Dtransposition -- standard transposition for Weyl algebra

Synopsis

Usage:

Dtransposition A
Inputs:
- A, a matrix, a matrix (between free modules), function, ideal, or chain complex of free modules over the Weyl algebra
Outputs:
- a matrix, the standard transpose of A as a matrix, function, ideal, or chain complex over the Weyl algebra

Description

The standard transposition is the involution of the Weyl algebra which sends x^ad^b to (-d)^bx^a. It provides the equivalence in the Weyl algebra between left and right D-modules.

i1 : makeWeylAlgebra(QQ[x,y])

o1 = QQ[x..y, dx, dy]

o1 : PolynomialRing, 2 differential variable(s)

i2 : L = x^2*dy + y*dy^2 + 3*dx^5*dy

        5      2         2
o2 = 3dx dy + x dy + y*dy

o2 : QQ[x..y, dx, dy]

i3 : Dtransposition L

        5      2         2
o3 = 3dx dy - x dy + y*dy  + 2dy

o3 : QQ[x..y, dx, dy]

Caveat

The standard transposition of a left ideal should be a right ideal, however M2 currently doesn't support right modules. Thus the output is left ideal generated by the transposition of the previous generators.

Ways to use Dtransposition :

Dtransposition(ChainComplex)
Dtransposition(Ideal)
Dtransposition(Matrix)
Dtransposition(RingElement)

For the programmer