Polynomial Weyl algebra presentation and normal-ordered basis

Order the basis of ${\mathfrak{h}}_n(k)$ as $X_1<\cdots <X_n<P_1<\cdots <P_n<Z$. By [F4], the elements $X^{\alpha }{P}^{\beta }{Z}^m$ form a $k$-basis of $U({\mathfrak{h}}_n(k))$. In particular, the powers $Z^m$ are linearly independent; let $k[Z]$ denote their polynomial subalgebra. Since $Z$ is central by [F1], grouping this basis by $\alpha ,\beta$ gives the direct-sum decomposition ${⨁}_{\alpha ,\beta \in {\mathbb{N}}^n}k[Z]X^{\alpha }{P}^{\beta }$ as a free $k[Z]$-module.

Send $X_i\mapsto x_i$, $P_i\mapsto p_i$, and $Z\mapsto \mathrm{\hslash }1$ in $W_n(k;\mathrm{\hslash })$. By [F1] and [F5], this linear assignment preserves the bracket on every basis pair, hence on all pairs by bilinearity. By [F9] it extends to a unital algebra map $T({\mathfrak{h}}_n(k))\to W_n(k;\mathrm{\hslash })$. The defining relators of $U({\mathfrak{h}}_n(k))$ in [F3] map to zero. The map's kernel is a two-sided ideal because it is an algebra homomorphism, so it contains their generated ideal by [F7], and [F10] factors the map through $U({\mathfrak{h}}_n(k))$. This induced algebra map sends $Z-\mathrm{\hslash }1$ to zero; its kernel is two-sided and contains the ideal $(Z-\mathrm{\hslash }1)$ generated by that central element by [F7], so a second application of [F10] gives $\varphi :Q_{\mathrm{\hslash }}\to W_n(k;\mathrm{\hslash })$. The factor maps remain $k$-linear because the quotient maps are surjective.

Orient the relations of [F5] by $x_j{x}_i\to x_i{x}_j$ and $p_j{p}_i\to p_i{p}_j$ when $j>i$, and $p_i{x}_j\to x_j{p}_i+\mathrm{\hslash }{\delta }_{ij}1$. For each word use the lexicographic measure (word length, number of inversions for $x_1<\cdots <x_n<p_1<\cdots <p_n$). Each swapped word has one fewer inversion, and each scalar correction has smaller length; hence every reduction terminates.

By step 1.1, $U({\mathfrak{h}}_n(k))$ is free as a $k[Z]$-module on $X^{\alpha }{P}^{\beta }$. Since $Z$ is central, the ideal $(Z-\mathrm{\hslash })U$ is the direct sum of $(Z-\mathrm{\hslash })k[Z]X^{\alpha }{P}^{\beta }$. For $f(Z)={\sum }_{m=0}^N{c}_m{Z}^m$, the identity $Z^m-{\mathrm{\hslash }}^m=(Z-\mathrm{\hslash }){\sum }_{r=0}^{m-1}{Z}^{m-1-r}{\mathrm{\hslash }}^r$ for $m\ge 1$ shows that $f(Z)-f(\mathrm{\hslash })$ is divisible by $Z-\mathrm{\hslash }$. The evaluation map ${\mathrm{ev}}_{\mathrm{\hslash }}:k[Z]\to k$, $f\mapsto f(\mathrm{\hslash })$ is surjective because it fixes constants, and the displayed identity shows its kernel is exactly $(Z-\mathrm{\hslash })$; hence $k[Z]\mathrm{/}(Z-\mathrm{\hslash })\cong k$. Therefore the classes of $X^{\alpha }{P}^{\beta }$ form a basis of $Q_{\mathrm{\hslash }}$.

The overlapping mixed reductions are $p_k{p}_j{x}_i$ with $k>j$ and $p_k{x}_j{x}_i$ with $j>i$. For $p_k{p}_j{x}_i$, reducing the left pair gives $p_k{p}_j{x}_i\to p_j{p}_k{x}_i\to p_j{x}_i{p}_k+\mathrm{\hslash }{\delta }_{ki}{p}_j\to x_i{p}_j{p}_k+\mathrm{\hslash }{\delta }_{ji}{p}_k+\mathrm{\hslash }{\delta }_{ki}{p}_j$, while reducing the right pair gives $p_k{p}_j{x}_i\to p_k{x}_i{p}_j+\mathrm{\hslash }{\delta }_{ji}{p}_k\to x_i{p}_k{p}_j+\mathrm{\hslash }{\delta }_{ki}{p}_j+\mathrm{\hslash }{\delta }_{ji}{p}_k\to x_i{p}_j{p}_k+\mathrm{\hslash }{\delta }_{ki}{p}_j+\mathrm{\hslash }{\delta }_{ji}{p}_k$; these final expressions agree. For $p_k{x}_j{x}_i$, reducing the left pair gives $p_k{x}_j{x}_i\to x_j{p}_k{x}_i+\mathrm{\hslash }{\delta }_{kj}{x}_i\to x_j{x}_i{p}_k+\mathrm{\hslash }{\delta }_{ki}{x}_j+\mathrm{\hslash }{\delta }_{kj}{x}_i\to x_i{x}_j{p}_k+\mathrm{\hslash }{\delta }_{ki}{x}_j+\mathrm{\hslash }{\delta }_{kj}{x}_i$, while reducing the right pair gives $p_k{x}_j{x}_i\to p_k{x}_i{x}_j\to x_i{p}_k{x}_j+\mathrm{\hslash }{\delta }_{ki}{x}_j\to x_i{x}_j{p}_k+\mathrm{\hslash }{\delta }_{kj}{x}_i+\mathrm{\hslash }{\delta }_{ki}{x}_j$; these final expressions also agree. For a strictly decreasing same-type triple, both routes $x_k{x}_j{x}_i\to x_j{x}_k{x}_i\to x_j{x}_i{x}_k\to x_i{x}_j{x}_k$ and $x_k{x}_j{x}_i\to x_k{x}_i{x}_j\to x_i{x}_k{x}_j\to x_i{x}_j{x}_k$ reach the same sorted word; the same calculation holds for $p$'s. Repeated indices remove the corresponding swap, and disjoint reductions commute by distributivity. Thus every local ambiguity joins.

By step 2.1, the central quotient $Q_{\mathrm{\hslash }}$ has the normal basis just described. In $Q_{\mathrm{\hslash }}$, the images of $X_i,P_i$ satisfy $[P_i,X_j]=\mathrm{\hslash }{\delta }_{ij}1$, and the same-type commutators vanish. Send the basis generators $x_i,p_i$ of $E_n$ to these images; [F9] extends this linear assignment to a unital algebra map $T(E_n)\to Q_{\mathrm{\hslash }}$. It kills the relators in [F5]. Its kernel is a two-sided ideal, so it contains their generated ideal by [F7], and [F10] factors it uniquely through the quotient to $\psi :W_n(k;\mathrm{\hslash })\to Q_{\mathrm{\hslash }}$. This factor map is $k$-linear because the quotient map is surjective.

Induct on the decreasing measure from step 1.3. If two reduction sequences first choose different redexes, step 2.2 joins those first reductions; every word in the joining expressions has smaller measure, so the induction hypothesis gives it a unique final reduction. Thus every word has one normal form. Because each defining relation is a permitted reduction in any left-right word context, confluence makes the normal-form map kill every such context and hence the whole ideal $J_{n,\mathrm{\hslash }}$ from [F5]. Each reduction also changes a word by an element of $J_{n,\mathrm{\hslash }}$, so every class is represented by its normal form. Therefore the quotient is spanned by normal words, and any linear relation among normal words in the quotient is fixed by the normal-form map while also lying in its kernel, so all its coefficients are zero.

The composites $\varphi \psi$ and $\psi \varphi$ fix every generator and the unit of their respective algebras. Those generators generate each quotient, so the composites are identity maps and $\varphi ,\psi$ are inverse isomorphisms. The basis from step 3.2 therefore transfers to $W_n(k;\mathrm{\hslash })$.

If $\mathrm{\hslash }\mathrm{\ne }0$, the assignments $x_i\mapsto x_i$, $p_i\mapsto \mathrm{\hslash }{p}_i$ preserve the relations from $W_n(k;\mathrm{\hslash })$ to $A_n(k)$; the inverse rescales $p_i$ by ${\mathrm{\hslash }}^{-1}$. If $\mathrm{\hslash }=0$, [F5] makes every pair of generators commute. Thus the parameter branches and normal basis stated in the theorem hold.