# lieAlgebra(...,Weights=>...) -- optional argument for lieAlgebra

## Synopsis

• Usage:
L = lieAlgebra(gen,Weights => w)
• Inputs:
• gen, a list,
• w, a list, of lists of weights or a list of positive integers
• w, an integer, 1, the same weight (1,0) is assigned to each generator
• Outputs:

## Description

This is an option to tell lieAlgebra to assign the given weights to the generators. A weight is a list of integers of a length that is the same for all generators, see degreeLength(LieAlgebra). The first component (also just called the degree) is positive, see firstDegree(LieElement). If the option LastWeightHomological is true, then the last component in a weight is the homological degree, which is non-negative and less than the first degree. If the option LastWeightHomological is false, then the program adds a last component 0 to the existing degrees. When the option is given as a list of integers $n1,n2,\ldots$, which is not possible when the option LastWeightHomological is true, then the program defines the weights for the generators to be $\{n1,0\},\ \{n2,0\},\ldots$. The default value is 1, which has the effect that all generators have weight $\{1,0\}$.

 i1 : describe lieAlgebra({a,b},Weights=>{{1,2},{2,3}}) o1 = generators => {a, b} Weights => {{1, 2, 0}, {2, 3, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i2 : describe lieAlgebra({a,b},Weights=>{{2,1},{3,2}}, LastWeightHomological=>true) o2 = generators => {a, b} Weights => {{2, 1}, {3, 2}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i3 : describe lieAlgebra{a,b} o3 = generators => {a, b} Weights => {{1, 0}, {1, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i4 : describe lieAlgebra({a,b},Weights=>{{1,0},{1,0}}) o4 = generators => {a, b} Weights => {{1, 0, 0}, {1, 0, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i5 : describe lieAlgebra({a,b},Weights=>{{1,0},{1,0}}, LastWeightHomological=>true) o5 = generators => {a, b} Weights => {{1, 0}, {1, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i6 : describe lieAlgebra({a,b},Weights=>{1,2}) o6 = generators => {a, b} Weights => {{1, 0}, {2, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0

## Further information

• Default value: 1
• Function: lieAlgebra -- make a free Lie algebra
• Option key: Weights -- assigning weights to the variables