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GradedLieAlgebras :: LieAlgebra / List

LieAlgebra / List -- A quotient Lie algebra

Synopsis

Description

The list I is extended by the action on I of the differential in L. The program converts the elements in the list, so that they will have type M=ambient(L) instead. The list of these converted elements may be looked upon by writing Q.relsLie.

i1 : M = lieAlgebra({a,b,c})

o1 = M

o1 : LieAlgebra
i2 : L = M/{a b}

o2 = L

o2 : LieAlgebra
i3 : Q = L/{a c}

o3 = Q

o3 : LieAlgebra
i4 : peekLie Q

o4 = gensLie => {a, b, c}
     genWeights => {{1, 0}, {1, 0}, {1, 0}}
     genSigns => {0, 0, 0}
     relsLie => { - (b a),  - (c a)}
     genDiffs => {0, 0, 0}
     field => QQ
     diffl => false
     compdeg => 1
i5 : ambient Q

o5 = M

o5 : LieAlgebra
i6 : class (Q.relsLie)_0

o6 = M

o6 : LieAlgebra

The Lie algebra Q below is M modulo the ideal in M generated by the elements in Q.relsLie and with differential defined by the induced differential on L.

i7 : L = lieAlgebra({a,b,c2,c3},genWeights=>{{1,0},{1,0},{2,1},{3,2}},
          genSigns=>{1,1,1,1},diffl=>true)

o7 = L

o7 : LieAlgebra
i8 : L = diffLieAlgebra{L.zz,L.zz,a a,b c2}/{a c2}
warning: relations have been added for the square of the differential to be zero

o8 = L

o8 : LieAlgebra
i9 : Q = L/{b c3}
warning: new generators for the ideal have been added to get invariance of the differential

o9 = Q

o9 : LieAlgebra
i10 : peekLie Q

o10 = gensLie => {a, b, c2, c3}
      genWeights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}}
      genSigns => {1, 1, 1, 1}
      relsLie => { - (b a a), (a c2), (b c3),  - (b b c2)}
      genDiffs => {0, 0, (a a), (b c2)}
      field => QQ
      diffl => true
      compdeg => 0
i11 : M = ambient Q

o11 = M

o11 : LieAlgebra
i12 : peekLie M

o12 = gensLie => {a, b, c2, c3}
      genWeights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}}
      genSigns => {1, 1, 1, 1}
      relsLie => {}
      genDiffs => {0, 0, 0, 0}
      field => QQ
      diffl => true
      compdeg => 4

See also