Singular Manual: tangentGens

| #### D.5.10.1 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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