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<p>A short exact sequence of Lie algebras <
br>
<
br>
</p>
<
div style=
"text-align: center;">0 → M → C → L → 0 <
br>
</
div>
<p><
br>
(over a field k) is said to be a <
span style=
"font-style: italic;">stem
extension</
span> of L if M lies both in the centre Z(C) and in the
derived subalgeba C<
sup>2</
sup>. If, in addition, the rank of the
vector space M is equal to the rank of the second Chevalley-Eilenberg
homology H<
sub>2</
sub>(L,k) then the Lie algebra C is said to be a <
span
style=
"font-style: italic;">cover</
span> of L.<
span
style=
"font-style: italic;"></
span><
br>
</p>
<
br>
Each finite dimensional Lie algebra L admits a cover C, and this cover
can be shown to be unique up to Lie isomorphism. <
br>
<
br>
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 <
span style=
"font-style: italic;">Lie
Epicentre</
span> of L, and this image is trivial if and only if such an
E exists.<
br>
<
br>
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 H<
sub>2</
sub>(L,k). </
td>
</
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<
tr>
<
td
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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<
tr>
<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
SCTL:=EmptySCTable(13,0,
"antisymmetric");;<
br>
gap> SetEntrySCTable( SCTL, 1, 6, [ 1, 7 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 8, [ 1, 9 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 10, [ 1, 11 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 12, [ 1, 13 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 7, [ -1, 6 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 9, [ -1, 8 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 11, [ -1, 10 ] );;<
br>
gap> SetEntrySCTable( SCTL, 1, 13, [ -1, 12 ] );;<
br>
gap> SetEntrySCTable( SCTL, 6, 7, [ 1, 2 ] );;<
br>
gap> SetEntrySCTable( SCTL, 8, 9, [ 1, 3 ] );;<
br>
gap> SetEntrySCTable( SCTL, 6, 9, [ -1, 5 ] );;<
br>
gap> SetEntrySCTable( SCTL, 7, 8, [ 1, 5 ] );;<
br>
gap> SetEntrySCTable( SCTL, 2, 8, [ 1, 12 ] );;<
br>
gap> SetEntrySCTable( SCTL, 2, 9, [ 1, 13 ] );;<
br>
gap> SetEntrySCTable( SCTL, 3, 6, [ 1, 10 ] );;<
br>
gap> SetEntrySCTable( SCTL, 3, 7, [ 1, 11 ] );;<
br>
gap> SetEntrySCTable( SCTL, 2, 3, [ 1, 4 ] );;<
br>
gap> SetEntrySCTable( SCTL, 5, 6, [ -1, 12 ] );;<
br>
gap> SetEntrySCTable( SCTL, 5, 7, [ -1, 13 ] );;<
br>
gap> SetEntrySCTable( SCTL, 5, 8, [ -1, 10 ] );;<
br>
gap> SetEntrySCTable( SCTL, 5, 9, [ -1, 11 ] );;<
br>
gap> SetEntrySCTable( SCTL, 6, 11, [ -1/2, 4 ] );;<
br>
gap> SetEntrySCTable( SCTL, 7, 10, [ 1/2, 4 ] );;<
br>
gap> SetEntrySCTable( SCTL, 8, 13, [ 1/2, 4 ] );;<
br>
gap> SetEntrySCTable( SCTL, 9, 12, [ -1/2, 4 ] );;<
br>
gap> L:=LieAlgebraByStructureConstants(Rationals,SCTL);;<
br>
<
br>
gap> LieAlgebraHomology(L,2);<
br>
2<
br>
gap> LeibnizAlgebraHomology(L,2);<
br>
6<
br>
<
br>
gap> C:=
Source(LieCoveringHomomorphism(L));;<
br>
gap> LieAlgebraHomology(C,2);<
br>
0<
br>
gap> LeibnizAlgebraHomology(C,2);<
br>
6<
br>
gap> Dimension(LieEpiCentre(L));<
br>
1<
br>
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