A review

Let us first recall the definition. (See Part I for details.)

DEFINITION 1.1   Let $k$ be a field. Let $n$ be a positive integer. A Weyl algebra $A_n(k)$ over a commutative ring $k$ is an algebra over $k$ generated by $2 n$ elements $\{\gamma_1,\gamma_2,\dots,\gamma_{2 n}\}$ with the ``canonical commutation relations''

$$[\gamma_i, \gamma_j](=\gamma_i \gamma_j -\gamma_j \gamma_i) =h_{i j} \qquad (1\leq i,j \leq 2 n).$$ (CCR)

Where $h$ is a non-degenerate anti-Hermitian $2 n \times 2 n$ matrix of the following form.

$$(h_{i j})= \begin{pmatrix} 0 & -1_n \\ 1_n & 0 \end{pmatrix}.$$

Throughout this section, the letter $h$ will always represent the matrix above and the letter $\bar{h}$ will always represent the inverse matrix of $h$ .