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A short exact sequence of Lie algebras 0 → M → C → L → 0 (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).
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: the second Chevalley-Eilenberg homology of L has rank 2. the second Leibniz homology of L has rank 6. the second Chevalley-Eilenberg homology of C is trivial. the second Leibniz homology of C has rank 6. the Lie algebra L is not isomorphic to any central quotient E/Z(E).
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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