# 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
*$\mathrm{|}I\mathrm{|}$-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

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

where the summation variable $\alpha $ is a multi-index with $|I|$
entries, $n$ is the degree of $D$, and ${f}_{\alpha}\in R$. The 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 |
---|---|

Canonical name | WeylAlgebra |

Date of creation | 2013-03-22 15:27:19 |

Last modified on | 2013-03-22 15:27:19 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 5 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 16S36 |

Classification | msc 16S32 |

Related topic | HeisenbergAlgebra |

Related topic | UniversalEnvelopingAlgebra |