Fourier -- Fourier transform for Weyl algebra

Usage:

Fourier A

FourierInverse A
Inputs:
- A, a matrix, a ring element, an ideal, or a chain complex, over the Weyl algebra
Outputs:
- a matrix, a ring element, an ideal, or a chain complex, the Fourier transform of A as a matrix, function, ideal, or chain complex over the Weyl algebra

Description

The Fourier transform is the automorphism of the Weyl algebra that sends x_i to -D_i and D_i to x_i. In order to compute the Fourier transform of the finitely generated module M, we compute the Fourier transform of the matrix A of which M is the cokernel.

i1 : makeWA(QQ[x,y])

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

o1 : PolynomialRing, 2 differential variable(s)

i2 : A = matrix{{2*x^2+1,y*dy},{9*x*dx, x*y*dx^2}}

o2 = | 2x2+1 ydy    |
     | 9xdx  xydx^2 |

                              2                       2
o2 : Matrix (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])

i3 : Fourier A

o3 = | 2dx^2+1 -ydy-1      |
     | -9xdx-9 x2dxdy+2xdy |

                              2                       2
o3 : Matrix (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])

i4 : L = x^2*dy + y*dy^2 + 3*dx^5*dy

        5      2         2
o4 = 3dx dy + x dy + y*dy

o4 : QQ[x..y, dx, dy]

i5 : Fourier L

       5        2    2
o5 = 3x y + y*dx  - y dy - 2y

o5 : QQ[x..y, dx, dy]

i6 : I = ideal(8*x*y*dy^3+y^5, dx^7+5)

             5          3    7
o6 = ideal (y  + 8x*y*dy , dx  + 5)

o6 : Ideal of QQ[x..y, dx, dy]

i7 : Fourier I

              3          5      2     7
o7 = ideal (8y dx*dy - dy  + 24y dx, x  + 5)

o7 : Ideal of QQ[x..y, dx, dy]

i8 : C = chainComplex{matrix{{x*dx, y^2+dx}},matrix{{dx*dy},{y^2*dy^3}}}

                       1                       2                       1
o8 = (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])
                                                      
     0                       1                       2

o8 : ChainComplex

i9 : FC = Fourier C

                       1                       2                       1
o9 = (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])  <-- (QQ[x..y, dx, dy])
                                                      
     0                       1                       2

o9 : ChainComplex

i10 : FC.dd

                            1                                          2
o10 = 0 : (QQ[x..y, dx, dy])  <--------------------- (QQ[x..y, dx, dy])  : 1
                                 | -xdx-1 dy^2+x |

                            2                                                1
      1 : (QQ[x..y, dx, dy])  <--------------------------- (QQ[x..y, dx, dy])  : 2
                                 {2} | xy              |
                                 {2} | y3dy^2+6y2dy+6y |

o10 : ChainComplexMap

Ways to use Fourier:

Fourier(ChainComplex)
Fourier(Ideal)
Fourier(Matrix)
Fourier(Module)
Fourier(RingElement)

For the programmer

The object Fourier is a method function.

The source of this document is in WeylAlgebras/DOC/basics.m2:156:0.

Fourier -- Fourier transform for Weyl algebra

Description

See also

Ways to use Fourier:

For the programmer