<Chapter><Heading> Lie commutators and nonabelian Lie tensors</Heading> Functions on this page are joint work with <B>Hamid Mohammadzadeh</B>, and implemented by him. <Section><Heading> </Heading>
<ManSection> <Func Name="LieCoveringHomomorphism" Arg="L"/> <Description> <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> <P/><B>Examples:</B> <URL><Link>../tutorial/chap7.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutLieCovers.html</Link><LinkText>2</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="LeibnizQuasiCoveringHomomorphism" Arg="L"/> <Description> <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> <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="LieEpiCentre" Arg="L"/> <Description> <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>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap7.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutLieCovers.html</Link><LinkText>2</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="LieExteriorSquare" Arg="L"/> <Description> <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> <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="LieTensorSquare" Arg="L"/> <Description> <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> <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="LieTensorCentre" Arg="L"/> <Description> <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. <P/><B>Examples:</B>
</Description> </ManSection> </Section> </Chapter>
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