  
  [1X16 [33X[0;0YLie commutators and nonabelian Lie tensors[133X[101X
  
  [33X[0;0YFunctions  on  this  page  are  joint  work  with  [12XHamid  Mohammadzadeh[112X, and
  implemented by him.[133X
  
  
  [1X16.1 [33X[0;0Y [133X[101X
  
  [1X16.1-1 LieCoveringHomomorphism[101X
  
  [33X[1;0Y[29X[2XLieCoveringHomomorphism[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  finite  dimensional  Lie  algebra  [22XL[122X  over a field, and returns a
  surjective Lie homomorphism [22Xphi : C→ L[122X where:[133X
  
  [30X    [33X[0;6Ythe  kernel  of  [22Xphi[122X  lies  in  both  the  centre of [22XC[122X and the derived
        subalgebra of [22XC[122X,[133X
  
  [30X    [33X[0;6Ythe  kernel  of [22Xphi[122X is a vector space of rank equal to the rank of the
        second Chevalley-Eilenberg homology of [22XL[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap7.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutLieCovers.html[107X) [133X
  
  [1X16.1-2 LeibnizQuasiCoveringHomomorphism[101X
  
  [33X[1;0Y[29X[2XLeibnizQuasiCoveringHomomorphism[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  finite  dimensional  Lie  algebra  [22XL[122X  over a field, and returns a
  surjective homomorphism [22Xphi : C→ L[122X of Leibniz algebras where:[133X
  
  [30X    [33X[0;6Ythe  kernel  of  [22Xphi[122X  lies  in  both  the  centre of [22XC[122X and the derived
        subalgebra of [22XC[122X,[133X
  
  [30X    [33X[0;6Ythe  kernel  of [22Xphi[122X is a vector space of rank equal to the rank of the
        kernel  [22XJ[122X  of  the homomorphism [22XL ⊗ L → L[122X from the tensor square to [22XL[122X.
        (We  note  that,  in  general,  [22XJ[122X  is  NOT equal to the second Leibniz
        homology of [22XL[122X.)[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X16.1-3 LieEpiCentre[101X
  
  [33X[1;0Y[29X[2XLieEpiCentre[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs a finite dimensional Lie algebra [22XL[122X over a field, and returns an ideal
  [22XZ^∗(L)[122X  of  the centre of [22XL[122X. The ideal [22XZ^∗(L)[122X is trivial if and only if [22XL[122X is
  isomorphic to a quotient [22XL=E/Z(E)[122X of some Lie algebra [22XE[122X by the centre of [22XE[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap7.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutLieCovers.html[107X) [133X
  
  [1X16.1-4 LieExteriorSquare[101X
  
  [33X[1;0Y[29X[2XLieExteriorSquare[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a finite dimensional Lie algebra [22XL[122X over a field. It returns a record
  [22XE[122X with the following components.[133X
  
  [30X    [33X[0;6Y[22XE.homomorphism[122X  is  a  Lie  homomorphism  [22Xµ  :  (L  ∧  L) ⟶ L[122X from the
        nonabelian  exterior  square  [22X(L ∧ L)[122X to [22XL[122X. The kernel of [22Xµ[122X is the Lie
        multiplier.[133X
  
  [30X    [33X[0;6Y[22XE.pairing(x,y)[122X  is  a  function  which  inputs  elements [22Xx, y[122X in [22XL[122X and
        returns [22X(x ∧ y)[122X in the exterior square [22X(L ∧ L)[122X .[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X16.1-5 LieTensorSquare[101X
  
  [33X[1;0Y[29X[2XLieTensorSquare[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a finite dimensional Lie algebra [22XL[122X over a field and returns a record
  [22XT[122X with the following components.[133X
  
  [30X    [33X[0;6Y[22XT.homomorphism[122X  is  a  Lie  homomorphism  [22Xµ  :  (L  ⊗  L) ⟶ L[122X from the
        nonabelian tensor square of [22XL[122X to [22XL[122X.[133X
  
  [30X    [33X[0;6Y[22XT.pairing(x,y)[122X  is  a function which inputs two elements [22Xx, y[122X in [22XL[122X and
        returns the tensor [22X(x ⊗ y)[122X in the tensor square [22X(L ⊗ L)[122X .[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X16.1-6 LieTensorCentre[101X
  
  [33X[1;0Y[29X[2XLieTensorCentre[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  finite  dimensional  Lie  algebra  [22XL[122X over a field and returns the
  largest  ideal  [22XN[122X  such  that  the induced homomorphism of nonabelian tensor
  squares [22X(L ⊗ L) ⟶ (L/N ⊗ L/N)[122X is an isomorphism.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
