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In this article we further the study of noncommutative numerical motives. Byexploring the change-of-coefficients mechanism, we start by improving some ofour previous main results. Then, making use of the notion of Schur-finiteness,we prove that the category NNum(k)_F of noncommutative numerical motives is(neutral) super-Tannakian. As in the commutative world, NNum(k)_F is notTannakian. In order to solve this problem we promote periodic cyclic homologyto a well-defined symmetric monoidal functor HP on the category ofnoncommutative Chow motives. This allows us to introduce the correctnoncommutative analogues C_NC and D_NC of Grothendieck's standard conjectures Cand D. Assuming C_NC, we prove that NNum(k)_F can be made into a Tannakiancategory NNum'(k)_F by modifying its symmetry isomorphism constraints. Byfurther assuming D_NC, we neutralize the Tannakian category NNum'(k)_F usingHP. Via the (super-)Tannakian formalism, we then obtain well-definednoncommutative motivic (super-)Galois groups. Finally, making use ofDeligne-Milne's theory of Tate triples, we construct explicit homomorphismsrelating these new noncommutative motivic (super-)Galois groups with theclassical ones.

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups