Weyl algebra

Abstract definition

Let $F$ be a field and $V$ be an $F$ -vector space with basis $\{P_i\}_{i\in I}\cup\{Q_i\}_{i\in I}$ , where $I$ is some non-empty index set. Let $T$ be the tensor algebra of $V$ and let $J$ be the ideal in $T$ generated by the set $\{P_i\otimes Q_j-Q_j\otimes P_i-\delta_{ij}\}_{i,j\in I}$ where $\delta$ is the Kronecker delta symbol. Then the quotient $T/J$ is the $|I|$ -th Weyl algebra.

A more concrete definition

If the field $F$ has characteristic zero we have the following more concrete definition. Let $R:=F[\{X_i\}_{i\in I}]$ be the polynomial ring over $F$ in indeterminates $X_i$ labeled by $I$ . For any $i\in I$ , let $\partial_i$ denote the partial differential operator with respect to $X_i$ . Then the $|I|$ -th Weyl algebra is the set $W$ of all differential operators of the form \begin{equation*} D=\Sum_{|\alpha|\leq n}f_\alpha\partial^\alpha \end{equation*}where the summation variable $\alpha$ is a multi-index with $|I|$ entries, $n$ is the degree of $D$ , and $f_\alpha\in R$ . The algebra structure is defined by the usual operator multiplication, where the coefficients $f_\alpha\in R$ are identified with the operators of left multiplication with them for conciseness of notation. Since the derivative of a polynomial is again a polynomial, it is clear that $W$ is closed under that multiplication.

The equivalence of these definitions can be seen by replacing the generators $Q_i$ with left multiplication by the indeterminates $X_i$ , the generators $P_i$ with the partial differential operator $\partial_i$ , and the tensor product with operator multiplication, and observing that $\partial_iX_j-X_j\partial_i=\delta_{ij}$ . If, however, the characteristic $p$ of $F$ is positive, the resulting homomorphism to $W$ is not injective, since for example the expressions $\partial_i^p$ and $X_i^n$ commute, while $P_i^{\otimes p}$ and $Q_i^{\otimes n}$ do not.