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15 Lie commutators and nonabelian Lie tensors

### 15 Lie commutators and nonabelian Lie tensors

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

`LieCoveringHomomorphism(L)`

Inputs a finite dimensional Lie algebra L over a field, and returns a surjective Lie homomorphism phi : C→ 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(L)`

Inputs a finite dimensional Lie algebra L over a field, and returns a surjective homomorphism phi : C→ 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 ⊗ L → L from the tensor square to L. (We note that, in general, J is NOT equal to the second Leibniz homology of L.)

`LieEpiCentre(L)` Inputs a finite dimensional Lie algebra L over a field, and returns an ideal Z^∗(L) of the centre of L. The ideal Z^∗(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(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 ∧ L) ⟶ L from the nonabelian exterior square (L ∧ 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 ∧ y) in the exterior square (L ∧ L) .

`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 ⊗ L) ⟶ 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 ⊗ y) in the tensor square (L ⊗ L) .

`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 ⊗ L) ⟶ (L/N ⊗ L/N) is an isomorphism.

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