Feedback Drifts of Polynomial Controls on Unitary Laws

Let $\mathbb{N}$ be the natural numbers, $d\in\mathbb{N}$, $[d]$ the initial segment determined by $d$, $W^{\circ}_{d}$ the set of cyclically reduced words, and $\mathcal{L}_{d}$ the set of unitary laws of $d$-tuples. Let $\mathcal{P}_{d}$ be the set of polynomial controls, $D^{i}_{w}$ the cyclic derivatives, products of word polynomials as in Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions §product, and $\lambda(X)$ the evaluation of a word polynomial $X$. Let $\mathbb{C}$ be the complex numbers; finite sums are those of Sum over a Finite Index Set.

(Feedback drift)¶ For $a\in\mathcal{P}_{d}$ and $\lambda\in\mathcal{L}_{d}$, the feedback drift $b_{a}(\lambda)$ is the map $W^{\circ}_{d}\to\mathbb{C}$ given by

$$b_{a}(\lambda)(w)=\sum_{i\in[d]}\lambda\bigl(a^{i}D^{i}_{w}\bigr).$$