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. If a symbol $a$ has been used as name for some output, then you must write a = symbol a to be able to use the symbol as a generator instead. Relations are introduced by the operator /, see LieAlgebra / List. It is also possible to define a Lie algebra modulo an ideal. See LieAlgebra / LieIdeal. A differential Lie algebra is defined by giving the value of the differential on the generators, see differentialLieAlgebra. If relations are introduced as a list, then the program adds relations to make the ideal of relations invariant under the differential. These non-normalized relations are obtained using ideal(LieAlgebra) and can also be seen using describe(LieAlgebra), see L2 below. The zero Lie algebra (over QQ) is defined as lieAlgebra\{\}.
i1 : F1 = lieAlgebra{a,b}
o1 = F1
o1 : LieAlgebra
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i2 : L1=F1/{a a b - b b a, a a a a b}
o2 = L1
o2 : LieAlgebra
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i3 : dims(1,6,L1)
o3 = {2, 1, 1, 1, 1, 0}
o3 : List
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i4 : describe L1
o4 = generators => {a, b}
Weights => {{1, 0}, {1, 0}}
Signs => {0, 0}
ideal => { - (a b a) - (b b a), - (a a a b a)}
ambient => F1
diff => {}
Field => QQ
computedDegree => 6
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i5 : F2 = lieAlgebra({a,b,c},Weights=>{{1,0},{1,0},{2,1}},
Signs=>{1,1,1},LastWeightHomological=>true)
o5 = F2
o5 : LieAlgebra
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i6 : D2 = differentialLieAlgebra{0_F2,0_F2,a a + b b}
o6 = D2
o6 : LieAlgebra
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i7 : L2=D2/{a b,a c}
o7 = L2
o7 : LieAlgebra
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i8 : describe L2
o8 = generators => {a, b, c}
Weights => {{1, 0}, {1, 0}, {2, 1}}
Signs => {1, 1, 1}
ideal => {(b a), (a c), - (a a a) - (a b b)}
ambient => F2
diff => {0, 0, (a a) + (b b)}
Field => QQ
computedDegree => 0
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i9 : dims(5,L2)
o9 = | 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
o9 : Matrix ZZ <--- ZZ
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i10 : describe lieAlgebra{}
o10 = generators => {}
Weights => {}
Signs => {}
ideal => {}
ambient => LieAlgebra{...10...}
diff => {}
Field => QQ
computedDegree => 0
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The object lieAlgebra is a method function with options.