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style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Covers of Lie algebras 
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 A short exact sequence of Lie algebras 
 
 
 <div style="text-align: center;">0 → M → C → L → 0 
 </div>
 
(over a field k) is said to be a stem
extension of L if M lies both in the centre Z(C) and in the
derived subalgeba C2. If, in addition, the rank of the
vector space M is equal to the rank of the second Chevalley-Eilenberg
homology H2(L,k) then the Lie algebra C is said to be a cover of L. 
 
 
Each finite dimensional Lie algebra L admits a cover C, and this cover
can be shown to be unique up to Lie isomorphism. 
 
The cover can be used to determine whether there exists a Lie algebra E
whose central quotient E/Z(E) is isomorphic to L. The image in L of the
centre of C is called the Lie
Epicentre of L, and this image is trivial if and only if such an
E exists. 
 
The cover can also be used to determine the stem extensions of L. It
can be shown that each stem extension is a quotient of the cover by an
ideal in the Lie multiplier H2(L,k). </td>
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style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands compute the cover C of the solvable but
non-nilpotent 13-dimensional Lie algebra L (over the rationals) that
was introduced by M. Wuestner in [ "An example of a nonsolvable Lie
algebra", Seminar Sophus Lie 2 (1992), 57-58 ]. They also show that: 
 <ul>
 <li>the second Chevalley-Eilenberg homology of L has rank
2.</li>
 <li>the second Leibniz homology of L has rank 6.</li>
 <li>the second Chevalley-Eilenberg homology of C is trivial.</li>
 <li>the second Leibniz homology of C has rank 6.</li>
 <li>the Lie algebra L is not isomorphic to any central quotient
E/Z(E). </li>
 </ul>
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style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
SCTL:=EmptySCTable(13,0,"antisymmetric");; 
gap> SetEntrySCTable( SCTL, 1, 6, [ 1, 7 ] );; 
gap> SetEntrySCTable( SCTL, 1, 8, [ 1, 9 ] );; 
gap> SetEntrySCTable( SCTL, 1, 10, [ 1, 11 ] );; 
gap> SetEntrySCTable( SCTL, 1, 12, [ 1, 13 ] );; 
gap> SetEntrySCTable( SCTL, 1, 7, [ -1, 6 ] );; 
gap> SetEntrySCTable( SCTL, 1, 9, [ -1, 8 ] );; 
gap> SetEntrySCTable( SCTL, 1, 11, [ -1, 10 ] );; 
gap> SetEntrySCTable( SCTL, 1, 13, [ -1, 12 ] );; 
gap> SetEntrySCTable( SCTL, 6, 7, [ 1, 2 ] );; 
gap> SetEntrySCTable( SCTL, 8, 9, [ 1, 3 ] );; 
gap> SetEntrySCTable( SCTL, 6, 9, [ -1, 5 ] );; 
gap> SetEntrySCTable( SCTL, 7, 8, [ 1, 5 ] );; 
gap> SetEntrySCTable( SCTL, 2, 8, [ 1, 12 ] );; 
gap> SetEntrySCTable( SCTL, 2, 9, [ 1, 13 ] );; 
gap> SetEntrySCTable( SCTL, 3, 6, [ 1, 10 ] );; 
gap> SetEntrySCTable( SCTL, 3, 7, [ 1, 11 ] );; 
gap> SetEntrySCTable( SCTL, 2, 3, [ 1, 4 ] );; 
gap> SetEntrySCTable( SCTL, 5, 6, [ -1, 12 ] );; 
gap> SetEntrySCTable( SCTL, 5, 7, [ -1, 13 ] );; 
gap> SetEntrySCTable( SCTL, 5, 8, [ -1, 10 ] );; 
gap> SetEntrySCTable( SCTL, 5, 9, [ -1, 11 ] );; 
gap> SetEntrySCTable( SCTL, 6, 11, [ -1/2, 4 ] );; 
gap> SetEntrySCTable( SCTL, 7, 10, [ 1/2, 4 ] );; 
gap> SetEntrySCTable( SCTL, 8, 13, [ 1/2, 4 ] );; 
gap> SetEntrySCTable( SCTL, 9, 12, [ -1/2, 4 ] );; 
gap> L:=LieAlgebraByStructureConstants(Rationals,SCTL);; 
 
gap> LieAlgebraHomology(L,2); 
2 
gap> LeibnizAlgebraHomology(L,2); 
6 
 
gap> C:=Source(LieCoveringHomomorphism(L));; 
gap> LieAlgebraHomology(C,2); 
0 
gap> LeibnizAlgebraHomology(C,2); 
6 
gap> Dimension(LieEpiCentre(L)); 
1 
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