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

- 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:
  - L, an instance of the type LieAlgebra,

## 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

## See also

- lieAlgebra -- make a free Lie algebra
- weight -- get the weight of a homogeneous element
- firstDegree(LieElement) -- get the degree of an element
- Second Lie algebra tutorial

## Functions with optional argument named Weights:

- `betti(...,Weights=>...)` -- see betti(BettiTally) -- view and set the weight vector of a Betti diagram
- lieAlgebra(...,Weights=>...) -- optional argument for lieAlgebra
- `minimalBetti(...,Weights=>...)` -- see minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
- monoid(...,Weights=>...) -- specify weights of the variables
- `newRing(...,Weights=>...)` -- see newRing -- make a copy of a ring, with some features changed
- `regularity(...,Weights=>...)` -- see regularity -- compute the Castelnuovo-Mumford regularity
- `symmetricAlgebra(...,Weights=>...)` -- see symmetricAlgebra -- the symmetric algebra of a module
- `tensor(Monoid,Monoid,Weights=>...)` -- see tensor(Monoid,Monoid) -- tensor product of monoids
- `tensor(Ring,Ring,Weights=>...)` -- see tensor(Monoid,Monoid) -- tensor product of monoids

## Further information

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:699:0.