The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables

Let $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, and let $W_{n}$ be the set of words in the letters $1,\dots,n$, with empty word $\varnothing$, concatenation $uv$ and reversal $w^{\mathrm{rev}}$. Let $\mathbb{C}$ be the field of complex numbers, with complex conjugation $z\mapsto\overline{z}$. Finite sets are as in Finite Set, and sums over a finite index set as in Sum over a Finite Index Set.

1. (Polynomials)¶ A noncommutative polynomial in the variables $x_{1},\dots,x_{n}$ is a map $p:W_{n}\to\mathbb{C}$ whose support $\operatorname{supp}p=\{w\in W_{n}:\ p(w)\neq0\}$ is finite; $p(w)$ is its coefficient at $w$. The set of these maps is written $\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$.

2. (Linear operations)¶ For $p,q\in\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ and $c\in\mathbb{C}$, the sum $p+q$ and the multiple $cp$ are the maps $w\mapsto p(w)+q(w)$ and $w\mapsto c\,p(w)$; their supports lie in $\operatorname{supp}p$ and in the union $\operatorname{supp}p\cup\operatorname{supp}q$, which is finite by claim 3 of Peeling an Element off a Finite Set, and Unions of Finite Sets, so they are polynomials by claim 3 of Basic Properties of Finite Sets. The zero polynomial $0$ is the map with all values $0$, and $p-q=p+(-1)q$.

3. (Monomials)¶ For $w\in W_{n}$ the monomial $x_{w}$ is the map with $x_{w}(w)=1$ and $x_{w}(v)=0$ for $v\neq w$; its support $\{w\}$ is finite by claim 2 of Basic Properties of Finite Sets, so $x_{w}$ is a polynomial. The unit is $1=x_{\varnothing}$, and for $j$ in the initial segment $[n]$ the variable $x_{j}$ is the monomial of the letter $(j)$.

4. (Product)¶ For $p,q\in\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ the product $pq$ is the map

where $F(w)$ is the set of factorisations $uv=w$, nonempty and finite by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §factorisations. If $(pq)(w)\neq0$ then some term $p(u)q(v)$ is nonzero, by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, so $p(u)\neq0$ and $q(v)\neq0$ by claim 1 of Zero Products and Elementary Identities in a Field, and $w=uv$ with $u\in\operatorname{supp}p$ and $v\in\operatorname{supp}q$; hence $\operatorname{supp}(pq)$ is finite by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §products and claim 3 of Basic Properties of Finite Sets, and $pq$ is a polynomial.

5. (Adjoint)¶ For $p\in\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ the adjoint $p^{*}$ is the map $p^{*}(w)=\overline{p(w^{\mathrm{rev}})}$. Its support is the image of $\operatorname{supp}p$ under reversal, by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal and claim 3 of Properties of Complex Conjugation and Modulus. This image is empty if $\operatorname{supp}p$ is empty, and finite by claim 4 of Basic Properties of Finite Sets otherwise; so $p^{*}$ is a polynomial.

6. (Self-adjoint polynomials)¶ A polynomial $p$ is self-adjoint if $p^{*}=p$. The set of self-adjoint polynomials is written $\mathbb{C}\langle x_{1},\dots,x_{n}\rangle_{\mathrm{sa}}$.