Lie commutators and nonabelian Lie tensors

Functions on this page are joint work with Hamid Mohammadzadeh, and implemented by him.
LieCoveringHomomorphism
LieCoveringHomomorphism(L)

Inputs a finite dimensional Lie algebra L over a field, and returns a surjective Lie homomorphism phi : C\rightarrow L where: the kernel of phi lies in both the centre of C and the derived subalgebra of C, the kernel of phi is a vector space of rank equal to the rank of the second Chevalley-Eilenberg homology of L.
LeibnizQuasiCoveringHomomorphism
LeibnizQuasiCoveringHomomorphism(L)

Inputs a finite dimensional Lie algebra L over a field, and returns a surjective homomorphism phi : C\rightarrow L of Leibniz algebras where: the kernel of phi lies in both the centre of C and the derived subalgebra of C, the kernel of phi is a vector space of rank equal to the rank of the kernel J of the homomorphism L \otimes L \rightarrow L from the tensor square to L. (We note that, in general, J is NOT equal to the second Leibniz homology of L.)
LieEpiCentre
LieEpiCentre(L)

Inputs a finite dimensional Lie algebra L over a field, and returns an ideal Z^\ast(L) of the centre of L. The ideal Z^\ast(L) is trivial if and only if L is isomorphic to a quotient L=E/Z(E) of some Lie algebra E by the centre of E.
LieExteriorSquare
LieExteriorSquare(L)

Inputs a finite dimensional Lie algebra L over a field. It returns a record E with the following components. E.homomorphism is a Lie homomorphism µ : (L \wedge L) \longrightarrow L from the nonabelian exterior square (L \wedge L) to L. The kernel of µ is the Lie multiplier. E.pairing(x,y) is a function which inputs elements x, y in L and returns (x \wedge y) in the exterior square (L \wedge L) .

LieTensorSquare
LieTensorSquare(L)

Inputs a finite dimensional Lie algebra L over a field and returns a record T with the following components. T.homomorphism is a Lie homomorphism µ : (L \otimes L) \longrightarrow L from the nonabelian tensor square of L to L. T.pairing(x,y) is a function which inputs two elements x, y in L and returns the tensor (x \otimes y) in the tensor square (L \otimes L) .
LieTensorCentre
LieTensorCentre(L)

Inputs a finite dimensional Lie algebra L over a field and returns the largest ideal N such that the induced homomorphism of nonabelian tensor squares (L \otimes L) \longrightarrow (L/N \otimes L/N) is an isomorphism.