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lieAlgebra -- make 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. 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
 
i2 : L1=F1/{a a b - b b a, a a a a b}

o2 = L1

o2 : LieAlgebra
 
i3 : dims(1,6,L1)

o3 = {2, 1, 1, 1, 1, 0}

o3 : List
 
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
 
i5 : F2 = lieAlgebra({a,b,c},Weights=>{{1,0},{1,0},{2,1}},
            Signs=>{1,1,1},LastWeightHomological=>true)

o5 = F2

o5 : LieAlgebra
 
i6 : D2 = differentialLieAlgebra{0_F2,0_F2,a a + b b}

o6 = D2

o6 : LieAlgebra
 
i7 : L2=D2/{a b,a c}

o7 = L2

o7 : LieAlgebra
 
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
 
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
 
i10 : describe lieAlgebra{}

o10 = generators => {}
      Weights => {}
      Signs => {}
      ideal => {}
      ambient => LieAlgebra{...10...}
      diff => {}
      Field => QQ
      computedDegree => 0

See also

Ways to use lieAlgebra :

For the programmer

The object lieAlgebra is a method function with options.