# isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed

## Synopsis

• Usage:
isTorusFixed I
• Inputs:
• I, an ideal, in a WeylAlgebra
• Outputs:
• , true if I is torus-fixed, false if not

## Description

There is a natural action of the n-dimensional algebraic torus on $D$ where $t \in (\mathbb{C}^*)^n$ acts on $\partial_i$ as $t_i\partial_i$ and on $x_i$ as $t_i^{-1}x_i$. The function isTorusFixed verifies whether a D-ideal is invariant under this action.

See [SST], just before Lemma 2.3.1.

 i1 : W = makeWA(QQ[x_1,x_2]) o1 = W o1 : PolynomialRing, 2 differential variables i2 : b = 2 o2 = 2 i3 : I = ideal(x_1*dx_1*(x_1*dx_1+b), x_1*dx_1*(x_2*dx_2+b), x_2*dx_2*(x_1*dx_1+b), x_2*dx_2*(x_2*dx_2+b)) 2 2 2 2 o3 = ideal (x dx + 3x dx , x x dx dx + 2x dx , x x dx dx + 2x dx , x dx + 1 1 1 1 1 2 1 2 1 1 1 2 1 2 2 2 2 2 ------------------------------------------------------------------------ 3x dx ) 2 2 o3 : Ideal of W i4 : isTorusFixed I o4 = true

## Ways to use isTorusFixed :

• "isTorusFixed(Ideal)"

## For the programmer

The object isTorusFixed is .