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# basis(List,VectorSpace) -- compute a basis

- Function: basis
- Usage:
    `b=basis(s,V)`
- Inputs:
  - s, a list, the weight
  - V, an instance of the type VectorSpace, an instance of type VectorSpace
- Optional inputs:
  - Degree => ..., default value null,
  - Limit => ..., default value infinity,
  - SourceRing => ..., default value null,
  - Strategy => ..., default value null,
  - Truncate => ..., default value false,
  - Variables => ..., default value null,
- Outputs:
  - b, a list, a basis for the part of $V$ of multi-degree $s$.

## Description

The optional inputs given above are not relevant for Lie algebras.

    i1 : L = lieAlgebra({a,b,c},Weights=>{{1,0},{2,1},{3,2}},
               Signs=>{1,1,1},LastWeightHomological=>true)

    o1 = L

    o1 : LieAlgebra

    i2 : D= differentialLieAlgebra({0_L,a a,a b})

    o2 = D

    o2 : LieAlgebra

    i3 : J=lieIdeal({b b + 4 a c})

    o3 = J

    o3 : FGLieIdeal

    i4 : Q=D/J

    o4 = Q

    o4 : LieAlgebra

    i5 : H=lieHomology Q

    o5 = H

    o5 : VectorSpace

    i6 : basis({5,3},H)

    o6 = {(b c)}

    o6 : List

    i7 : Z=cycles Q

    o7 = Z

    o7 : LieSubAlgebra

    i8 : basis({5,3},Z)

    o8 = {(b c)}

    o8 : List

## See also

- dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- basis(ZZ,LieAlgebra) -- compute a basis
- basis(ZZ,ZZ,LieAlgebra) -- compute a basis
- basis(List,LieAlgebra) -- compute a basis
- basis(ZZ,ExtAlgebra) -- compute a basis
- basis(ZZ,ZZ,ExtAlgebra) -- compute a basis
- basis(List,ExtAlgebra) -- compute a basis
- basis(ZZ,VectorSpace) -- compute a basis
- basis(ZZ,ZZ,VectorSpace) -- compute a basis

## Ways to use this method:

- basis(List,VectorSpace) -- compute a basis

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc2.m2:376:0.