# Dtransposition -- standard transposition for Weyl algebra

## Synopsis

• Usage:
Dtransposition A
• Inputs:
• A, , a matrix (between free modules), function, ideal, or chain complex of free modules over the Weyl algebra
• Outputs:
• , 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 xadb to (-d)bxa. 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.

• WeylAlgebra -- specify differential operators in the ring

## Ways to use Dtransposition :

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

## For the programmer

The object Dtransposition is .