# 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

 $D=\sum\limits_{|\alpha|\leq n}f_{\alpha}\partial^{\alpha}$

where the summation variable $\alpha$ is a multi-index with $|I|$ entries, $n$ is the degree of $D$, and $f_{\alpha}\in R$. The 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 $\partial_{i}X_{j}-X_{j}\partial_{i}=\delta_{ij}$. 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.

Title Weyl algebra WeylAlgebra 2013-03-22 15:27:19 2013-03-22 15:27:19 GrafZahl (9234) GrafZahl (9234) 5 GrafZahl (9234) Definition msc 16S36 msc 16S32 HeisenbergAlgebra UniversalEnvelopingAlgebra