GAP (HAP commands) - Chapter 16: Lie commutators and nonabelian Lie tensors

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16 Lie commutators and nonabelian Lie tensors
16.1
16.1-1 LieCoveringHomomorphism
16.1-2 LeibnizQuasiCoveringHomomorphism
16.1-3 LieEpiCentre
16.1-4 LieExteriorSquare
16.1-5 LieTensorSquare
16.1-6 LieTensorCentre

### 16 Lie commutators and nonabelian Lie tensors

Functions on this page are joint work with Hamid Mohammadzadeh, 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 \(phi : C\rightarrow L\) 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 \(L\).

Examples: 1

##### 16.1-2 LeibnizQuasiCoveringHomomorphism

`‣ LeibnizQuasiCoveringHomomorphism`( L ) | ( function )

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 \(L\).)

Examples:

##### 16.1-3 LieEpiCentre

`‣ LieEpiCentre`( L ) | ( function )

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 \(E\).

Examples: 1

##### 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 \(µ : (L \wedge L) \longrightarrow L\) 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 \((L \wedge L)\) .

Examples:

##### 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 \(µ : (L \otimes L) \longrightarrow L\) 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 \((L \otimes L)\) .

Examples:

##### 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 \((L \otimes L) \longrightarrow (L/N \otimes L/N)\) is an isomorphism.

Examples:

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