A note on free divergence-free vector fields

Abstract

We exhibit an orthonormal basis of cyclic gradients and a (non-orthogonal) basis of the homogeneous free divergence-free vector field on the full Fock space and determine the dimension of Voiculescu's free divergence-free vector field of degree k or less. Moreover, we also give a concrete formula for the orthogonal projection onto the space of cyclic gradients as well as the free Leray projection.

arXiv comment8 pages. We improved the introduction and added one remark in the last part. We also added examples of vectors in the free divergence-free vector field, inspired by the classical case