Fuzzy complex Grassmannians and quantization of line bundles

Abstract

We construct by purely representation-theoretic methods fuzzy versions of an arbitrary complex Grassmannian M = Gr n (C n+m ), i.e., a sequence of matrix algebras tending SU (n + m)-equivariantly to the algebra of smooth functions on M . We also show that this approximation can be interpreted in terms of the Berezin-Toeplitz quantization of M . Furthermore, we use branching rules to prove that the quantization of every complex line bundle over M is given by a SU (n + m)-equivariant truncation of the space of its L 2 -sections.

Noncommutative deformations of locally symmetric Kähler manifolds

“…In addition, recurrence relations to construct star products for G 2,2 was derived. Deformation quantization of Grassmann manifolds and flag manifolds were studied in [17,10,11,24]. Complex Grassmann manifold G p,q is defined as a set of the whole p dimensional part vector space of p + q dimensional vector space V .…”

Polarization Tensors for the fuzzy Grassmannians $Gr_{2;n}^F$ and Grassmannian harmonics on $Gr_{2;n}$

“…) and their fuzzy versions Gr F 2;n ; we give their U(n)-invariant Laplacian and eigenfunctions in a Schwinger-Fock formalism, extending the one used for fuzzy complex projective spaces in [7]. Already in [8] the construction of the fuzzy Grassmannians Gr F 2;n was given along with a star product and the corresponding coherent state map to functions, the relevant coherent states were constructed in [9] and a Berezin-Toeplitz version was first given in [10]. The approach used here is different to that taken in [8] in that it is based on composite pseudo-oscillators and a double Fock vacuum.…”

Vector bundles for “Matrix algebras converge to the sphere”

“…But some of the detailed estimates needed to prove convergence require fairly complicated considerations (see Section 11) concerning highest weights for representations of compact semi-simple Lie groups. It appears to me that it would be quite challenging to carry out those details for the general case, though I expect that some restricted cases, such as matrix-algebra approximations for complex projective spaces [8,46,27,12,6], are quite feasible to deal with.…”