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A note on free divergence-free vector fields

Hyuga Ito, Akihiro Miyagawa

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.

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.
Paper Structure (3 sections, 9 theorems, 37 equations)

This paper contains 3 sections, 9 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The dimension of homogeneous free divergence-free vector field

Key Result

Theorem 2.1

(v02) We have $(\delta^s\mathbb{C}^s_{\langle n\rangle})[1\oplus\cdots\oplus1] =(\delta^l\mathbb{C}^l_{\langle n\rangle})[1\oplus\cdots\oplus1]$ in $\mathcal{F}(\mathbb{C}^n)^n$.

Theorems & Definitions (18)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 8 more