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GradedLieAlgebras :: cyclesBasisLie

cyclesBasisLie -- a basis for the cycles of a Lie algebra

Synopsis

Description

i1 : L=lieAlgebra({a,b,c,r3,r4},genWeights =>
          {{1,0},{1,0},{2,0},{3,1},{4,1}},
          diffl=>true,
          genSigns=>{1,1,0,0,1})

o1 = L

o1 : LieAlgebra
i2 : L=diffLieAlgebra{L.zz,L.zz,L.zz,b c - a c,a a c - 2 b b b a}

o2 = L

o2 : LieAlgebra
i3 : cyclesTableLie 7

o3 = | 2 4 4 7 16 34 68 |
     | 0 0 0 0 0  0  3  |
     | 0 0 0 0 0  0  0  |
     | 0 0 0 0 0  0  0  |
     | 0 0 0 0 0  0  0  |
     | 0 0 0 0 0  0  0  |
     | 0 0 0 0 0  0  0  |

              7        7
o3 : Matrix ZZ  <--- ZZ
i4 : ba = cyclesBasisLie(7,1)

o4 = { - (c a a r3) + (c b a r3) + (1/2)(a c a r3) + (1/2)(a c b r3) +
     ------------------------------------------------------------------------
     (1/2)(a a c r3) - (1/2)(a b c r3) - (b c a r3),  - (c a b r3) + (c b b
     ------------------------------------------------------------------------
     r3) + (a c b r3) - (1/2)(b c a r3) - (1/2)(b c b r3) + (1/2)(b a c r3) -
     ------------------------------------------------------------------------
     (1/2)(b b c r3),  - (c a r4) - (c a a r3) + (c b r4) + (a c r4) + (a a c
     ------------------------------------------------------------------------
     r3) + 2 (a b b b r3) - (b c r4) + 2 (b a b b r3) - 2 (b b a b r3) - 2 (b
     ------------------------------------------------------------------------
     b b a r3)}

o4 : List
i5 : indexFormLie ba

      1            1            1                        1                 
o5 = {-mb        - -mb        + -mb        - mb        + -mb        - mb   
      2  {7, 82}   2  {7, 84}   2  {7, 92}     {7, 93}   2  {7, 94}     {7,
     ------------------------------------------------------------------------
                        1            1            1                       
          + mb        , -mb        - -mb        - -mb        + mb        -
     100}     {7, 101}  2  {7, 83}   2  {7, 85}   2  {7, 93}     {7, 94}  
     ------------------------------------------------------------------------
     1
     -mb        - mb         + mb        , - 2mb        - 2mb        + 2mb   
     2  {7, 95}     {7, 102}     {7, 103}       {7, 57}      {7, 61}      {7,
     ------------------------------------------------------------------------
         + 2mb        + mb        + mb        - mb        - mb         -
     63}      {7, 64}     {7, 82}     {7, 96}     {7, 97}     {7, 100}  
     ------------------------------------------------------------------------
     mb         + mb        }
       {7, 104}     {7, 105}

o5 : List

See also

Ways to use cyclesBasisLie :