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GradedLieAlgebras :: idealBasisLie

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

Synopsis

Description

A basis is given in the specified degree or multidegree. Observe that if the Lie algebra has no differential, then an extra homological degree=0 is added to the given weights of the generators.

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

o1 = L

o1 : LieAlgebra
i2 : computeLie 4

o2 = {3, 4, 5, 12}

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

o3 = {{1, 2}, {[c, b, b, a], [b, c, b, a]}}

o3 : List
i4 : i=idealBasisLie(5,{[a,a],d})

o4 = {[c, c, b, a, a], {{2, 1}, {[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, 1},
     ------------------------------------------------------------------------
     {[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 : idealLie(5,{[a,a],d})

o6 = {0, 1, 1, 4, 11}

o6 : List
i7 : weightLie i

o7 = {{5, 4, 0}, {5, 4, 0}, {5, 2, 0}, {5, 2, 0}, {5, 2, 0}, {5, 2, 0}, {5,
     ------------------------------------------------------------------------
     2, 0}, {5, 0, 0}, {5, 0, 0}, {5, 0, 0}, {5, 0, 0}}

o7 : List
i8 : idealBasisLie({5,4,0},{[a,a],d})

o8 = {[c, c, b, a, a], {{2, 1}, {[c, b, c, b, a], [c, c, b, b, a]}}}

o8 : List
i9 : indexFormLie oo

o9 = {mb      , mb        + 2mb       }
        {5, 7}    {5, 13}      {5, 15}

o9 : List
i10 : indexFormLie i

o10 = {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}

o10 : List

See also

Ways to use idealBasisLie :