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 = ZZ/101[x,dx,t,WeylAlgebra => {x=>dx}]



o1 = R



o1 : PolynomialRing

i2 : dx*x



o2 = x*dx + 1



o2 : R

i3 : dx*x^5



      5       4

o3 = x dx + 5x 



o3 : R