


A short exact sequence of Lie algebras 0 → M → C → L → 0
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 H_{2}(L,k). 

The
following commands compute the cover C of the solvable but
nonnilpotent 13dimensional 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), 5758 ]. They also show that:


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 

