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GradedLieAlgebras :: diffLieAlgebra

diffLieAlgebra -- A differential Lie algebra

Synopsis

Description

The input should be a list of LieElement in the current Lie algebra F and this list consists of the differentials of the generators, F.zz is used for the zero element. The option diffl for F must have the value true. The current Lie algebra F must be free when diffLieAlgebra is used. The program adds relations to the Lie algebra to get the square of the differential to be zero and a warning is printed if this is done. See L2 below. It is also possible to use the key F.genDiffs, but then there is no checking done.

i1 : F1=lieAlgebra({a,b,c,r3,r4,r42},
        genWeights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
        genSigns=>{0,0,0,1,1,0},diffl=>true)

o1 = F1

o1 : LieAlgebra
i2 : L1=diffLieAlgebra{F1.zz,F1.zz,F1.zz,a c,a a c,r4 - a r3}/{b c - a c,a b,b r4 - a r4}

o2 = L1

o2 : LieAlgebra
i3 : peekLie L1

o3 = gensLie => {a, b, c, r3, r4, r42}
     genWeights => {{1, 0}, {1, 0}, {2, 0}, {3, 1}, {4, 1}, {4, 2}}
     genSigns => {0, 0, 0, 1, 1, 0}
     relsLie => { - (a c) + (b c),  - (b a),  - (a r4) + (b r4)}
     genDiffs => {0, 0, 0, (b c), (b b c), r4 - (a r3)}
     field => QQ
     diffl => true
     compdeg => 5
i4 : F2=lieAlgebra({a,b,c2,c3,c5},genSigns=>{0,0,1,0,1},
        genWeights=>{{1,0},{1,0},{2,1},{3,2},{5,3}},diffl=>true)

o4 = F2

o4 : LieAlgebra
i5 : L2=diffLieAlgebra{F2.zz,F2.zz,a b,a c2,a b c3}
warning: relations have been added for the square of the differential to be zero

o5 = L2

o5 : LieAlgebra
i6 : peekLie L2

o6 = gensLie => {a, b, c2, c3, c5}
     genWeights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}, {5, 3}}
     genSigns => {0, 0, 1, 0, 1}
     relsLie => { - (a b a), (a b a c2)}
     genDiffs => {0, 0,  - (b a), (a c2), (a b c3)}
     field => QQ
     diffl => true
     compdeg => 5

See also