Classication results for cocalibrated structures

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, J_m(λ)denotes a complex Jordan block of size m withλ∈ C on the diagonal and further that the 1s in J_m(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 b_ij = 0 exceptb21= 1 or b31= 1, respectively.

4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 84 (b) If F=R, then g admits a cocalibratedG2-structure if and only if g admits a cocali-brated G^∗₂-structure such that u is non-degenerate and this is the case if and only if there exists a non-degenerate ω∈Λ²u^∗ such that ad(e7)|_u∈sp(u, ω).

(c) If F=C, theng admits a cocalibrated(G₂)

C-structure such thatu is non-degenerate if and only if there exists a non-degenerate ω∈Λ²u^∗ such that ad(e7)|_u ∈sp(u, ω). (d) If F = R (resp. F = C), then g admits a cocalibrated G^∗₂-structure (resp. (G2)_C

-structure) such that u is degenerate if and only if there exists a two-dimensional subspace V2, a complementary four-dimensional subspace V4 and a non-degenerate two-formω ∈Λ²V₄^∗ on V4 such that

ad(e₇)|_u ∈

f ∈gl(u)

f|_V2 =f₂+h, f|_V4 =−tr(f2)

2 id_V4+f₄, f₂ ∈gl(V₂), h∈hom(V₂, V₄), f₄ ∈sp(V₄, ω)}.

Proof. Fix e7 ∈ g\u, let e⁷ ∈ u⁰ be that element with e⁷(e7) = 1 and identify as usual Λ^ke₇⁰withΛ^ku^∗using the decompositiong=u⊕span(e₇). Letϕ∈Λ³g^∗ be a cocalibrated G₂-structure (resp. cocalibrated (G2)_C-structure). There exists ρ ∈ Λ³u^∗ and Ω ∈ Λ⁴u^∗ with ?_ϕϕ = ρ∧e⁷+ Ω. Using Lemma 4.3 we get, as in the proof of Theorem 4.7, that Ω ∈ Λ⁴e70 ∼= Λ⁴u^∗ is closed. Proposition 2.48 and Proposition 2.49 tell us that the model tensor of Ω is ¹₂ω²₀ ∈ Λ⁴ F⁶∗

or −¹₂ω₀² ∈ Λ⁴ F⁶∗

if u is not degenerate and e¹²³⁴+e¹²⁵⁶ ∈Λ⁴ F⁶∗

if uis degenerate. So the existence of a cocalibratedG₂-structure (resp. cocalibrated (G2)_C-structure) with non-degenerate u implies the existence of a closed four-form Ω ∈ Λ⁴e₇⁰ ∼= Λ⁴u^∗ with model tensor ¹₂ω²₀ ∈Λ⁴ F⁶∗

. With the use of Proposition 2.48, we can argue, similarly as in the proof of Theorem 4.7, that the existence of a cocalibrated G₂-structure (resp. cocalibrated (G₂)_C-structure) with non-degenerate u is even equivalent to the existence of a closed four-form Ω∈ Λ⁴e₇⁰ ∼= Λ⁴u^∗ with model tensor ¹₂ω²₀ ∈Λ⁴ F⁶∗

. In particular, (a) follows. Analogously, we get that the existence of a cocalibrated G^∗₂-structure (resp. cocalibrated (G₂)_C-structure) with degenerate u is equivalent to the existence of a closed four-form Ω˜ ∈ Λ⁴e70 ∼= Λ⁴u^∗ with model tensor e¹²³⁴+e¹²⁵⁶ ∈Λ⁴ F⁶∗

. Using Lemma 4.3 we see that, both in the non-degenerate as in the degenerate case, the four-form Ω is closed if and only if ad(e7)|_u ∈ L(GL(u)Ω). By Lemma 2.4, Ω∈ Λ⁴u^∗ has model tensor ¹₂ω²₀ ∈ Λ⁴ F⁶∗

if and only if there exits a non-degenerateω ∈Λ²u^∗ withΩ = ¹₂ω². By Proposition 2.5, the stabiliser groupGL(u)_Ω ofΩ is then equal toSp(u, ω) and (b) and (c) follow. (d) follows form the concrete form of the stabiliser of e¹²³⁴+e¹²⁵⁶ ∈Λ⁴ F⁶∗

, which is given in Proposition 2.5.

Remark 4.13. Regarding Proposition 4.12, it might be of interest to know whether or not the existence of a cocalibrated G₂-structure always implies the existence of a cocalibrated G^∗₂-structure. We suppose not, but cannot provide a concrete counterexample.

4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 85 To transfer the conditions on ad(e7)|_u in Proposition 4.12, which are equivalent to the existence of a cocalibrated structure, in terms of the complex Jordan normal form of ad(e7)|_u, we need to recall some well-known results, see e.g. [DPWZ], on the complex Jordan normal forms of elements in sp(2n,F)⊆gl(2n,F):

Proposition 4.14. Let(V, ω)be aF-symplectic vector space. Then a linear transformation f ∈GL(V) is conjugate under the action of GL(V) to an element in sp(V, ω) if and only if the complex Jordan normal form of f has the property that for all m∈Nand all 06=λ the number of Jordan blocks of size m with λon the diagonal equals the number of Jordan blocks of size m with −λon the diagonal and the number of Jordan blocks of size 2m−1 with 0 on the diagonal is even.

Proposition 4.12 and Proposition 4.14 allow us to prove

Theorem 4.15. Let g be a seven-dimensional almost Abelian F-Lie algebra and u be a codimension one Abelian ideal.

(a) If F=R, then the following are equivalent:

(i) g admits a cocalibrated G2-structure.

(ii) g admits a cocalibrated G^∗₂-structure such that the subspaceu is non-degenerate with respect to the induced pseudo-Euclidean metric on g.

(iii) For any e₇ ∈ g\u, ad(e₇)|_u ∈ gl(u) is in sp(u, ω), ω ∈ Λ²u^∗ being a non-degenerate two-form on u.

(iv) For any e₇ ∈g\u, the complex Jordan normal form of ad(e₇)|_u has the property that for all m ∈N and all λ6= 0 the number of Jordan blocks of size m with λ on the diagonal is the same as the number of Jordan blocks of size m with −λ on the diagonal and the number of Jordan blocks of size 2m−1 with 0 on the diagonal is even.

(b) If F=C, then the following are equivalent

(i) gadmits a cocalibrated(G₂)_C-structure such that the subspaceu is non-degener-ate with respect to the induced non-degenernon-degener-ate complex symmetric bilinear form on g.

(ii) For any e₇ ∈ g\u, ad(e₇)|_u ∈ gl(u) is in sp(u, ω), ω ∈ Λ²u^∗ being a non-degenerate two-form on u.

(iii) For anye7 ∈g\u, the complex Jordan normal form of ad(e7)|_u has the property that for all m ∈N and all λ6= 0 the number of Jordan blocks of size m with λ on the diagonal is the same as the number of Jordan blocks of size m with −λ on the diagonal and the number of Jordan blocks of size 2m−1 with 0 on the diagonal is even.

4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 86 (c) If F=R(resp. F=C), then the following are equivalent:

(i) g admits a cocalibrated G^∗₂-structure (resp. cocalibrated (G2)_C-structure).

(ii) For anye7 ∈g\u, there exists a two-dimensional subspace V2, a complementary four-dimensional subspace V₄ and a non-degenerate two-form ω ∈Λ²V₄^∗ on V₄ such that ad(e7)|_u ∈gl(u) is in

{f ∈gl(u)|f|_V2 =f₂+h, f₂ ∈gl(V₂), h∈hom(V₂, V₄), f|_V4 =−tr(f2)

2 id_V4 +f4, f4 ∈sp(V4, ω)

.

(iii) For any e7 ∈ g\u the complex Jordan normal form of ad(e7)|_u ∈ gl(u) has the property that there exists a partition of {1, . . . ,6} into three subsets I₁, I₂, I₃, each of cardinality two, such that the following is true:

(1) P

i∈I₁λi =P

i∈I₂λi =−P

i∈I₃λi.

(2) If there are i₁ ∈ I₁, i₂ ∈ I₂ such that JB(i₁) = JB(i₂) then λ_i1 = λ_i2 =

−

P

i∈I3λi

2 or JB(j1) = JB(j2) for the uniquely determined jk∈Ik such that {i_k, j_k}=I_k,k= 1,2.

(3) If there exists i2 ∈ I2 such that JB(j) = JB(i2) for all j ∈ I1 or if there existsi₁ ∈I₁ such that JB(j) = JB(i₁) for allj ∈I₂, then λ_j =−

P

i∈I3λi

2

for all j∈I₁∪I₂ and JB(j) = JB(k) for all j, k∈I₁∪I₂.

Proof. (a) and (b) follow directly from Proposition 4.12 and Proposition 4.14. For the proof of (c), note that Proposition 4.12 shows that (ii) implies (i) Moreover, by Proposition 4.12, the implication "(i)⇒ (ii)" follows if we are able to show that the existence of a cocalibrated G^∗₂-structure (resp. cocalibrated(G2)_C-structure) with non-degenerate u on g ,implies condition (ii). Since (a) (resp. (b)) is already proved, we may also proceed as follows to nish the entire proof:

• First step: Show that condition (iv) in (a) (resp. (iii) in (b)) implies condition (iii) in (c).

• Second step: Show that the conditions (ii) and (iii) in (c) are equivalent.

First step:

LetA∈C^6×6 be a matrix in complex Jordan normal form such that for allm∈Nand all06=λ∈Cthe number of Jordan blocks of sizemwithλon the diagonal is the same as the number of Jordan blocks of sizemwith−λon the diagonal and the number of Jordan blocks of size 2m−1 with 0 on the diagonal is even. Number consecutively the diagonal elements of the complex Jordan normal form by λ1, . . . , λ6. The assumptions onA imply that we can portion {1, . . . ,6} as follows into three subsetsI1, I2, I3 of cardinality two:

4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 87

• We can group the Jordan blocks with non-zero diagonal elements into pairs of Jordan blocks of the same size withλand−λ,λ6= 0on the diagonal. Construct now subsets I1, . . . , Ir of cardinality two by going successively through all these pairs of Jordan blocks and putting successively the two indices corresponding to the rst,. . .,l-th,. . . diagonal element in the two Jordan blocks in one I_k. By the index i corresponding to the l-th diagonal element in a certain Jordan block we mean that i∈ {1, . . . ,6}

such the i-th diagonal element of the big matrix A is the l-th diagonal element in the Jordan block.

• Similarly, we can group the Jordan blocks with zero on the diagonal and of odd size into pairs of the same size and construct subsets I_r+1, . . . , I_s taking successively all these pairs of Jordan blocks and putting again the two indices corresponding to the rst,. . .,l-th, . . .diagonal element in the two Jordan block in oneI_k.

• Finally, we construct subsets Is+1, . . . , I3 by taking successively the Jordan blocks with0in the diagonal of even size and putting together the two indices corresponding to the (2l−1)-th and 2l-th diagonal element.

By construction, P

i∈I_kλi = 0 for all k = 1,2,3 and so condition (1) in Theorem 4.15 (c) (iii) is fullled. Moreover, if i₁ ∈ I₁, i₂ ∈ I₂ are such that JB(i₁) = JB(i₂), then by construction also JB(j₁) = JB(j₂) for the unique j_k ∈ I_k such that I_k = {i_k, j_k} for k= 1,2. This show that condition (2) in Theorem 4.15 (c) (iii) is fullled. Finally, we argue that also condition (3) in Theorem 4.15 (c) (iii) is satised. Therefore, assume, without loss of generality, that there is i2 ∈I2 such that JB(i1) = JB(j1) = JB(i2), {i₁, j1}=I1. Then λ_i1 +λ_j1 = 0 and λ_i1 =λ_j1 =λ_i2 imply 0 =λ_i1 =λ_j1 =λ_i2. By construction, the identity JB(i1) = JB(j1) = JB(i2) implies that JB(j2) = JB(i2) for j2 ∈I2,j2 6=i2. But then also λ_j2 = 0 and the rst part is proved.

Second step:

For this part of the proof, we remind the reader that we follow that standard convention on the form of Jordan blocks which puts the 1s on the superdiagonal. We rst show that condition (ii) implies condition (iii) in Theorem 4.15 (c). Let f := ad(e7)|_u,e7 ∈g\u. By assumption, we have a four-dimensional invariant subspaceV₄ ⊆u and a two-dimensional complementary subspaceV2 ⊆u such thatf|_V2 =f2+h,f2 ∈gl(V2),h∈hom(V2, V4) and f|_V4 =f4− ^tr(f₂²⁾id_V4 with f4 ∈ sp(V4, ω) for some non-degenerate two-form ω on V4. To simplify the way of speaking, we say in the following that certain vectors u₁, . . . , u_s are a Jordan basis of a linear map if there is a permutation making them into a Jordan basis.

Choose a Jordan basisv₁, . . . , v₄ off₄ and denote byµ₁, . . . , µ₄ the corresponding diagonal elements. Then Proposition 4.14 tells us that, without loss of generality, µ1 = −µ₂ and µ₃ = −µ₄. Set λ_i := µ_i− ^tr(f₂²⁾. The vectors v₁, . . . , v₄ are also a Jordan basis of f|_V4. Moreover, v_i and v_j are in one Jordan block for f₄ withµ_i on the diagonal if and only if

4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 88

v_i and v_j are in one Jordan block for f₄− ^tr(f₂²⁾id_V4 with λ_i on the diagonal. By [GLR, Theorem 4.1.4], there is a Jordan basisw1, . . . , w6off such that for alli, j∈ {1, . . . ,4}the vectors v_i and v_j are in the same Jordan block for f₄−^tr(f₂²⁾id_V4 withλ_i on the diagonal if and only if wi and wj are in the same Jordan block for f with λi on the diagonal.

Since the characteristic polynomial off is the product of the characteristic polynomials of f₄−^tr(f₂²⁾id_V4 andf₂, the Jordan basis vectorsw₅ orw₆ are in Jordan blocks withλ₅ orλ₆ on the diagonal, respectively, where λ5, λ6 are the roots of the characteristic polynomial of f₂. In particular,tr(f₂) =λ₅+λ₆. This allows us now to prove that the conditions (1) - (3) in Theorem 4.15 (c) are fullled for the sets Ik:={2k−1,2k},k= 1,2,3:

• We get

λ₁+λ₂ =µ₁+µ₂−tr(f₂) =−λ₅−λ₆, λ₃+λ₄ =µ₃+µ₄−tr(f₂) =−λ₅−λ₆, which is exactly condition (1).

• If w_i1 and w_i2 are in one Jordan block for f with λ_i1 = λ_i2 on the diagonal for i₁ ∈ {1,2}, i₂ ∈ {3,4}, then v_i1 and v_i2 are in one Jordan block for f₄ with µ_i1 = λi1+^λ5+λ₂ ⁶ on the diagonal. We may haveµi1 =µi2 = 0and soλi1 =λi2 =−^λ5+λ₂ ⁶. If this is not the case, Proposition 4.14 implies that f₄ has to contain two Jordan blocks of size two, one with µi1 and the other with −µ_i1 on the diagonal and so v_j1, v_j2 are in one Jordan block, j₁, j₂ such that {i₁, j₁} = {1,2}, {i₂, j₂} ={3,4}. Hence, wj1, wj2 are in one Jordan block. Thus, condition (2) is satised.

• If w₁, w₂ and w_i2 for some i₂ ∈ {3,4} or w_i1, w₃ and w₄ for some i₁ ∈ {1,2} are in one Jordan block for f withλon the diagonal, then v1,v2 andvi2 or vi1, v3 and v4

are in one Jordan block forf₄ withλ+^λ5+λ₂ ⁶ on the diagonal. But then Proposition 4.14 tells us that v1, v2, v3 and v4 are in one Jordan block for f4 with 0 on the diagonal. Hence, w1,w2,w3, w4 are in one Jordan block for f with−^λ5+λ₂ ⁶ on the diagonal. This is condition (3).

Finally, we show that condition (iii) implies condition (ii) in Theorem 4.15 (c). Let A∈C^6×6 be in complex Jordan normal form and assume that it fulls all the conditions in Theorem 4.15 (c) (iii). Let I1, I2 and I3 be a partition of {1, . . . ,6} as in condition (iii). We may assume that JB(i_k) = JB(i3) for i_k ∈ I_k, k= 1,2, i3 ∈ I3 implies i_k < i3

simply by redening I_k and I₃ if this is not the case (note therefore that λ_ik =λ_i3). Set V2 := span(ei|i ∈ I3) and V4 := span(ej|j ∈ I1 ∪I2). Due to our assumption, V4 is an invariant subspace for A. That means there are A₂ ∈ gl(V₂), H ∈ hom(V₂, V₄) and A4 ∈gl(V4) such thatA|_V2 =A2+H and A|_V4 =A4. Moreover, A4 is in complex Jordan normal form and so B :=A₄+^tr(A₂²⁾I₄ is also in complex Jordan normal form. We claim that B is conjugate to an element in sp(4,F). Therefore, we have to check that B fulls

4.3. CLASSIFICATION RESULTS FOR COCALIBRATED STRUCTURES 89 all the conditions in Proposition 4.14. We use the conditions (1) - (3) in Theorem 4.15 (c) (iii) to get information on the structure ofB. First, the identitytr(A₂) =P

i∈I₃λ_i shows that the diagonal elements ofB are given byµj =λj+

P

i∈I3λi

2 ,j ∈I1∪I2. Hence, we get the following properties of B:

(A) Condition (1) states that P

j∈I_kµj =P

j∈I_kλj +P

i∈I3λi= 0 for k= 1,2.

(B) Condition (2) implies that ifB contains a Jordan block of size2withµ=λ+

P

i∈I3λi

2

on the diagonal, then λ= −

P

i∈I3λi

2 , i.e. µ = 0, or µ 6= 0 and there is a dierent Jordan block of size 2. Property (A) implies that the value on the diagonal in this other Jordan block of size2 has then to be equal to−µ.

(C) Condition (3) states that there cannot be any Jordan block of size 3 inB and there can only be a Jordan block of size4 inB if the diagonal elements are equal to 0. Regarding (A) - (C), the Jordan blocks of size greater than one in B obviously full all conditions in Proposition 4.14. To discuss those of size one, note that if there is at least one Jordan block of size one with µ6= 0on the diagonal, then (B) and (C) directly imply that all the Jordan blocks in B with non-zero value on the diagonal must be of size one.

Hence, (A) implies that the number of Jordan blocks of size one withµ6= 0on the diagonal equals the number of Jordan blocks of size one with−µon the diagonal.

Thus, we are left with the Jordan blocks of size one with 0 on the diagonal and have to show that their number is even. Suppose that their number is odd, i.e. it is one or three.

If it was one, then (A) and (C) show that there is a Jordan block of size two with0on the diagonal. But then there has to be exactly one Jordan block of size one with a non-zero value on the diagonal, which we just excluded. If the number of Jordan blocks of size one with 0 on the diagonal was three, we again get that there is exactly one Jordan block of size one with a non-zero value on the diagonal. Thus, the number of Jordan blocks of size one with 0 on the diagonal has to be even and the statement is proved.

Remark 4.16. • Theorem 4.15 (a) implies that seven-dimensional almost Abelian Lie algebras admitting a cocalibrated G2-structure are necessarily unimodular. This is not true for arbitrary seven-dimensional Lie algebras, cf. Theorem 5.18.

• The admittance of a cocalibrated G2-structure on an almost Abelian Lie algebra puts restrictions on the Lie algebra Betti numbers hⁱ(g). Since g is unimodular, we have h⁷(g) = 1. Moreover, Theorem 4.15 (a) implies the existence of a closed two-form ω ∈ Λ²e₇⁰ of length three. Hence, h²ⁱ(g) > 0 for i = 1,2,3. A more thorough discussion of condition (iv) in Theorem 4.15 (a) implies the existence of three linearly independent closed two-forms in Λ²e₇⁰. Hence, h²(g) ≥ 3 and also h³(g) ≥3 since there have to be three linearly independent non-exact closed two-forms inΛ²e₇⁰.

4.4. CLASSIFICATION RESULTS FOR PARALLEL STRUCTURES 90

• However, whether a seven-dimensional almost Abelian Lie algebra admits a cocali-brated G₂-structure or not cannot be decided solely by the Lie algebra cohomology.

Therefore, note that the seven-dimensional almost Abelian Lie algebra g=R⁶o Re7

with ad(e7)|_u = diag(1,−1,2,−2,4,−4) has the same Lie algebra cohomology as the one with ad(e₇)|_u = diag(1,−1,−1,−1,4,−2), namely

h¹(g), h²(g), h³(g), h⁴(g), h⁵(g), h⁶(g), h⁷(g)

= (1,3,3,3,3,1,1).

Theorem 4.15 (a) implies that the rst Lie algebra admits a cocalibratedG₂-structure while the second does not.

We like to note the following consequences of Theorem 4.15 and Proposition 4.12.

Corollary 4.17. Let g be a real seven-dimensional almost Abelian Lie algebra and u be a codimension one Abelian ideal.

(a) If g admits a cocalibrated G^∗₂-structure with non-degenerate u, then it also admits a cocalibrated G^∗₂-structure with degenerate u.

(b) g admits a cocalibrated G^∗₂-structure if and only if g_C admits a cocalibrated (G2)_C -structure.

Remark 4.18. An interesting open question one may ask is if the analogue of Corollary 4.17 (b) holds for all real seven-dimensional Lie algebras. We do not think so but cannot give a concrete counterexample.

We end this section by noting what Theorem 4.15 implies for the nilpotent almost Abelian Lie algebras. As in the case of a calibrated G₂-structure, the interest stems from the fact that we get compact nilmanifolds with cocalibratedG₂-structures.

Corollary 4.19. Letgbe a nilpotentF-Lie algebra of dimension seven with six-dimensional Abelian ideal u. Then:

(a) If F=R, then g admits a cocalibrated G2-structure if and only if g∈/

A_4,1⊕R³,n_6,1⊕R,n_6,2⊕R .

(b) If F=R, then g admits a cocalibrated G^∗₂-structure.

(c) If F=C, then g admits a cocalibrated (G2)_C-structure.

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