# lieAlgebra -- make a free Lie algebra

## Synopsis

• Usage:
L=lieAlgebra(liegens)
• Inputs:
• liegens, a list, of elements of class Symbol or IndexedVariable
• Optional inputs:
• Field => ..., default value QQ, optional argument for lieAlgebra
• LastWeightHomological => ..., default value false, optional argument for lieAlgebra
• Signs => ..., default value 0, optional argument for lieAlgebra
• Weights => ..., default value 1, optional argument for lieAlgebra
• Outputs:

## 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