Polynomial Weyl algebra and its central parameter

Let $k$ be a field, $n\ge 1$, and $\mathrm{\hslash }\in k$. Let $E_n$ be the $2n$-dimensional $k$-vector space with basis $x_1,\dots ,x_n,p_1,\dots ,p_n$, and form its tensor algebra $T(E_n)$ (Tensor algebra of a vector space). Let $J_{n,\mathrm{\hslash }}$ be the two-sided ideal generated by

$x_i{x}_j-x_j{x}_i,p_i{p}_j-p_j{p}_i,p_i{x}_j-x_j{p}_i-\mathrm{\hslash }{\delta }_{ij}1(1\le i,j\le n)\mathrm{.}$

The polynomial Weyl algebra with central parameter $\mathrm{\hslash }$ is

$W_n(k;\mathrm{\hslash }):=T(E_n)\mathrm{/}{J}_{n,\mathrm{\hslash }},$

where the quotient is the ring quotient from The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$ and $J_{n,\mathrm{\hslash }}$ is the generated ideal from The ideal generated by a subset and principal ideals. The standard $n$th polynomial Weyl algebra is $A_n(k):=W_n(k;1)$. The phrase “polynomial Weyl algebra” here means this algebraic associative quotient; no topology, completion, or unitary exponential generators are part of the definition.

The exact presentation, normal-ordered basis, and equivalence with the central quotient of $U({\mathfrak{h}}_n(k))$ are established in Polynomial Weyl algebra presentation and normal-ordered basis ↗.