  
  [1X15 [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
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLieCoveringHomomorphism(L)[110X[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
  [33X[0;0Ythe kernel of [22Xphi[122X lies in both the centre of [22XC[122X and the derived subalgebra of
  [22XC[122X,[133X
  [33X[0;0Ythe  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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLeibnizQuasiCoveringHomomorphism(L)[110X[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
  [33X[0;0Ythe kernel of [22Xphi[122X lies in both the centre of [22XC[122X and the derived subalgebra of
  [22XC[122X,[133X
  [33X[0;0Ythe  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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLieEpiCentre(L)[110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLieExteriorSquare(L) [110X[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
  [33X[0;0Y[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
  [33X[0;0Y[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLieTensorSquare(L) [110X[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
  [33X[0;0Y[22XT.homomorphism[122X  is  a  Lie  homomorphism [22Xµ : (L ⊗ L) ⟶ L[122X from the nonabelian
  tensor square of [22XL[122X to [22XL[122X.[133X
  [33X[0;0Y[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLieTensorCentre(L) [110X[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
  
