nLab
noncommutative motive

The motives in algebraic geometry can be adapted to derived noncommutative geometry. Such a theory has been developed by Maxim Kontsevich. There is a remarkable observation that the category of Chow motives can be after localizing at the Lefschetz motive can be embedded into the category of noncommutative motives. More recently this direction has been systematically studied by Cisinski and Tabuada.

There is another approach by Arne Ostvaer.

In noncommutative geometry a la Connes, Connes and Marcolli have also introduced some motivic ideas. Marcolli also has most recent collaboration with Tabuada on the algebraic side.

References

• Denis-Charles Cisinski, Gonçalo Tabuada, Symmetric monoidal structure on Non-commutative motives, arxiv/1001.0228
• Goncalo Tabuada: A guided tour through the garden of noncommutative motives, arxiv1108.3787; Bivariant cyclic cohomology and Connes’ bilinear pairings in Non-commutative motives, arxiv/1005.2336; Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives, arxiv/1101.0731; Chow motives versus non-commutative motives, arxiv/1103.0200; A guided tour through the garden of noncommutative motives, arxiv/1108.3787; Galois descent of additive invariants, arxiv/1301.1928
• Matilde Marcolli, Goncalo Tabuada, Kontsevich’s noncommutative numerical motives, arxiv/1108.3785; Noncommutative motives, numerical equivalence, and semi-simplicity, arxiv/1105.2950; Noncommutative numerical motives, Tannakian structures, and motivic Galois groups, arxiv/1110.2438
• Ivo Dell’Ambrogio, Gonçalo Tabuada, Tensor triangular geometry of non-commutative motives, arxiv/1104.2761

category: algebraic geometry, noncommutative geometry