <\/jats:p>","DOI":"10.4171\/jems\/598","type":"journal-article","created":{"date-parts":[[2016,2,16]],"date-time":"2016-02-16T17:45:13Z","timestamp":1455644713000},"page":"623-655","source":"Crossref","is-referenced-by-count":7,"title":["Noncommutative numerical motives, Tannakian structures, and motivic Galois groups"],"prefix":"10.4171","volume":"18","author":[{"given":"Matilde","family":"Marcolli","sequence":"first","affiliation":[{"name":"California Institute of Technology, Pasadena, United States"}]},{"given":"Gon\u00e7alo","family":"Tabuada","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, USA"}]}],"member":"2673","container-title":["Journal of the European Mathematical Society"],"original-title":[],"link":[{"URL":"http:\/\/www.ems-ph.org\/fulltext\/10.4171\/JEMS\/598","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T13:58:49Z","timestamp":1762869529000},"score":1,"resource":{"primary":{"URL":"https:\/\/ems.press\/doi\/10.4171\/jems\/598"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,16]]},"references-count":0,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.4171\/jems\/598","relation":{},"ISSN":["1435-9855","1435-9863"],"issn-type":[{"value":"1435-9855","type":"print"},{"value":"1435-9863","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2,16]]}}}