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-emptyindex 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$\mathrm{|}I\mathrm{|}$-th Weyl algebra.

A more concrete definition

If the field $F$ has characteristic zero we have the following moreconcrete definition. Let $R:=F[{\{X_i\}}_{i\in I}]$ be the polynomialring over $F$ in indeterminates $X_i$ labeled by $I$. For any $i\in I$, let ${\partial }_i$ denote the partial differential operator withrespect to $X_i$. Then the $|I|$-th Weyl algebra is the set $W$ of alldifferential operators of the form

$$D=\sum _{|\alpha |\le n}{f}_{\alpha }{\partial }^{\alpha }$$

where the summation variable $\alpha$ is a multi-index with $|I|$entries, $n$ is the degree of $D$, and $f_{\alpha }\in R$. The algebrastructure is defined by the usual operator multiplication, where thecoefficients $f_{\alpha }\in R$ are identified with the operators of leftmultiplication with them for conciseness of notation. Since thederivative 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 thegenerators $Q_i$ 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, andobserving that ${\partial }_i{X}_j-X_j{\partial }_i={\delta }_{ij}$. If, however,the characteristic $p$ of $F$ is positive, the resulting homomorphismto $W$ 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.