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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''

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

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

$\displaystyle (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$ .

Subsections

2007-12-11