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D.5.3.1 tangentGens
Procedure from library orbitparam.lib (see orbitparam_lib).
- Usage:
- tangentGens(L,v); L list, v matrix.
- Assume:
- L is a list of strictly upper triangular n x n matrices of same size.
The vector space <L> genererated by the elements of L should be closed
under the Lie bracket.
v is matrix of constants of size n x 1.
- Return:
- list, with four entries
- first entry is the dimension of the orbit of under the action of exp(<L>)
- second entry is a list generators of the tangent space of the orbit of v at v
under the action of exp(<L>). If the characteristic p of the ground field is positive, then n has to be smaller than p. The generators are elements of <L>.
- third entry is the list of matrices with the coefficient to obtain the generators
as a linear combination of the elements of L
- fourth entry is list of integers with entries in v which can be made zero by the
action of exp(<L>)
- Theory:
- We apply the theorem of Chevalley-Rosenlicht.
Example:
| LIB "orbitparam.lib";
ring R = 0,(x),dp;
matrix L1[3][3] = 0,1,0, 0,0,0, 0,0,0;
matrix L2[3][3] = 0,0,1, 0,0,0, 0,0,0;
matrix L3[3][3] = 0,1,1, 0,0,1, 0,0,0;
list L = L1,L2,L3;
matrix v[3][1] = 1,2,3;
tangentGens(L,v);
==> [1]:
==> 2
==> [2]:
==> [1]:
==> _[1,1]=0
==> _[1,2]=1/3
==> _[1,3]=1/3
==> _[2,1]=0
==> _[2,2]=0
==> _[2,3]=1/3
==> _[3,1]=0
==> _[3,2]=0
==> _[3,3]=0
==> [2]:
==> _[1,1]=0
==> _[1,2]=1/2
==> _[1,3]=0
==> _[2,1]=0
==> _[2,2]=0
==> _[2,3]=0
==> _[3,1]=0
==> _[3,2]=0
==> _[3,3]=0
==> [3]:
==> [1]:
==> _[1,1]=0
==> _[2,1]=0
==> _[3,1]=1/3
==> [2]:
==> _[1,1]=1/2
==> _[2,1]=0
==> _[3,1]=0
==> [4]:
==> [1]:
==> 2
==> [2]:
==> 1
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