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GradedLieAlgebras :: idealBasisLie

idealBasisLie -- computes a basis of a Lie ideal in a given degree or multi-degree

Synopsis

Description

The ideal is the least subspace containing the generators of the ideal and which is closed under Lie multiplication by the Lie generators and closed under application of the differential. A basis is given in the specified degree or multi-degree.

i1 : L = lieAlgebra({a,b,c},genSigns=>{1,0,1},genWeights=>{{1,0},{1,0},{1,2}})/{c a}

o1 = L

o1 : LieAlgebra
i2 : dimsLie 5

o2 = {3, 4, 5, 12, 24}

o2 : List
i3 : d4=defLie (mb_{4,5}+2*mb_{4,6})

o3 = 2 (b c b a) + (c b b a)

o3 : L
i4 : ib=idealBasisLie(5,{a a,d4})

o4 = {(c c b a a), 2 (c b c b a) + (c c b b a), (c b b a a), (c a b b a), (c
     ------------------------------------------------------------------------
     a b a a), (b c b a a), 2 (b b c b a) + (b c b b a), (b b b a a), (b a b
     ------------------------------------------------------------------------
     a a), (a b b a a), (a a b a a)}

o4 : List
i5 : length oo

o5 = 11
i6 : indexFormLie ib

o6 = {mb      , mb        + 2mb       , mb      , mb      , mb      , mb   
        {5, 7}    {5, 13}      {5, 15}    {5, 5}    {5, 9}    {5, 2}    {5,
     ------------------------------------------------------------------------
       , mb        + 2mb       , mb      , mb      , mb      , mb      }
     6}    {5, 12}      {5, 14}    {5, 4}    {5, 1}    {5, 3}    {5, 0}

o6 : List
i7 : idealBasisLie({5,4,0},{a a,d4})

o7 = {(c c b a a), 2 (c b c b a) + (c c b b a)}

o7 : List
i8 : indexFormLie oo

o8 = {mb      , mb        + 2mb       }
        {5, 7}    {5, 13}      {5, 15}

o8 : List
i9 : F = lieAlgebra({a,b},genWeights=>{{1,0},{2,1}},genSigns=>{1,1},diffl=>true)

o9 = F

o9 : LieAlgebra
i10 : Q = diffLieAlgebra{F.zz,a a}

o10 = Q

o10 : LieAlgebra
i11 : idealBasisLie(4,{b b})
warning: new generators for the ideal have been added to get invariance of the differential

o11 = {(a a b), (b b)}

o11 : List

See also

Ways to use idealBasisLie :