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GradedLieAlgebras :: lieAlgebra

lieAlgebra -- constructing a free Lie algebra

Synopsis

Description

A generator may be of class Symbol or IndexedVariable. The same name for a generator can be used in several Lie algebras and also as name for a variable in a polynomial ring. Relations are introduced by the operator /, see LieAlgebra / List. A differential Lie algebra is defined by giving the value of the differential on the generators, see diffLieAlgebra. Lie multiplication is given by LieElement LieElement. The zero Lie algebra is defined as lieAlgebra{}.

i1 : L1 = lieAlgebra{a,b}/{a a b - b b a, a a a a b}

o1 = L1

o1 : LieAlgebra
i2 : dimsLie 6

o2 = {2, 1, 1, 1, 1, 0}

o2 : List
i3 : peekLie L1

o3 = gensLie => {a, b}
     genWeights => {{1, 0}, {1, 0}}
     genSigns => {0, 0}
     relsLie => { - (a b a) - (b b a),  - (a a a b a)}
     genDiffs => {0, 0}
     field => QQ
     diffl => false
     compdeg => 6
i4 : F2 = lieAlgebra({a,b,c},genWeights=>{{1,0},{1,0},{2,1}},
                    genSigns=>{1,1,1},diffl=>true)

o4 = F2

o4 : LieAlgebra
i5 : L2 = diffLieAlgebra{F2.zz,F2.zz,a a + b b}/{a b,a c}

o5 = L2

o5 : LieAlgebra
i6 : peekLie L2

o6 = gensLie => {a, b, c}
     genWeights => {{1, 0}, {1, 0}, {2, 1}}
     genSigns => {1, 1, 1}
     relsLie => {(b a), (a c)}
     genDiffs => {0, 0, (a a) + (b b)}
     field => QQ
     diffl => true
     compdeg => 3
i7 : dimTableLie 5

o7 = | 2 2 0 0 0 |
     | 0 1 1 1 1 |
     | 0 0 0 1 1 |
     | 0 0 0 0 0 |
     | 0 0 0 0 0 |

              5        5
o7 : Matrix ZZ  <--- ZZ
i8 : peekLie lieAlgebra{}

o8 = gensLie => {}
     genWeights => {}
     genSigns => {}
     relsLie => {}
     genDiffs => {}
     field => QQ
     diffl => false
     compdeg => 0

See also

Ways to use lieAlgebra :