Editing Weyl algebra (section)

== Representation theory ==
{{further|Stone–von Neumann theorem}}

=== Zero characteristic ===
In the case that the ground field {{mvar|F}} has characteristic zero, the ''n''th Weyl algebra is a [[simple ring|simple]] [[Noetherian ring|Noetherian]] [[domain (ring theory)|domain]].{{sfn | Coutinho | 1995 | p=70}}  It has [[global dimension]] ''n'', in contrast to the ring it deforms, Sym(''V''), which has global dimension 2''n''.

It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of ''σ''(''q'') and ''σ''(''Y'') for some finite-dimensional representation ''σ'' (where {{nowrap|1=[''q'',''p''] = 1}}).
: <math> \mathrm{tr}([\sigma(q),\sigma(Y)])=\mathrm{tr}(1)~.</math>
Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations.  To any finitely generated ''A<sub>n</sub>''-module ''M'', there is a corresponding subvariety Char(''M'') of {{nowrap|''V'' × ''V''<sup>∗</sup>}} called the 'characteristic variety'{{what|date=August 2016}} whose size roughly corresponds to the size{{what|date=August 2016}} of ''M'' (a finite-dimensional module would have zero-dimensional characteristic variety).  Then [[Bernstein's inequality (mathematical analysis)|Bernstein's inequality]] states that for ''M'' non-zero,
: <math>\dim(\operatorname{char}(M))\geq n</math>
An even stronger statement is [[Gabber's theorem]], which states that Char(''M'') is a [[Lagrangian submanifold|co-isotropic]] subvariety of {{nowrap|''V'' × ''V''<sup>∗</sup>}} for the natural symplectic form.

=== Positive characteristic ===
The situation is considerably different in the case of a Weyl algebra over a field of [[characteristic (algebra)|characteristic]] {{nowrap|''p'' > 0}}.

In this case, for any element ''D'' of the Weyl algebra, the element ''D<sup>p</sup>'' is central, and so the Weyl algebra has a very large center.  In fact, it is a finitely generated module over its center; even more so, it is an [[Azumaya algebra]] over its center.  As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension ''p''.