Weyl Algebra - Properties of The Weyl Algebra

Properties of The Weyl Algebra

In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain. It has global dimension n, in contrast to the ring it deforms, Sym(V), which has global dimension 2n.

It has no finite dimensional representations; although this follows from simplicity, it can be more directly shown by taking the trace σ(X) and σ(Y) for some finite dimensional representation σ (where = 1).

Since the trace of a commutator is zero, and the trace of the identity is the dimension of the matrix, the representation must be zero dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations. To any f.g. A_n-module M, there is a corresponding subvariety Char(M) of V × V* called the 'characteristic variety' whose size roughly corresponds to the size of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,

An even stronger statement is Gabber's theorem, which states that Char(M) is a co-isotropic subvariety of V × V* for the natural symplectic form.