### 16 Lie commutators and nonabelian Lie tensors

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

#### 16.1

##### 16.1-1 LieCoveringHomomorphism
 ‣ LieCoveringHomomorphism( L ) ( function )

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.

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##### 16.1-2 LeibnizQuasiCoveringHomomorphism
 ‣ LeibnizQuasiCoveringHomomorphism( L ) ( function )

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.)

##### 16.1-3 LieEpiCentre
 ‣ LieEpiCentre( L ) ( function )

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.

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##### 16.1-4 LieExteriorSquare
 ‣ LieExteriorSquare( L ) ( function )

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) .

##### 16.1-5 LieTensorSquare
 ‣ LieTensorSquare( L ) ( function )

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) .

##### 16.1-6 LieTensorCentre
 ‣ LieTensorCentre( L ) ( function )

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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