Matrix models for noncommutative algebraic manifolds

International audienceWe discuss the notion of matrix model, pi : C(X) -> M-K(C(T)), for algebraic submanifolds of the free complex sphere, X subset of S-C,+(N-1) When K is an element of N is fixed there is a universal such model, which factorizes as pi : C(X) -> C(X-(K)) subset of M-K(C(T)). We have X-(1) = X-class and, under a mild assumption, inclusions X-(1) subset of X-(2) subset of X-(3) subset of ... subset of X. Our main results concern X-(2), X-(3), X-(4),..., their relation with various half-classical versions of X, and lead to the construction of families of higher half-liberations of the complex spheres and of the unitary groups, all having faithful matrix models