<Chapter><Heading> Lie commutators and nonabelian Lie tensors</Heading>

<Table Align="|l|" >

<Row>
<Item>
Functions on this page are joint work with Hamid Mohammadzadeh, and implemented by him.
</Item>
</Row>

<Row>
<Item>
<Index> LieCoveringHomomorphism</Index>
<C>
LieCoveringHomomorphism(L)</C>
<P/>

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

<Row>
<Item>
<Index> LeibnizQuasiCoveringHomomorphism</Index>
<C>
LeibnizQuasiCoveringHomomorphism(L)</C>
<P/>

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




<Row>
<Item>
<Index> LieEpiCentre</Index>
<C>
LieEpiCentre(L)</C>
<P/>

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

<Row>
<Item>
<Index> LieExteriorSquare</Index>
<C>
LieExteriorSquare(L)
</C>
<P/>

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

</Item>
</Row>


<Row>
<Item>
<Index> LieTensorSquare</Index>
<C>
LieTensorSquare(L)
</C>
<P/>

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


<Row>
<Item>
<Index> LieTensorCentre</Index>
<C>
LieTensorCentre(L)
</C>
<P/>

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


</Table>
</Chapter>


