Editing Weyl algebra (section)

== Generalizations ==
The [[Ideal (ring theory)|ideals]] and automorphisms of <math>A_1</math> have been well-studied.{{sfn|Berest|Wilson|2000|pp=127–147}}{{sfn|Cannings|Holland|1994|pp=116–141}} The [[moduli space]] for its right ideal is known.{{sfn|Lebruyn|1995|pp=32–48}} However, the case for <math>A_n</math> is considerably harder and is related to the [[Jacobian conjecture]].{{sfn|Coutinho|1995|loc=section 4.4}}

For more details about this quantization in the case ''n'' = 1 (and an extension using the [[Fourier transform]] to a class of integrable functions larger than the polynomial functions), see [[Wigner–Weyl transform]].

Weyl algebras and Clifford algebras admit a further structure of a [[*-algebra]], and can be unified as even and odd terms of a [[superalgebra]], as discussed in [[CCR and CAR algebras]].

=== Affine varieties ===
Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring
: <math>R = \frac{\mathbb{C}[x_1,\ldots,x_n]}{I}.</math>
Then a differential operator is defined as a composition of <math>\mathbb{C}</math>-linear derivations of <math>R</math>. This can be described explicitly as the quotient ring
: <math> \text{Diff}(R) = \frac{\{ D \in A_n\colon D(I) \subseteq I \}}{ I\cdot A_n}.</math>