Matrix models for noncommutative algebraic manifolds

We discuss the notion of matrix model, π : C(X) → MK(C(T )), for algebraic submanifolds of the free complex sphere, X ⊂ SN−1 C,+ . When K ∈ N is fixed there is a universal such model, which factorizes as π : C(X)→ C(X) ⊂MK(C(T )). We have X = Xclass and, under a mild assumption, inclusions X (1) ⊂ X ⊂ X ⊂ . . . ⊂ X. Our main results concern X, X, X, . . ., 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.