4.2. CLASSIFICATION RESULTS FOR CALIBRATED STRUCTURES 77
m with λ on the diagonal. Hence, not all complex almost Abelian Lie algebras are complexications of real almost Abelian Lie algebras.
• Proposition 4.4 gives us a classication of the real and the complex almost Abelian Lie algebras. One may, in principal, write down a complete list in each dimension as follows. One considers, step-by-step, all possible sizes of the Jordan blocks in the complex Jordan form for ϕ(en), chooses the diagonal elements in each Jordan blocks as parameters and restricts these parameters in such a way that they are non-isomorphic for dierent parameter values but still give all isomorphism classes using the conditions given in Proposition 4.4.
• Proposition 4.4 may be reformulated in the way that the isomorphism classes of n -dimensional almost Abelian F-Lie algebras which are not Abelian are in one-to-one correspondence to the orbits ofPGL(n−1,F) on the projective spaceP(End(Fn−1)). This is a stratied space with the largest strata having codimension (n−2).
4.2. CLASSIFICATION RESULTS FOR CALIBRATED STRUCTURES 78 Proof. The group GL(6,R)ρ0 for F = R has been determined in [V]. We repeat the arguments to determine also the complex stabiliser ofρ0. We do the computation for the real and complex case in parallel. A short computation shows thatV0 := span(e4, e5, e6) = n
v∈F6
(vyρ0)2= 0
o. Moreover, if A∈GL(6,F)ρ0 and v∈V0, we get A(v)yρ0 =ρ0(A(v),·,·) = (A.ρ0)(A(v),·,·) =A.(vyρ0) and so A(v) ∈ V0. Hence, we may write the dual map A∗ : F6∗
→ F6∗
as At = A∗ = B BC
0 D
!
for B, D ∈ GL(3,F) and C ∈gl(3,F) with respect to the ordered basis (e1, . . . , e6). ApplyingA∗ to ρwe get
e126−e135+e234=A∗ e126−e135+e234
=Be12∧De6−Be13∧De5+Be23∧De6+ tr(C)Be123
=Be12∧De6−Be13∧De5+Be23∧De6+ tr(C) det(B)e123 and sotr(C) = 0. We setV1 := span(e1, e2, e3). We have an isomorphismF : Λ2V1∗∧V0∗ → End (V0, V1) given by F(ω∧α)(v) = α(v)·ωy (e123) for ω ∈ Λ2V1∗, α ∈ V0∗ and v ∈ V0. We identify End (V0, V1) withgl(3,R) via the basis(e4, e5, e6) ofV0 and (e1, e2, e3) of V1. Then the identities F(ρ0) =I3 and
F
Be12∧De6−Be13∧De5+Be23∧De6
= det(B)B−tDt are true. Thus, D= det(B)B and so
GL(6,F)ρ0 ⊆
( A 0 BA det(A)A
!
A∈GL(3,F), B ∈sl(3,F) )
The converse inclusion follows by inverting the above calculations. The computation of the associated Lie algebra L(GL(6,F)ρ0) is straightforward.
We are now able to prove
Theorem 4.7. Let g be a seven-dimensional real almost Abelian Lie algebra and u be a six-dimensional Abelian ideal in g.
(a) The following are equivalent:
(i) g admits a calibrated G2-structure.
(ii) g admits a calibrated G∗2-structure such that u has signature (2,4) with respect to the induced pseudo-Euclidean metric ong.
4.2. CLASSIFICATION RESULTS FOR CALIBRATED STRUCTURES 79 (iii) For anye7 ∈g\u, there exist A, B ∈ sl(3,R) and an ordered basis (e1, . . . , e6) of u such that the transformation matrix of ad(e7)|u with respect to(e1, . . . , e6) is given by
A B
−B A
!
(iv) For any e7 ∈g\u, the complex Jordan normal form of ad(e7)|u is given, up to a permutation of the complex Jordan blocks, by diag J, J
for some trace-free matrix J ∈C3×3 in complex Jordan normal form.
(b) The following are equivalent:
(i) g admits a calibrated G∗2-structure such that u has signature (3,3) with respect to the induced pseudo-Euclidean metric ong.
(ii) For any e7 ∈ g\u, there exists a vector space decomposition g = V ⊕W of g into three-dimensional subspacesV,W such that
ad(e7)|u ∈ {f ∈gl(g)|f|V =fV, f|W =fW, fV ∈sl(V), fW ∈sl(W)} (iii) For any e7 ∈ g\u, the complex Jordan normal form of ad(e7)u is given, up
to a permutation of the complex Jordan blocks, by diag (J1, J2) for trace-free matrices J1, J2 ∈ C3×3 which are complex Jordan normals form of real three-by-three matrices. That means, for i= 1,2, Ji contains no Jordan block with a non-real number on the diagonal or exactly two Jordan blocks of size 1 with a non-real number and its complex conjugate on the diagonal, respectively.
(c) The following are equivalent:
(i) g admits a calibrated G∗2-structure such that u is degenerate with respect to the induced pseudo-Euclidean metric on g.
(ii) For any e7 ∈ g\u, there exists an ordered basis (e1, . . . , e6) of u, A ∈ gl(3,R) andB ∈sl(3,R) such that the transformation matrix ofad(e7)|u with respect to (e1, . . . , e6) is given by
A 0
B A−tr(A)I3
!
Proof. Choose e7 ∈g\u. Lete7 ∈u0 with e7(e7) = 1 and identifyΛke70 withΛku∗ using the decompositiong=u⊕span(e7). If we say in the following that an element of Λku∗ is closed, we always mean that the corresponding form inΛke70 is closed with respect to the dierential of g.
4.2. CLASSIFICATION RESULTS FOR CALIBRATED STRUCTURES 80 Let ϕ∈Λ3g∗ be a calibrated G2-structure. There are uniqueω∈Λ2u∗,ρ∈Λ3u∗ with ϕ=ω∧e7+ρ. Lemma 4.3 implies
0 =dϕ=d(ω∧e7+ρ) =dρ.
Proposition 2.48 tells us that the model tensor of ρ is ρ−1 if = 1 and u has signature (2,4)or if =−1, thatρ has model tensorρ1 if = 1and u has signature (3,3)and that ρ has model tensorρ0 if u is degenerate.
Conversely, letρ∈Λ3e70 ∼= Λ3u∗be closed with model tensorρ−1. Choose an arbitrary G2-structureϕ˜∈Λ3g∗ and an arbitrary G∗2-structure ϕˇ ∈Λ3g∗ such thatu has signature (2,4)with respect to the induced pseudo-Euclidean metric. We decomposeϕ˜= ˜ω∧e7+ ˜ρ,
ˇ
ϕ = ˇω∧e7+ ˇρ with ω,˜ ωˇ ∈ Λ2u∗ and ρ,˜ ρˇ∈ Λ3u∗. By Proposition 2.48, both ρ˜ and ρˇ have model tensorρ−1. Hence, there are isomorphismsF ,˜ Fˇ :u→uwithF˜∗ρ˜=ρ= ˇF∗ρˇ. We dene isomorphisms G,˜ Gˇ :g → g by G|˜ u := ˜F, Gˇu := ˇF and G(e˜ 7) := e7 =: ˇG(e7). ThenG˜∗ϕ˜is aG2-structure withG˜∗ϕ|˜u=ρand the closure ofρand Lemma 4.3 show that G˜∗ϕ˜ is closed. Moreover, by the same arguments Gˇ∗ϕˇ is a calibrated G∗2-structure with Gˇ∗ϕ|ˇu =ρ. SinceGˇ is an isometry between(g, gGˇ∗ϕˇ)and(g, gϕˇ), the signature ofuis(2,4) with respect to gGˇ∗ϕˇ. Similarly, we see that for each closed ρ ∈ Λ3u∗ with model tensor ρ1 there exists a calibrated G∗2-structure ϕˆ ∈ Λ3g∗ with ϕ|ˆu = ρ and u having signature (3,3) with respect to gϕˆ and the analogous statement for closed ρ with model tensor ρ0
and calibrated G∗2-structures with degenerateuis true.
Summarizing, the existence of a calibrated G2-structure such that u has the desired property is equivalent to the existence of a closed three-form ρ ∈ Λ3e70 ∼= Λ3u∗ with the corresponding model tensor mentioned above. By Lemma 4.3, the closure ofρis equivalent to ad(e7)|u ∈L(GL(V)ρ). The identity component of the stabiliser of ρ, ∈ {−1,1}, is, according to Lemma 2.19, equal toSL(3,C)⊆GL(6,R). This gives us the equivalence of (i)-(iii) in (a) and of (i) and (ii) in (b). The stabiliser ofρ0is given in Lemma 4.6 and we get the equivalence of (i) and (ii) in (c). The equivalence of (iii) and (iv) in (a) follows from the fact that L(GL(6,R)ρ−1) = i(sl(3,C)) for some injective R-algebra homomorphism i:gl(3,C)→gl(6,C)and that ifJ is a complex Jordan normal form forA∈gl(3,C), then diag J, J
is a complex Jordan normal form fori(A). The equivalence of (ii) and (iii) in (b) is obvious.
Remark 4.8. In Section 4.3 we show that a seven-dimensional almost Abelian Lie algebra g with codimension one Abelian idealu admits a cocalibratedG∗2-structures such that u has signature(2,4)if and only ifgadmits a cocalibratedG∗2-structure such thatuhas signature (3,3). Moreover, the existence of a cocalibratedG∗2-structure with non-degenerate uimplies the existence of a cocalibratedG∗2-structure with degenerate u. The corresponding relations do not hold for calibrated G∗2-structures:
4.2. CLASSIFICATION RESULTS FOR CALIBRATED STRUCTURES 81
• If the complex Jordan normal form of ad(e7)|u is given by diag(1 +i,1−i,2 + 2i,2− 2i,−3−3i,−3 + 3i), then Theorem 4.7 shows that gadmits a calibratedG∗2-structure such that u has signature (2,4) but neither one such that u has signature (3,3) nor one such that u is degenerate.
• If the complex Jordan normal form ofad(e7)|u is given bydiag(1,2,−3,4,5,−9), then Theorem 4.7 shows that g admits a calibratedG∗2-structure such that u has signature (3,3)but neither one such thatuhas signature(2,4)nor one such thatuis degenerate.
• If the complex Jordan normal form of ad(e7)|u is given by diag(1,2,3,−5,−4,−3), then Theorem 4.7 shows thatgadmits a calibrated G∗2-structure with degenerateubut neither one where u has signature (2,4)nor one where u has signature (3,3). Remark 4.9. Recently, results on the existence of calibratedG2-structures on Lie algebras have been obtained. Namely, [CF] gives a full classication of the seven-dimensional nilpo-tent Lie algebras admitting a calibrated G2-structure. Moreover, [FMOU] determines all the six-dimensional solvable Lie algebrashadmitting a so-called symplectic half-atSU(3) -structure. There is an analogous relation between symplectic half-at SU(3)-structures on h and calibrated G2-structures on h⊕R as between half-at SU(3)-structures on h and cocalibrated G2-structures on h⊕R, cf. [FMOU]. Thus, the results obtained in [FMOU]
give us a full list of the seven-dimensional solvable Lie algebras of the form h⊕R ad-mitting a calibrated G2-structure such that the splitting h⊕R is orthogonal. The results in [FMOU] show that a six-dimensional almost Abelian Lie algebra hadmits a symplectic half-atSU(3)-structure if and only if h⊕Radmits a calibratedG2-structure. Analogously to the proof of Theorem 6.7, one may give a direct proof of this assertion.
For (G2)C-structures we obtain the following result:
Theorem 4.10. Let g be a complex seven-dimensional almost Abelian Lie algebra andu be a six-dimensional Abelian ideal in g.
(a) The following are equivalent:
(i) g admits a calibrated(G2)
C-structure such that uis non-degenerate with respect to the induced non-degenerate symmetric complex bilinear form on g.
(ii) For any e7 ∈ g\u, there exists a vector space decomposition g = V ⊕W into three-dimensional subspaces V andW such that
ad(e7)|u∈ {f ∈gl(u)|f|V =fV, f|W =fW, fV ∈sl(V), fW ∈sl(W)}. (iii) For anye7 ∈g\u, the complex Jordan normal form ofad(e7)u is given, up to a
permutation of the complex Jordan blocks, bydiag (J1, J2)for trace-free matrices J1, J2∈C3×3 in complex Jordan normal form.
4.2. CLASSIFICATION RESULTS FOR CALIBRATED STRUCTURES 82 (b) The following are equivalent:
(i) g admits a calibrated (G2)C-structure such that u is degenerate with respect to the induced non-degenerate symmetric complex bilinear form ong.
(ii) For any e7 ∈ g\u, there exists an ordered basis (e1, . . . , e6) of u, A ∈ gl(3,C) andB ∈sl(3,C) such that the transformation matrix ofad(e7)|u with respect to (e1, . . . , e6) is given by
A 0
B A−tr(A)I3
!
Proof. The proof is completely analogous to the proof of Theorem 4.7. Using Proposition 2.49 and Lemma 4.3 we see as in the proof of Theorem 4.7 that the existence of a calibrated (G2)C-structure with non-degenerate u (resp. degenerateu) is equivalent to the existence of a closed three-form ρ ∈ Λ3e70 ∼= Λ3u∗ with model tensor ρ1 (resp. ρ0), e7 ∈ g\u, and that this is equivalent toad(e7)|u∈L(GL(u)ρ). The stabiliser ofρ1 is given in Proposition 2.19 and the one ofρ0 in Lemma 4.6. This establishes the equivalence of (i) and (ii) both in (a) and (b). The equivalence of (ii) and (iii) in (a) is obvious.
We nish this section and use our results to determine the seven-dimensional nilpotent almost Abelian Lie algebra admitting calibrated G2-/(G2)C-structures. Note that the classication for the G2-case already has been done in [CF]. Note further that a seven-dimensional almost AbelianF-Lie algebra with six-dimensional Abelian idealuis nilpotent if and only if ad(e7)|u is nilpotent for e7 ∈ g\u and this is the case if and only if the diagonal elements in the complex Jordan normal form are all 0. Thus, for each partition n1+. . .+nk = 6 of 6 with n1, . . . , nk ∈ {1, . . . ,6}, n1 ≥ . . . ≥ nk, there is exactly one nilpotent almost Abelian Lie algebra, namely that one whose complex Jordan normal form has Jordan blocks of sizesn1, . . . , nk, and these are all nilpotent seven-dimensional almost AbelianF-Lie algebras. Therefore, in total we have11such nilpotent Lie algebras for both F=R and F =C. All of them have rational structure constants so each of them admits a co-compact lattice. Hence, if g admits a calibrated G2-structure, we get a compact nilmanifold with calibrated G2-structure.
We obtain the following result, where we refer to the appendix for the names of the appearing Lie algebras.
Corollary 4.11. Let g be a seven-dimensional nilpotent almost Abelian F-Lie algebra.
Then:
(a) If F = R, then g admits a calibrated G2-structure if and only if g ∈ {R7, A5,1 ⊕ R2,n7,2}.
(b) If F=R, then g admits a calibratedG∗2-structure.
4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 83 (c) If F=C, then g admits a calibrated(G2)C-structure.
Proof. Theorem 4.7 shows thatgadmits a calibratedG2-structure if and only if the Jordan blocks ofad(e7)|u have the sizes(1,1,1,1,1,1),(2,2,1,1)or(3,3). Hence, (a) follows. The proof of (c) is analogous to the one of (b) and we only show (b). Theorem 4.7 (a) and (b) show thatgadmits a calibratedG∗2-structure with non-degenerateuif and only if the sizes of the Jordan blocks in the complex Jordan normal form are (1,1,1,1,1,1), (2,1,1,1,1), (2,2,1,1), (3,1,1,1), (3,2,1) or (3,3). To prove the assertion, it suces, according to Theorem 4.7 (c), to give examples ofA∈gl(3,R)andB ∈sl(3,R)such that A 0
B A
! has complex Jordan normal form with only zeros on the diagonal and with Jordan blocks of sizes(2,2,2),(4,1,1),(4,2),(5,1)and (6). Recall that by our convention, Jm(λ)denotes a complex Jordan block of size m withλ∈ C on the diagonal and further that the 1s in Jm(λ) are on the superdiagonal. A complex Jordan normal form with Jordan blocks of sizes(2,2,2)and only zeros on the diagonal may be achieved withA= 0 andB ∈sl(3,R) of rank three. Those where the blocks have the sizes(4,1,1)or(4,2)may be achieved with A = diag(J2(0),0)and B = diag
1 0 1 b
!
,−1−b
!
with b =−1 or b= 1, respectively.
The sizes (5,1) or (6) may be achieved with A = J3(0) and B ∈ sl(3,R) with bij = 0 exceptb21= 1 or b31= 1, respectively.