nLab Kontsevich-Rosenberg principle

Kontsevich-Rosenberg principle is a heuristic that a good analogue of a commutative geometric structure on a noncommutative space (associative algebras or related objects) should induce its classical counterpart on the associated commutative representation schemes (or derived representation schemes). An example is a double Poisson structure of Michel Van den Bergh which induces ordinary Poisson bracket on representation spaces.

• M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Gelfand Math. Sem., Birkhäuser Boston 2000; arXiv:math/9812158 (published version is slightly updated with respect to arXiv version)

This came as continuation/generalization of some ideas from

• Maxim Kontsevich, Formal (non)-commutative symplectic geometry, The Gelfand Mathematical. Seminars, 1990-1992, Ed. L.Corwin, I.Gelfand, J.Lepowsky, Birkhauser 1993, 173-187, pdf

Some formalizations of the principle include so called Van den Bergh’s functor.

• Yu. Berest, X. Chen, F. Eshmatov, A. Ramadoss, Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras, Contemp. Math. 583 (2012) 219–246 arXiv:1202.2717
• Yu. Berest, G. Felder, A. Ramadoss, Derived representation schemes and noncommutative geometry, arXiv:1304.5314
• George Khachatryan, Derived representation schemes and non-commutative geometry, Cornell PhD thesis under guidance of Yuri Berest online
• Yuri Berest, George Khachatryan, Ajay Ramadoss, Derived representation schemes and cyclic homology, Adv. Math. 245, (2013) 625–689 arXiv:1112.1449
• David Fernández, The Kontsevich-Rosenberg principle for bi-symplectic forms, arXiv:1708.02650
• Stefano D’Alesio, Noncommutative derived Poisson reduction, arXiv:2012.04451
• R. Bocklandt, L. Le Bruyn, Necklace Lie algebras and noncommutative symplectic geometry, Math Z 240, 141–167 (2002) doi
• H. Zhao, Commutativity of quantization and reduction for quiver representations, Math. Z. 301, 3525–3554 (2022). doi
• Maxime Fairon, David Fernández, On the noncommutative Poisson geometry of certain wild character varieties, arXiv:2103.10117; Euler continuants in noncommutative quasi-Poisson geometry, arXiv:2105.04858