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 which 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, which we do 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

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