Weyl algebras

Weyl algebras

A Weyl algebra is the non-commutative algebra of algebraic differential operators on a polynomial ring. To each variable x corresponds the operator dx that differentiates with respect to that variable. The evident commutation relation takes the form dx*x == x*dx + 1.

We can give any names we like to the variables in a Weyl algebra, provided we specify the correspondence between the variables and the derivatives, with the WeylAlgebra option, as follows.

i1 : R = QQ[x,y,dx,dy,t,WeylAlgebra => {x=>dx, y=>dy}]



o1 = R



o1 : PolynomialRing

i2 : dx*dy*x*y



o2 = x*y*dx*dy + x*dx + y*dy + 1



o2 : R

i3 : dx*x^5



      5       4

o3 = x dx + 5x



o3 : R

All modules over Weyl algebras are, in Macaulay 2, right modules. This means that multiplication of matrices is from the opposite side:

i4 : dx*x



o4 = x*dx + 1



o4 : R

i5 : matrix{{dx}} * matrix{{x}}



o5 = | xdx |



             1       1

o5 : Matrix R  <--- R

All Groebner basis and related computations work over this ring. For an extensive collection of D-module routines (A D-module is a module over a Weyl algebra), see D-modules.