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Who calls this the “Kontsevich-Rosenberg principle”? It sounds like the most evident consistency requirement for decent definitions.
Most of the references quoted and many others.
Edit: few of the references have this in the abstract or even the title.
2 — it is very nontrivial principle as it is not easy to satisfy by naive definitions (widely used for other purposes in various flavours of ncg). For example, most of the notions used for deformed algebras like quantum group theory, noncommutative projective geometry a la Artin.Zhang-Smith do NOT satisfy the principle. It is a specific kind of noncommutative geometry applying mainly to algebras which are close to free algebra (when formal smoothness applies), rather than q-deformations and alike.
Proofs that some constructions do satisfy this principle is often quite nontirivial and sometimes you are forced to go to the derived world to satisfy the principle. Berest and collaborators spend more than 10 years of research to develop that derived side of the picture.
Moreover, the van den Bergh functor (and its derived version) is a very specific construction.
Of course, the way commutative geometry relates to noncommutative can be of very many kinds. Noncommutative world can contain commutative as a full subcategory of some kind but it can be merely an analogue in which commutative world behaves differently internally (for example products are different). Also in some sorts one requires the consistency with parameter going to zero. This is different.
Here is an instructive example. Double Poisson brackets satisfy the principle.
However, if you want to apply the formalism of double Poisson brackets to commutative algebras then usually there applies a no go theorem
Double Poisson structures (à la Van den Bergh) on commutative algebras are considered. The main result shows that there are no non-trivial such structures on polynomial algebras of Krull dimension greater than one. For an arbitrary commutative algebra , this places significant restrictions on possible double Poisson structures. Exotic double Poisson structures are exhibited by the case of the polynomial algebra on a single generator, previously considered by Van den Bergh.
On the other hand, there are no go theorems for extending usual Poisson algebra definition to associative algebras (for nc prime algebras only trivial case).
Recipe for “inducing” the structure to representation schemes is not necessarily identity if you start with a commutative algebra.
Maybe a vision would be a better word than principle: the vision is that there is a full fledged noncommutative geometry of associative algebras satisfying this principle. Regarding that many standard constructions in other flavours of ncg did not satisfy this would be discourageing in mid 1990s. The example of quite systematic Kontsevich’s formal noncommutative symplectic geometry helped crystalize this, and the motivation was completely different coming from invariants of low dimensional topology, expansion for Chern-Simons theory and some related characteristic classes, while on the formalism side the inspiration was formal differential geometry of Gelfand and Kazhdan. For examples, necklace Lie algebras are an analogue of Poisson reduction.
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