Algebraic invariants for orbits of k-forms

cor-responding dual (n−k)-form ρˆ of a stable k-form ρ ∈ Λ^kV^∗ are given for all the cases relevant in this thesis.

Remark 1.38. A priori, the dual (n−k)-form ρˆ of a stable k-form ρ has nothing to do with the Hodge dual ?ρ of ρ with respect to some non-degenerate bilinear form g. In particular, we do not need any pseudo-Euclidean metric to dene ρˆ. But it turns out that in all cases where the stabiliser of ρ is a subgroup ofO(p, n−p) there is a tight connection to the Hodge dual dened by the induced bilinear form g.

1.4 Algebraic invariants for orbits of k -forms

In this section, we deal with certain algebraic invariants ofk-vectors on a nite-dimensional F-vector space. These invariants give us information on the structure of particulark-forms and are used to obtain obstructions to the existence of cocalibrated structures in Chapter 5. They have partly been introduced by Westwick in [W3], where he used them to classify the orbits of three-vectors on a seven-dimensional real vector space under the natural action of GL(V). We recall this classication but formulate it for three-forms, as we also dene the invariants directly fork-forms and not fork-vectors as Westwick did. If we deal withk-vectors, we consider them implicitly ask-forms onV^∗ via the natural isomorphism between V and V^∗∗. For more background on some of the invariants and other related results, we also refer the reader to [Gu], [BG1], [Cap], [W1] and [W2].

Denition 1.39. Let V be an n-dimensional F-vector space and let k ≥ 1. The Grass-man cone G_k(V^∗) consists of all decomposable k-forms on V, i.e. of all those k-forms ψ ∈ Λ^kV^∗ such that there are k one-forms α1, . . . , α_k with ψ = α1 ∧ . . .∧ α_k. The length l(ψ) of an arbitrary k-form ψ ∈ Λ^kV^∗ is dened as the minimal number m of de-composable k-forms ψ1, . . . , ψm which is needed to writeψ as the sum of ψ1, . . . , ψm, i.e.

l(ψ) := min{m∈N0|∃ψ₁, . . . , ψ_m ∈G_k(V^∗) :ψ=Pm

i=1ψ_i}. The rank rk(ψ) of ψ is the dimension of the subspace

[ψ] :=\ n

ψ∈Λ^kU|U is a subspace of V^∗o

or, equivalently, the rank of the linear map T : V → Λ^k−1V^∗, T(v) = vyψ. [ψ] is also called the support (ofψ). Note that by denition l(0) = 0 and rk(0) = 0.

Next, let ψ∈Λ^kV^∗, ψ6= 0, be given. Choose v /∈ker(T) and a subspace W of V such that W ⊕span(v)⊕kerT =V is a direct vector space sum. We get a natural (k−1)-form ρ(v, W) := (vyψ)|_W ∈Λ^k−1W^∗ and a natural k-form Ω(W) :=ψ|_W ∈Λ^kW^∗on W. From this construction, we obtain two more algebraic invariantsr(ψ)andm(ψ)by looking at the

1.4. ALGEBRAIC INVARIANTS FOR ORBITS OFK-FORMS 18

lengths of Ω(W)andρ(v, W) and minimizing over all possiblev, W. More exactly, we set:

r(ψ) := minn

l(Ω)|Ω = Ω(W)∈Λ^kW^∗, dim(W) = (rk(ψ)−1), W ∩kerT ={0}o , m(ψ) := min

n

l(ρ)|ρ=ρ(v, W)∈Λ^k−1W^∗, v /∈kerT,W ⊕span(v)⊕kerT =V o

. For completeness, we setr(0) := 0 andm(0) := 0.

Remark 1.40. • If ψ ∈Λ^kV^∗ and α ∈V^∗ such that α /∈ [ψ], then l(ψ) = l(ψ∧α), cf. [BuGl, (2.2)].

• On a 2m-dimensional F-vector space, non-degenerate two-forms are exactly those with full rank2m. Hence, another way of generalizing the concept of non-degeneracy to forms of higher degree on ann-dimensional vector space is to callk-forms with full rank n non-degenerate. This generalisation has been done in [MaSw3], where also various other generalisations of non-degeneracy to higher forms are discussed.

• In [Cap], an algebraic invariant for k-forms ψ∈Λ^kV^∗, called B-longueur, was con-sidered. Therefore, let B be the set of all bases of V. For a xed b∈B, set

lb(ψ) := min (

m∈N0

ψ=

m

X

i=1

ψi s.t. ∀j∈ {1, . . . , m}: ψj =λjαj1 ∧. . .∧αjk, λ_j ∈F, α_j1, . . . , α_jm ∈b}

The B-longueur of ψ is dened as min{l_b(ψ)|b ∈ B}. Of course, the B-longueur is greater or equal to the irreducible length of ψ and, in general, they do not coincide.

E.g. the B-longueur of the three-form ρ−1 ∈ Λ³ R⁶∗

dened in Equation (1.8) is four and the length of it is three, cf. [Cap].

• An equivalent description of the numbers r(ψ) andm(ψ) is obtained as follows:

Let α ∈[ψ], α 6= 0 and U be a complement of span(α) in [ψ]. Denote by ρ(α, U) ∈ Λ^k−1U and Ω(α, U)∈Λ^kU the unique three- and four-form onV such that

ψ=ρ(α, U)∧α+ Ω(α, U).

Then

r(ψ) = min{l(Ω)|Ω = Ω(α, U)∈Λ^kU, α∈[ψ]\{0}, U⊕span(α) = [ψ]}, m(ψ) = min{l(ρ)|ρ=ρ(α, U)∈Λ^k−1U, α∈[ψ]\{0}, U⊕span(α) = [ψ]}.

We will mostly work with this description.

For a k-form ψ, a given v ∈ V\{0} and a given subspace W of V with span(v)⊕ W = V, the (k−1)-form ρ(v, W) := (vyψ)|_W depends on both v and W. However, in the following sense it essentially only depends on v, and, in particular, the values of the algebraic invariants only depend onv:

1.4. ALGEBRAIC INVARIANTS FOR ORBITS OFK-FORMS 19 Remark 1.41. Let ψ∈Λ^kV^∗ be ak-form and setT :V → Λ^k−1V^∗, T(w) :=wyψ. Let v /∈kerT and let W₁, W₂ be two subspaces of V such that V = span(v)⊕W_i⊕kerT for i= 1,2. Setρ(v, Wi) := (vyψ)|_Wi fori= 1,2and denote bypr_W2 :V →W2 the projection of V ontoW2 alongspan(v)⊕kerT. Thenf :W1 →W2,f := pr_W2|_W1 is an isomorphism with f^∗ρ(v, W₂) =ρ(v, W₁).

If f : W → V is a linear isomorphism, then the induced map f^∗ : Λ^kV^∗ → Λ^kW^∗ is a linear isomorphism which obviously preserves the length of a k-form and also all the other algebraic invariants rk,r and m. In particular, these algebraic invariants are really invariants ofGL(V)-orbits inΛ^kV^∗. Essentially there is only one more map which preserves the length [W1], namely a dual isomorphism. Note that we use a slightly dierent denition of a dual isomorphism as the one given e.g. in [KPRS].

Denition 1.42. LetV be ann-dimensional vector space andvol∈ΛⁿV^∗\{0}be a volume form. Then, for allk∈ {1, . . . , n−1}, the map δ: Λ^kV →Λ^n−kV^∗ dened by

δ(X) :=Xyvol

for X ∈ Λ^kV is called a dual isomorphism. Note that any other dual isomorphism is a non-zero multiple of δ.

Lemma 1.43. Let V be an n-dimensional F-vector space, k∈ {1, . . . , n−1}, δ : Λ^kV → Λ^n−kV^∗ be a dual isomorphism. Then l(X) = l(δ(X)) for all X ∈ Λ^kV. Hence, if

? : Λ^∗V^∗ → Λ^∗V^∗ is a Hodge star operator on V, then l(ψ) = l(?ψ) for all k-forms ψ∈Λ^kV^∗.

Proof. Let δ : Λ^kV → Λ^n−kV^∗, δ(X) := Xyvol be a dual isomorphism with vol ∈ ΛⁿV^∗\{0}. The image of a non-zero decomposable k-vector Y = v1 ∧. . .∧v_k on V is a non-zero (n−k)-form Ωwhich lies inΛ^n−k[Y]⁰. Since the dimension of the annihilator [Y]⁰ is n−k,Ω has to be decomposable. Hence, l(X) ≥ l(δ(X)) for all X ∈ Λ^kV. The inverse map of δis also a dual isomorphism and we get the equalityl(X) =l(δ(X))for all X∈Λ^kV. The statement for the Hodge dual follows since by Remark 1.28 the Hodge dual is the composition of a dual isomorphism with a linear isomorphism of the form f^∗. Remark 1.44. In [Fre1], the author of this thesis showed that r(δ(X)) = m(X) and m(δ(X)) =r(X) ifr(X)>0 andrk(X) =nand the result is used to determine the values of the invariants for the orbits of Hodge duals of G₂-structures and of (G2)_C-structures.

In this thesis, we use a dierent approach which also determines the model tensors of the induced three- and four-forms on a codimension one subspace ofV, see Section 2.4.

We end this section by recalling the classication of real three-forms in seven dimension by Westwick [W3] and also the classication of complex three-forms in seven dimensions

1.4. ALGEBRAIC INVARIANTS FOR ORBITS OFK-FORMS 20 [Gu]. We add the values of the algebraic invariantsrk, l, r, min the real case determined by Westwick [W3] and the values of the algebraic invariants rk, l determined in [W2], [Cap], [Gu].

Proposition 1.45. Letψ∈Λ³V^∗be a three-form on a seven-dimensional real vector space V and Ψ∈Λ³W^∗ be a three-form on a seven-dimensional complex vector space W.

(a) ψ is equivalent to exactly one of the following three-forms on R⁷: Table 1.1: Real three-forms in seven dimensions

ψ (rk(ψ), l(ψ), m(ψ), r(ψ)) (rk(ψ_C), l(ψ_C))

Q1 0 (0,0,0,0) (0,0)

Q2 e¹²³ (3,1,1,0) (3,1)

Q3 e¹²³+e¹⁴⁵ (5,2,1,0) (5,2)

ρ1 e¹³⁵+e¹⁴⁶+e²³⁶+e²⁴⁵ (6,2,1,1) (6,2)

ρ−1 e¹³⁵−e¹⁴⁶−e²³⁶−e²⁴⁵ (6,3,2,2) (6,2)

ρ0 e¹²⁶−e¹³⁵+e²³⁴ (6,3,1,1) (6,3)

P₁ e¹²³+e¹⁴⁵+e²⁶⁷ (7,3,1,1) (7,3)

R e¹²³+e¹⁴⁵+e¹⁶⁷+e²⁴⁶−e²⁵⁷ (7,4,1,2) (7,3) P₂ e¹²³+e²³⁷+e²⁶⁷−e³⁵⁷+e⁴⁵⁶+e⁵⁶⁷ (7,3,1,2) (7,3) S e¹⁴⁵+e¹⁶⁷+e²⁴⁶−e²⁵⁷+e³⁴⁷+e³⁵⁶ (7,4,2,3) (7,3)

P3 e¹²³+e¹⁴⁵+e¹⁶⁷ (7,3,1,0) (7,3)

P4 e¹²³+e¹⁴⁵+e¹⁶⁷+e²⁴⁶ (7,4,1,1) (7,4)

ϕ1 −e¹²³+e¹⁴⁵−e¹⁶⁷+e²⁴⁶+e²⁵⁷+ e³⁴⁷−e³⁵⁶

(7,4,2,2) (7,4)

ϕ₋₁ −e¹²³−e¹⁴⁵+e¹⁶⁷−e²⁴⁶−e²⁵⁷− e³⁴⁷+e³⁵⁶

(7,5,3,3) (7,4)

(b) Let ψ₁, ψ₂ ∈Λ³ R⁷∗

be two dierent three-forms in Table 1.1. Then the complex-linear extensions (ψ1)_C ∈Λ³ C⁷∗

and (ψ2)_C∈Λ³ C⁷∗

are equivalent if and only if {ψ₁, ψ2} ∈ {{ρ₁, ρ−1},{P₁, R},{P₂, S},{ϕ₁, ϕ−1}}. Moreover, Ψ is equivalent to the complex-linear extension of one of the three-forms in Table 1.1.

Note that Table 1.1 implies

Corollary 1.46. Let V be a seven-dimensional real vector space. For ψ₁, ψ₂ ∈ Λ³V^∗ we haveψ1 ∈GL(V)·ψ2 if and only if

(ρ(ψ₁), l(ψ₁), r(ψ₁), m(ψ₁)) = (ρ(ψ₂), l(ψ₂), r(ψ₂), m(ψ₂)).

Chapter 2

Interesting examples of G -structures

2.1 G -structures related to two-forms

In this section, we look at two-forms and(n−2)-forms on ann-dimensionalF-vector space V. We classify them up to equivalence and compute all the stabiliser subgroups. If n= 4, we characterise subspace of Λ²V^∗ in which each non-zero element has length two.

We start with two-forms. For those forms, the length is enough to distinguish them up to equivalence.

Lemma 2.1. Let V be an n-dimensional F-vector space and let ω ∈ Λ²V^∗. Then ω has length l if and only if ω^l 6= 0 and ω^l+1 = 0 and this is equivalent to the existence of 2l linearly independent one-forms α1, . . . , α_2l ∈ V^∗ such that ω =Pl

i=1α2i−1∧α2i. In the case l=_n

2

this is also equivalent to the stability of ω and if additionallyn is even, also to the non-degeneracy of ω. Moreover, the mapΛ²V^∗→N0, ω7→l(ω) induces a bijection between the GL(V)-orbits of two-forms on V and

0, . . . ,_n

2 .

Proof. The last assertion in Lemma 2.1 follows from the previous ones. Hence, we only have to prove them. A proof of the rst equivalence may be found in [BuGl, Theorem 2.11]. Ifω^l 6= 0and ω^l+1= 0, then, by the rst equivalence, ω has length l. Hence, there exist ω_i ∈ G₂(V^∗), i = 1, . . . , l, with ω = Pl

i=1ω_i. We may choose one-forms α_j ∈ V^∗, j= 1, . . . ,2l such thatωi =α2i−1∧α2i. Then

α1∧. . .∧α2l=ω1∧. . .∧ωl= ω^l l! 6= 0.

Thus, α1, . . . , α2l are linearly independent andω has the stated form. Conversely, if ω = Pl

i=1α2i−1∧α2i for linearly independent α1, . . . , α_2l ∈V^∗, thenω^l=l!α1∧. . .∧α_2l6= 0 and ω^l+1 = 0. Moreover, such an ω has rank land so it is stable if and only if l=_n

2

by Example 1.30 and non-degenerate if additionally nis even.

For the construction of cocalibratedG2-structures in Chapter 5, we needk-dimensional

2.1. G-STRUCTURES RELATED TO TWO-FORMS 22 subspaces of the two-forms on a real four-dimensional vector space,k∈ {0,1,2,3}, in which each non-zero element is of length two. Such subspaces can be characterised as follows.

Lemma 2.2. Let V be a real four-dimensional vector space, k∈ {0,1,2,3}, ω1, . . . , ω_k ∈ Λ²V^∗ be arbitrary two-forms on V, τ ∈ Λ⁴V^∗\{0} and π be an arbitrary permutation of {1,2,3}. Set W := span(ω1, . . . , ω_k), ω˜1 := e¹²+e³⁴ ∈ Λ² R⁴∗

, ω˜2 := e¹³−e²⁴ ∈ Λ² R⁴∗

,ω˜₃ :=e¹⁴+e²³∈Λ² R⁴∗

. Moreover, dene the symmetric matrixH= (h_ij)_ij ∈ R^k×k by ωi∧ωj =hijτ fori, j = 1, . . . , k. Then the following are equivalent:

(i) W isk-dimensional and each element inW\{0} has length two.

(ii) There is an isomorphism u : V → R⁴ such that

u^∗ω˜_π(i)|i= 1, . . . , k is a basis of W.

(iii) H is denite.

(iv) There exists a Euclidean metric and an orientation on V such that W is a subspace of the space of all self-dual two-forms onV.

Proof. Condition (i) implies condition (ii) by [W3, Theorem 3.1] and [W3, Theorem 3.2].

The converse direction follows since ω˜_i∧ω˜_j = 0for i6=j and soω²6= 0 for allω∈W\{0}

if

u^∗ω˜_π(i)|i= 1, . . . , k is a basis of W. Since ω˜1,ω˜2,ω˜3 form a basis of the self-dual two-forms onR⁴ with respect to the standard Euclidean metric and orientation, we get the equivalence of (ii) and (iv). To prove the equivalence of (i) and (iii), letω =Pk

i=1aiωi ∈W witha:= (a1, . . . , a_k)^t6= 0. By Lemma 2.1,ω has length two if and only if

06=ω² =

k

X

i,j=1

a_ih_ija_jτ =a^tHa τ,

i.e. if and only if a^tHa6= 0. Hence, all elements inW\{0} have length two if and only if H is denite.

Next, we compute the stabiliser subgroup of a two-form of length l under the natural action of the general linear group.

Proposition 2.3. LetV be ann-dimensionalF-vector space andω ∈Λ²V^∗ be a two-form of length l. Set ker(ω) := {v∈V|ω(v,·) = 0} and choose some complement W of ker(ω) in V. Then the stabiliser subgroup GL(V)ω of ω under GL(V) is given by

GL(V)_ω=

f ∈GL(V)

f|_W =f₁+h, f|_ker(ω)=f₂, f₁ ∈Sp(ω|_W, W), h∈hom(W,ker(ω)), f₂ ∈GL(ker(ω))}.

∼= (Sp(2l,F)×GL(n−2l,F))o F^2l×(n−2l)

2.1. G-STRUCTURES RELATED TO TWO-FORMS 23 Proof. Assume that f ∈GL(V) stabilisesω. Then it also stabilisesker(ω), i.e. f|_ker(ω) ∈ GL(ker(ω)). The two-formω|_W onW is non-degenerate and so we must havef|_W =f₁+h withf1 ∈Sp(ω|_W, W), h ∈hom(W,ker(ω)). Hence,f has the stated form. Conversely, it is obvious that elements inGL(V)of the form as in the assertion stabilise ω.

The results for two-forms imply the following results on the equivalence classes of (n−2)-forms:

Lemma 2.4. Let V be an n-dimensional F-vector space, Ω∈ Λⁿ⁻²V be an (n−2)-form on V andl∈

0, . . . ,_n

2 . In a wedge product, denote by αb a one-form which is omitted in this product. Then the following statements are true.

(a) Ω is stable if and only if l=_n

2

.

(b) Let F=C or (l, n)6= (m,2m) for all odd m∈N. Then Ωhas length l if and only if there exists a basis α₁, . . . , α_n of V^∗ such that

Ω =

l

X

i=1

α1∧. . .α[2i−1∧αc2i∧. . .∧αn.

(c) Letl= 2mfor somem∈N. Assume thatF=Corm is even. ThenΩhas length m if and only if there exists a non-degenerate two-formω∈Λ²V^∗ such that Ω = _(m−1)!^ωm−1 . (d) Let F =R and n = 2m for some odd m ∈ N. Then Ω has length m if and only if

there exists a basis β1, . . . , βn of V^∗ such that Ω =±

l

X

i=1

β₁∧. . .β[2i−1∧βc_2i∧. . .∧β_n.

This is the case if and only if there exists a non-degenerate two-form ω∈Λ²V^∗ such that Ω = ±_(m−1)!^ωm−1 . Moreover, for each non-degenerate two-form ω on V, _(m−1)!^ωm−1 is not equivalent to −_(m−1)!^ωm−1 .

(e) If F=R and n= 2m for some odd m, the map Λⁿ⁻²V^∗→ N0, Ω7→l(Ω) induces a surjection between the orbits of (n−2)-forms on V and the set {0, . . . , m} such that each element in {0, . . . , m−1} has exactly one preimage and m has two preimages.

In all other cases, the mapΛⁿ⁻²V^∗→N0, Ω7→l(Ω) induces a bijection between the orbits of (n−2)-forms on V and the set

0, . . . ,_n

2 .

Proof. Let V be an n-dimensional F-vector space. (a) follows directly from Proposition 1.33, Lemma 1.43 and Lemma 2.1. Moreover, note (e) follows directly from (b),(c) and (d). So it remains to prove (b), (c) and (d). Ifn= 2m, Lemma 2.1 gives the identity

( _m X

i=1

α₁∧. . .α[2i−1∧αc_2i∧. . .∧α_n

α₁, . . . , α_2m basis of V )

=

ω^m−1 (m−1)!

ω ∈Λ²V^∗ non-degenerate

.

2.1. G-STRUCTURES RELATED TO TWO-FORMS 24 Choose a Hodge star operator ? : Λ²V^∗ → Λⁿ⁻²V^∗, where in the case F = R we choose the dening non-degenerate symmetric bilinear form to be positive denite. Furthermore, choose an ordered basis (f1, . . . , fn) ofV which is oriented and orthonormal with respect to the structures which dene ?. For F=C, oriented means vol(f1, . . . , fn) = 1. Then

Ω_l :=?

l

X

i=1

f²ⁱ⁻¹∧f²ⁱ

!

=

l

X

i=1

f¹∧. . .f[²ⁱ⁻¹∧fc²ⁱ∧. . .∧fⁿ∈Λⁿ⁻²V^∗ for l = 0, . . . ,_n

2

. By Lemma 1.43, Ω_l has lengthl. Moreover, if F=C, Lemma 2.2 and Proposition 1.33 show that

Ωl

l∈

0, . . . ,_n

2

is a system of representatives of the orbits of(n−2)-forms onV. Hence, the statements forF=Cfollow. Assume for the rest of the proof thatF=R. In this case, Lemma 2.2 and Proposition 1.33 show that the set n

Ω_l

−Ω_l∈GL(V)Ω_l, l∈n

0, . . . ,jn 2

ko o∪n

Ω_l,−Ω_l

−Ω_l6∈GL(V)Ω_l, l∈n

0, . . . ,jn 2

ko o

is a system of representatives for the orbits of (n−2)-forms on V. Hence, the statement follows if we can show thatΩl is not equivalent to−Ω_l if and only if(l, n) = (m,2m) for some odd m.

Consider rst the case 2l < n. ThenF.Ω_l=−Ω_l forF ∈GL(V)dened byF(e_i) :=e_i fori= 1, . . . , n−1andF(en) :=−e_nand soΩ_land−Ω_lare equivalent. Ifn= 2m,meven and l =m, then G.Ω_m =−Ω_m for G∈GL(V) dened by G(e2i−1) := e_2i and G(e_2i) :=

−e_2i−1 for i = 1, . . . , m and so Ωm is equivalent to −Ω_m. To nish the proof, we show thatΩ_m is not equivalent to−Ω_m if n= 2m andmis odd. Assume the contrary, i.e. that there is some h ∈ GL(V) with h.Ωm = −Ω_m. Consider the GL(V)-module isomorphism κ: Λ^2m−2V^∗ →Λ²V ⊗Λ^2mV^∗ dened forΩ∈Λ^2m−2V^∗ byκ(Ω) :=X⊗ν withX∈Λ²V, ν ∈Λ^2mV^∗ such thatXyν= Ω. ThenΛ^2mV⊗ Λ^2mV^∗⊗m ∼= Λ^2mV^∗⊗(m−1)

asGL(V) -modules and so κ(Ωm)^m ∈ Λ^2mV^∗⊗(m−1)

. Thus, 1

det(h)^m−1κ(Ω_m)^m=h.(κ(Ω_m)^m) =κ(h.Ω_m)^m= (−1)^mκ(Ω_m)^m =−κ(Ω_m)^m since m is odd. A short computation shows that κ(Ωm)^m 6= 0. Hence, det(h)^m−1 =−1, which is impossible sincem−1is even. Thus,Ω_mis not equivalent to−Ω_min this case.

Next, we compute the stabiliser groups of an(n−2)-form of lengthl. For the statement, note that by denition,det(id₀) = 1 for the only linear endomorphismid₀:{0} → {0} on the0-dimensional vector space {0} and sosgn(det(id0)) = 1.

Proposition 2.5. Let V be ann-dimensionalF-vector space and Ω∈Λⁿ⁻²V^∗ be of length l ∈

1, . . . ,_n

2 . Consider the map F : V^∗ → Λⁿ⁻¹V^∗, F(α) = Ω∧α and set V₁ :=

ker(F)⁰ ⊆ V^∗∗ ∼= V. Choose some complement V2 of V1 in V. Then Ω = _(l−1)!^ωl−1 ∧ν for a non-degenerate two-form ω ∈ Λ²V₁^∗ and ν ∈ Λ^n−2lV₂^∗, where we use the decomposition V₁⊕V₂ to identify (V₁⊕V₂)^∗ with V₁^∗⊕V₂^∗. Moreover:

2.1. G-STRUCTURES RELATED TO TWO-FORMS 25

(a) If F=C, then GL(V)Ω =

n

f ∈GL(V)

f|_V1 =λf1, f|_V2 =f2+h, λ^2−2l= det(f2),

f1 ∈Sp(V1, ω), λ∈C^∗, f2 ∈GL(V2), h∈hom(V2, V1)}. (b) If F=Rand l is even, then

GL(V)_Ω=n

f ∈GL(V)

f|_V1 =|det(f₂)|^2−2l1 f₁, f|_V2 =f₂+h, f₁∈GL(V₁), f₁.ω= sgn(det(f₂))ω, f₂ ∈GL(V₂), h∈hom(V₂, V₁)}. (c) If F=Rand l6= 1 is odd, then

GL(V)_Ω =n

f ∈GL(V)

f|_V1 = det(f₂)^2−2l1 f₁, f|_V2 =f₂+h, f₁.ω=ω, ∈ {−1,1}, f₂∈GL⁺(V₂), h∈hom(V₂, V₁) . and if l= 1, then

GL(V)_Ω ={f ∈GL(V)|f|_V1 =f₁, f|_V2 =f₂+h, f₁ ∈GL(V₁), f2 ∈SL(V2), h∈hom(V2, V1)}.

Proof. By Lemma 2.4, we may assume for the computation of the stabiliser ofΩthat there exists a basis F₁, . . . , F_n ofV such that in the dual basisF¹, . . . , Fⁿ we have

Ω =

l

X

i=1

F¹∧. . .F\²ⁱ⁻¹∧dF²ⁱ∧. . .∧Fⁿ. (2.1) Then V₁ = span(F₁, . . . , F_2l). After possibly redening F_2l+1, . . . , F_n, we may assume that V2 = span(F2l+1, . . . , Fn). Then Ω = _(l−1)!^ωl−1 ∧ν for ω := Pl

i=1F²ⁱ⁻¹ ∧F²ⁱ, ν :=

F^2l+1 ∧ . . .∧Fⁿ and ω ∈ Λ²V₁^∗ is non-degenerate. Choose a Hodge star operator ? such that(F₁, . . . , F_n) is an oriented orthonormal basis and such that in the real case the corresponding non-degenerate symmetric bilinear form is positive denite. Note that then Ω =?ω. IfF=C, then Proposition 1.33 and Proposition 2.3 imply

GL(V)_Ω = λh^−t

λⁿ⁻² = det(h), h|_V1 =h1+g, h|_V2 =h2, λ∈C^∗, h1 ∈Sp(ω, V1), g∈hom(V1, V2), h2∈GL(V2)}.

SetH :=λh^−t. Now our choice of an orthonormal basis shows thatV₁⊥V₂ andSp(ω, V₁), considered as a subgroup of GL(V), is closed under transposition. Hence, H|_V1 = λH₁ and H|_V2 = H2 +G with H1 ∈ Sp(ω, V1), G ∈ hom(V2, V1) and H2 ∈ GL(V2) such that det(H₂) = det λh⁻¹₂

= _det(h^λn−2l

2) = _det(h)^λn−2l =λ^2−2l. This proves that the stabiliser is contained in the corresponding group in the statement. Conversely, a short computation shows that actually all elements in the group given in the statement stabilise Ω.

2.1. G-STRUCTURES RELATED TO TWO-FORMS 26

Now we come to the real case. By Proposition 1.33, we have GL(V)Ω=

h^−t

h∈GL(V), h.ω= ω det(h)

.

Let h ∈ GL(V) with h.ω = _det(h)^ω . Obviously, we have h|_V1 = h1 +g, h|_V2 = h2 for certain h_i ∈ GL(V_i) for i = 1,2 and g ∈ hom(V₁, V₂). For H₁ := √ ^h1

|det(h)| we have det(H1) = _|^det(h_det(h)|¹⁾l, H1.ω = sgn(det(h))ω and so _det(H¹ ₁₎ω^l = H1.ω^l = sgn(det(h))^lω^l. Thus

sgn(det(h₁))|det(h₁)|^1−l|det(h₂)|^−l= det(h₁)

|det(h)|^l = det(H₁) = sgn(det(h))^−l. (2.2) Forl even, sgn(det(h))^−l= 1 and Equation (2.2) yields sgn(det(h₁)) = 1 and |det(h₁)|=

|det(h₂)|^1−ll . Hence, sgn(det(h)) = sgn(det(h₂)), |det(h)| = |det(h₂)|^1−l1 and so h₁ =

|det(h2)|^2−2l1 H1. Since the transposition is the usual one if we identify GL(V) with GL(n,R) via the ordered basis (F₁, . . . , F_n), we obtain

GL(V)_Ω⊆n

f ∈GL(V)

f|_V1 =|det(f₂)|^2−2l1 f₁, f|_V2 =f₂+h, f₁ ∈GL(V₁), f1.ω= sgn(det(f2))ω, f2 ∈GL(V2), h∈hom(V2, V1)}. The converse inclusion follows by direct calculation.

For l odd, l 6= 1, Equation (2.2) gives us sgn(det(h₁)) = sgn(det(h)) and |det(h₁)|=

|det(h2)|^1−ll . Thus, sgn(det(h2)) = 1, |det(h)|=|det(h2)|^1−l1 = det(h2)^1−l1 and so h1 = det(h₂)^2−2l1 H₁ withH₁.ω= ωfor some ∈ {−1,1}. This shows

GL(V)_Ω⊆n

f ∈GL(V)

f|_V1 = det(f₂)^2−2l1 f₁, f|_V2 =f₂+h, f₁.ω=ω, ∈ {−1,1}, f₂ ∈GL⁺(V₂), h∈hom(V₂, V₁) .

Again the converse inclusion follows by direct calculation. The stabiliser in the case l= 1 is obviously as in the statement.

Next, we dene aGL⁺(V)-equivariant mapφ: Λ²V^∗ →Λ^2mV^∗ as in Proposition 1.37.

Denition 2.6. LetV be a2m-dimensional real vector space. Letω∈Λ²V^∗be a two-form on V. We dene φ: Λ²V^∗→Λ^2mV^∗ by

φ: Λ²V^∗ →Λ^2mV^∗, φ(ω) := ω^m m!.

φ is even GL(V)-equivariant. Since the stable two-forms are exactly the non-degenerate ones, the set φ⁻¹(0) is, in fact, the set of all non-stable two-forms. Moreover, the dual stable (2m−2)-form ωˆ is given by ωˆ = _(m−1)!^ωm−1 .

We end this section by dening aGL⁺(V)-equivariant mapφ: Λ^2m−2V^∗ →Λ^2mV^∗ as in Proposition 1.37. We omit the calculations necessary to check the claimed properties and instead refer to [CLSS] for some more details.

2.2. -COMPLEX STRUCTURES 27 Remark 2.7. • Let Ω∈Λ^2m−2V^∗ be a stable(2m−2)-form on a2m-dimensional real vector space and m be even. Using the GL(V)-module isomorphism κ: Λ^2m−2V^∗ ∼= Λ²V ⊗Λ^2mV^∗, we may consider κ(Ω)^m as an element in (Λ^2mV^∗)^⊗(m−1). We set

φ: Λ^2m−2V^∗→Λ^2mV^∗, φ(Ω) :=

κ(Ω)^m m!

_m−1¹ .

φ is GL(V)-equivariant. Moreover, φ(Ω) 6= 0 holds if and only if Ω is stable. By Lemma 2.4 there exists a stable two-form ω ∈ Λ²V^∗ such that Ω = _(m−1)!^ωm−1 . One can compute that φ(Ω) = φ(ω) and that the dual two-form Ωˆ is equal to _m−1^ω . In particular, ω ∈Λ²V^∗ with Ω = _(m−1)!^ωm−1 is unique.

• LetΩ∈Λ^2m−2V^∗ be a stable(2m−2)-form on a2m-dimensional oriented real vector space and now let m be odd. In this case, we set

φ: Λ^2m−2V^∗ →Λ^2mV^∗, φ(Ω) :=

κ(Ω)^m m!

1 m−1

.

The map φ is GL⁺(V)-equivariant, and again φ(Ω) 6= 0 holds if and only if Ω is stable. For odd m, Lemma 2.4 yields the existence a stable two-form ω ∈ Λ²V^∗ which induces the given orientation and ∈ {−1,1} such thatΩ =_(m−1)!^ωm−1 . One can compute that φ(Ω) = φ(ω) and that the dual two-form Ωˆ is given by _m−1^ω . Hence, ω∈Λ²V^∗ with the property that it induces the given orientation and that there exists ∈ {−1,1} with Ω =_(m−1)!^ωm−1 is unique.

2.2 -complex structures

In this section, we deal with complex and para-complex structures on2m-dimensional real vector spaces. We unify the language as in [SHPhD] and speak of -complex structures, where = −1 refers to complex and = 1 to para-complex structures. After the basic denitions, we recall the well-known decompositions of the -complex k-forms induced by an -complex structure J. Next, we discuss -complex volume forms. We show how one can reconstructJ from such a volume form and that in the case of oddmwe only need the real part of the volume form for the reconstruction ofJ. Lastly, we consider the particular case m = 3 and relate our results to the formalism of stable forms introduced in Section 1.3. Throughout this section, we follow closely [SHPhD]. More background on complex structures and related subjects may be found in any textbook on complex geometry like [Huy] or [Wells]. For para-complex structures, we refer the reader to [Kr].

We start with the main denitions of this section.

Denition 2.8. Let ∈ {−1,1}.

2.2. -COMPLEX STRUCTURES 28

• Let V be a 2m-dimensional real vector space. An -complex structure on V is an endomorphism J ∈ End(V) such that J² = id_V and such that if = 1 we have dim(V+) = dim(V−) =mfor V+:=Eig(J,1)andV−:=Eig(J,−1). A(−1)-complex structure is a complex structure in the usual sense and a1-complex structure is also called a para-complex structure.

• The -complex numbers are dened as the real unital associative algebra generated by 1 and the symboli subject to the relation i² =·1 and are denoted byC. C−1 =C are the usual complex numbers and the real unital associative algebraC1 are the para-complex numbers already mentioned Example 1.15 (d). From time to time, we writei instead ofi−1 andeinstead ofi1. We have C ∼=R⊕Ri as real vector spaces. Thus, we may write an elementz∈C asz=a+bi with a, b∈R. Re(z) :=ais called the real part of z and Im(z) := b is called the imaginary part of z. Moreover, the map z7→z:= a−bi is called the -complex conjugation and z the -complex conjugate of z. For = −1, · is the usual complex conjugation and z the usual complex conjugate of z and for = 1 we call · also the para-complex conjugation and z the para-complex conjugate of z. Note that the notation is in accordance with the one of Section 1.2 if we consider C as a composition algebra with the pseudo-Euclidean metric g(z, w) :=zw.

• If V is a real n-dimensional vector space, then the free C-module V_C := V ⊗_RC

is called the -complexication ofV. The (−1)-complexication is simply the usual complexication and the 1-complexication is also called para-complexication. To simplify the notation, we say that a free C1-module V is a C1-vector space.

Remark 2.9. • C1 contains zero divisors, namely exactly thez∈C1\{0}withzz= 0.

• If J is an -complex structure on an n-dimensional real vector space V, then n = 2m for some m ∈ N and V is a C-vector space via (a+bi)·v := av+bJ v for a, b ∈ R and v ∈ V. Note that for = 1, V is, in fact, a C1-vector space since any real basis v₁, . . . , v_m of V₊ and any real basisw₁, . . . , w_m ofV− give theC1-basis v1+w1, . . . , vm+wm of V. Conversely, if W is an m-dimensional C-vector space, then W is a 2m-dimensional real vector space and the multiplication with i is an -complex structure on W. In this sense -complex structures on even-dimensional real vector spaces are the same as nite-dimensionalC-vector spaces. Moreover, all C1-modules which are nite-dimensional real vector spaces are free.

The following example is the main example of an -complex structure. We will use it as a model tensor in the following.

2.2. -COMPLEX STRUCTURES 29 Example 2.10. An -complex structure on R^2m is given by

J:=

m

X

i=1

e²ⁱ⊗e2i−1+e²ⁱ⁻¹⊗e_2i

(2.3) We use these tensors as model tensors to identify -complex structures with GL(2m,R)J -structures.

For the use as a model tensor, we have to determine the stabiliser group of J.

Denition 2.11. The -complex general linear group GL(m,C) ⊆ GL(2m,R) is de-ned as the stabiliser of J ∈ End R^2m

. For = −1 it is given by the usual complex general linear group GL(m,C) embedded as a subgroup of GL(2m,R). With respect to the ordered basis (e1, e3, . . . , e2m−1, e2, . . . , e2m) of R^2m, GL(m,C) is given by the sub-group

( A B

−B A

!

A+iB ∈GL(m,C) )

of GL(2m,R). For = 1, GL(m,C1) is also called the para-complex general linear group and is given byGL(m,C1) = GL

R^2m

+

× GL

R^2m

−

∼= GL(m,R) × GL(m,R) with R^2m

+ := Eig(J1,1) and R^2m

− :=

Eig(J₁,−1).

The splitting of the -complexication V_C into the eigenspaces of the C-linear ex-tension of an -complex structure J ∈ End(V) gives us corresponding splittings of the -complexk-forms.

Denition 2.12. Let (V, J)be a 2m-dimensional vector spaceV with an -complex struc-tureJ. We consider the-complexicationV_C and the C-linear extensionJ_C ∈End(V_C) of J, which is dened forv₁+iv₂ ∈V_C, v₁, v₂ ∈V, by

J_C(v₁+iv₂) :=J(v₁) +iJ(v₂).

We set V^1,0 := Eig(J_C, i) ⊆ V_C and V^0,1 := Eig(J_C,−i) ⊆ V_C and observe that V_C =V^1,0⊕V^0,1 asC-vector spaces,V^1,0 ={w=v+iJ v∈V_C|v∈V}andV^0,1=V^1,0.

We have a natural C-isomorphism (V_C)^∗ = (V^∗)

C. Using this isomorphism, we simply write V^∗

C and get the decomposition V^∗

C = (V^∗)^1,0 ⊕ (V^∗)^0,1 with (V^∗)^1,0 :=

Eig(J^∗

C, i) = (V^0,1)⁰ and (V^∗)^0,1 := Eig(J^∗

C,−i) = (V^1,0)⁰. More explicitly, the two spaces are given by

(V^∗)^1,0 ={α+iJ^∗α|α∈V^∗}, (V^∗)^0,1 ={α−iJ^∗α|α∈V^∗}= (V^∗)^1,0. For p, q∈N0 we set

Λ^p,qV^∗ := Λ^p(V^∗)^1,0∧Λ^q(V^∗)^0,1 ⊆Λ^p+qV_C^∗ (2.4)

2.2. -COMPLEX STRUCTURES 30 and call the elements in Λ^p,qV^∗ (p, q)-forms or forms of type (p, q). We have Λ^p,qV^∗ = Λ^q,pV^∗ and Λ^kV^∗

C =Pk

p=0Λ^p,k−pV^∗. For p6=q, we set

[[Λ^p,qV^∗]] := Λ^kV^∗∩(Λ^p,qV^∗⊕Λ^q,pV^∗) (2.5) and call the elements in [[Λ^p,qV^∗]] real forms of type(p, q) and(q, p). Moreover, we set

[Λ^p,pV^∗] := Λ^kV^∗∩Λ^p,pV^∗ (2.6) and call the elements in [Λ^p,pV^∗]real forms of type (p, p). Note that we have [[Λ^p,qV^∗]]⊗ C= Λ^p,qV^∗⊕Λ^q,pV^∗, [Λ^p,pV^∗]⊗C= Λ^p,pV^∗,

[[Λ^p,qV^∗]] ={α+α|α∈Λ^p,qV^∗}, [Λ^p,pV^∗] ={α+α|α∈Λ^p,pV^∗} (2.7) and

Λ^2lV^∗ =

l−1

M

p=0

[[Λ^p,2l−pV^∗]]⊕[Λ^l,lV^∗], Λ^2l+1V^∗=

l

M

p=0

[[Λ^p,2l+1−pV^∗]]. (2.8) Remark 2.13. • In the para-complex case there is the natural decompositionV =V₊⊕ V− and the corresponding decomposition V^∗ := (V^∗)₊⊕(V^∗)₋. Then [[Λ^p,qV^∗]] = Λ^p(V^∗)₊∧Λ^q(V^∗)₋⊕Λ^q(V^∗)₊∧Λ^q(V^∗)₋ and[Λ^p,pV^∗] = Λ^p(V^∗)₊∧Λ^p(V^∗)₋.

• AlthoughEig(J_C1, λ)is a well-denedC1-submodule ofV_C1 for allλ∈C1, an element in V_C1 may have more than one eigenvalue with respect to J_C1. E.g. if v∈V₊, then v+ev ∈ V_C1 has both eigenvalue e = i1 and 1 with respect to J_C1, which stems from the fact that v+ev is linearly dependent in the C1-vector space V_C1 and that e−1∈C1 is a null-vector.

As remarked in Section 1.1, volume forms are nothing but SL(n,R)-structures. A nat-ural question to ask is what kind of tensors are related toSL(m,C)-structures,SL(m,C)⊆ GL(2m,R), and what are the corresponding objects in the para-complex case.

Denition 2.14. Let V be a 2m-dimensional real vector space and J be an -complex structure on V. An -complex m-form Ψ∈Λ^mV_C is called non-degenerate if Ψ∧Ψ6= 0.

An-complex(m,0)-formΨis called -complex volume form ifΨis non-degenerate. Note that each non-zero complex m-form is non-degenerate.

The basic example of an -complex volume form which we will use as a model tensor is the following.

Example 2.15. An -complex volume form on (R^2m, J) is given by Ψ:= e¹+ie²

∧. . .∧ e^2m−1+ie^2m

(2.9)

2.2. -COMPLEX STRUCTURES 31 Remark 2.16. Note that in the para-complex case there are degenerate para-complex (m,0)-forms. An example is given by (1 +e)Ψ₁ on R^2m, Ψ₁ as in Equation (2.9). Note further that by induction on m one may show that for all m∈N the identity

Ψ₁= f^1...m+f^m+1...2m

+e f^1...m−f^m+1...2m is true, where fⁱ := ^e2i−1m√^+e2i

2 and f^m+i := ^e2i−1m√^−e2i

2 for i= 1, . . . m. Moreover, (R^m)^∗₊ = span(f¹, . . . , f^m) and (R^m)^∗₋ = span(f^m+1, . . . , f^2m).

Denition 2.17. The -complex special linear groupSL(m,C)is dened asGL(2m,R)Ψ

with the -complex volume form Ψ on R^2m dened in Equation (2.9). Here, GL(2m,R) acts on (R^2m)_C ∼= R^2m ⊕iR^2m in the natural way on each of the summands R^2m and iR^2m. So GL(2m,R)Ψ = GL(2m,R)_Re(Ψ) ∩GL(2m,R)_Im(Ψ) by denition. Ob-viously, SL(m,C−1) = SL(m,C)⊆GL(2m,R)and from Remark 2.16 we get SL(m,C1) = SL

R^2m

+

×SL

R^2m

−

∼= SL(m,R)×SL(m,R).

If an -complex m-form Ψ ∈ Λ^mV_C^∗ is an -complex volume form with respect to an -complex structureJ onV, then Ψhas model tensor Ψ. Conversely, suppose that we do not have an-complex structureJ but we have an-complexm-formΨ∈Λ^mV^∗

Csuch that u^∗

CΨ = Ψfor some real isomorphismu:R^2m →V. The inclusionSL(m,C)⊆GL(m,C) implies thatΨinduces an-complex structureJ such thatΨis an-complex volume form with respect to this structure. We also callΨ∈Λ^mV_C^∗ with model tensorΨ an-complex volume form. Since SL(m,C) ⊆ SL(2m,R), Ψ also induces a real volume form. The following lemma gives a concrete description of these constructions.

Proposition 2.18. Let V be a 2m-dimensional real vector space and m≥2. Then

φ: Λ^mV_C^∗ →Λ^2mV^∗, φ(Ψ) :=







1

4Ψ∧Ψ, if m is even,

1

4iΨ∧Ψ, if m is odd. maps the -complex volume forms Ψ∈Λ^mV^∗

C to real volume forms on V. Moreover, any -complex volume formΨ∈Λ^mV_C^∗ induces a unique -complex structure J such that Ψis an-complex volume form in the sense of Denition 2.14 with respect toJ. If we denote by κ: Λ^2m−1V^∗ →V⊗Λ^2mV^∗the naturalGL(V)-module isomorphism given byκ(ψ) :=w⊗ν for ψ∈Λ^2m−1V^∗ and w∈V andν ∈Λ^2mV^∗ with wyν=ψ, J is dened by

J(v)Φ(Ψ) =







κ vyRe(Ψ)∧Im(Ψ)

if m is even, κ vyRe(Ψ)∧Re(Ψ)

if m is odd,

(2.10)

for v∈V.

Proof. For the proof one may consult, e.g., [SHPhD, Proposition 1.4].

2.2. -COMPLEX STRUCTURES 32 If m ≥ 3 is odd, we show below in Proposition 2.21 that on an oriented vector space we may recover the complex structure J from the real part of an-complex volume form.

To understand the construction abstractly on the level of enlargements ofG-structures, we have to compute the stabiliser in GL⁺(2m,R)of Re(Ψ). We do this for arbitrary m≥3. Lemma 2.19. Let m ≥ 3, Ψ ∈ Λ^mC^2m be the -complex m-form dened in Equation (2.9) andA:= diag(1,−1,1,−1, . . . ,1,−1), B := diag(−I₂, I2m−2)A∈GL(2m,R). Then:

GL(2m,R)_Re(Ψ)= SL(m,C)o{A, I_2m}, GL(2m,R)_Im(Ψ)= SL(m,C)o{B, I_2m}. Proof. For the proof note thatSL(m,C) = GL(2m,R)_Re(Ψ)∩GL(2m,R)_Im(Ψ).

We only show GL(2m,R)_Re(Ψ) = SL(m,C)o{A, I_2m}. The computation of the stabiliser ofIm(Ψ)is completely analogous. First, let =−1. Letg∈GL(2m,R)_Re(Ψ−1). The identity Re(Ψ−1) = ^Ψ₂⁻¹ + ^Ψ₂⁻¹ = ^g∗Ψ₂⁻¹ + ^g∗Ψ₂⁻¹ is true. Now ^Ψ−1₂ , ^g∗Ψ₂⁻¹, ^Ψ₂⁻¹ and

g^∗Ψ−1

2 are all decomposable as complex m-forms, ^Ψ₂⁻¹ ∧ ^Ψ₂⁻¹ 6= 0 and ^g∗Ψ₂⁻¹ ∧^g∗Ψ₂⁻¹ 6= 0.

Since m≥3, [BuGl, Theorem 4.4] states that g^∗Ψ−1= Ψ−1,g^∗Ψ−1=g^∗Ψ−1 or g^∗Ψ−1 = Ψ−1, g^∗Ψ−1 = Ψ−1. Since A^∗Ψ−1 = Ψ−1, A^∗Ψ−1 = Ψ−1, the assertion follows. Next, let = 1 and g ∈ GL(2m,R)_Re(Ψ1). By Remark 2.16, there is a basis f¹, . . . , f^2m of V^∗ such that Re(Ψ₁) = f^1...m+f^m+1...2m. Again [BuGl, Theorem 4.4] shows that due to m≥3,g^∗ either stabilises both f¹∧. . .∧f^m andf^m+1∧. . .∧f^2m or it interchanges the two decomposable forms. Moreover, if it stabilises both, it also stabilises Ψ₁. Hence, the statement follows from the fact thatA^∗ interchangesf¹∧. . .∧f^mandf^m+1∧. . .∧f^2m.

Letm= 2l−1≥3be odd. ThenA, B∈GL(2m,R)as in Lemma 2.19 has determinant

−1and so GL⁺(2m,R)_Re(Ψ)= GL⁺(2m,R)_Im(Ψ)= SL(m,C). This motivates us to call an m-form ρ ∈ Λ^mV^∗ on a 2m-dimensional oriented real vector space V with model tensor Re(Ψ) or Im(Ψ) an -complex volume form if m = 2l−1 ≥ 3. By enlarging the correspondingSL(m,C)-structure, such aρ induces an-complex structureJ_ρ, a real volume form φ(ρ) on V and an -complex volume form Ψ with respect to Jρ such that Re(Ψ) =ρ orIm(Ψ) =ρ, respectively. The construction of J_ρis as follows.

Denition 2.20. Letm= 2l−1be odd andρ∈Λ^mV^∗be anm-form on a2m-dimensional oriented real vector space V. We dene K_ρ:V →V ⊗Λ^2mV^∗ by

K_ρ(v) :=κ((vyρ)∧ρ),

where κ : Λ^2m−1V^∗ → V ⊗Λ^2mV^∗ is the natural isomorphism whose inverse is given by κ⁻¹(v⊗ν) = vyν for v ∈ V and ν ∈ Λ^2mV^∗. Note that (K_ρ⊗id_Λ2mV^∗)◦K_ρ : V → V ⊗(Λ^2mV^∗)^⊗2. Thus, we have

λ(ρ) := 1

2mtr ((Kρ⊗id_Λ^2m_V^∗)◦Kρ)∈(Λ^2mV^∗)^⊗2. (2.11)

2.2. -COMPLEX STRUCTURES 33 We dene the map φ: Λ^mV^∗ →Λ^2mV^∗ by

φ(ρ) :=p

|λ(ρ)| ∈Λ^2mV^∗. (2.12) If φ(ρ)6= 0, we dene J_ρ:V →V via

J_ρφ(ρ) =K_ρ. (2.13)

For m= 2l−1odd, one sees that J^∗Ψ =i^l−1Ψ implies J^∗Re(Ψ) =^lIm(Ψ) and J^∗Im(Ψ) = ^l−1Re(Ψ). Thus, ^l−1J

∗

Im(Ψ) = ^−2(l−1)2Re(Ψ) = Re(Ψ). Hence, if an m-form ρ∈Λ^mV^∗ on a 2m-dimensional real vector space V has model tensorIm(Ψ) it also has model tensor Re(Ψ). This is why we restrict ourselves to the model tensor Re(Ψ) in the following proposition.

Proposition 2.21. LetV be a2m-dimensional oriented real vector space andm= 2l−1≥ 3be odd. The mapφ: Λ^mV^∗→Λ^2mV^∗ dened in Equation (2.12) isGL⁺(V)-equivariant.

Assume thatρ∈Λ^mV^∗ has model tensorRe(Ψ). Thenφ(ρ)6= 0,J_ρas dened in Equation (2.13) is an -complex structure on V and

Ψ :=ρ+i^lJ_ρ^∗ρ∈Λ^mV_C^∗ (2.14) is an -complex volume form with respect to J_ρ with Re(Ψ) = ρ and φ(Ψ) = φ(ρ). Fur-thermore, λ(ρ) > 0 for λ: Λ^mV^∗ → Λ^2mV^∗⊗2

dened in Equation (2.11). Moreover, Ψ is the unique -complex structure Ψ˜ ∈Λ^mV^∗

C such that Re Ψ˜

= ρ and such that the orientation induced by Ψ˜ is the given one. Furthermore, we have

φ(ρ) = ^l

2J_ρ^∗ρ∧ρ (2.15)

and

J_ρ^∗α(v)φ(ρ) =α∧(vyρ)∧ρ (2.16) for all α∈V^∗.

Proof. TheGL⁺(V)-equivariance ofφis obvious. Letρ∈Λ^mV^∗have model tensorRe(Ψ). If u : Rⁿ → V is such that u^∗ρ = Re(Ψ), then Ψ := u⁻¹

C

∗

Ψ ∈ Λ^mV_C is an -complex volume form with Re(Ψ) = ρ. By choosing Re(Ψ)−iIm(Ψ) instead of Ψ = Re(Ψ)+iIm(Ψ), we may assume thatΨinduces the same orientation asρ. Note therefore that φ(Ψ) = ¹₂Im(Ψ)∧Re(Ψ). By Proposition 2.18 we have J_Ψφ(Ψ) = K_ρ, where J_Ψ is the -complex structure induced by Ψ. Since J_Ψ² = id_V, we get λ(ρ) = φ(Ψ)². Thus, λ(ρ) = φ(Ψ)² > 0 and φ(ρ) = φ(Ψ) 6= 0. Hence, Jρ = JΨ and Jρ is an -complex structure. Since Ψ is an (m,0)-form with respect to J_Ψ = J_ρ, the calculations directly above Proposition 2.21 showIm(Ψ) =^lJ_ρ^∗ρand soφ(ρ) =φ(Ψ) = ₂^lJ_ρ^∗ρ∧ρ. SinceIm(Ψ) is determined byρ,Ψ is unique. Moreover, using Equation (2.13), we may calculate

J_ρ^∗α

(v)φ(ρ) =J_ρ(v)yα∧φ(ρ) =α∧J_ρ(v)yφ(ρ) =α∧κ⁻¹(K_ρ(v)) =α∧(vyρ)∧ρ for all α∈V^∗.

2.3. (SPECIAL)-HERMITIAN STRUCTURES 34 We take a closer look at the case m= 3.

Proposition 2.22. (a) Let V be a six-dimensional oriented real vector space. Then φ(ρ) 6= 0 if and only if ρ ∈ Λ³V^∗ is stable. This is the case if and only if ρ is an -complex volume form for some∈ {−1,1}. Moreover, if ρ is stable, then the dual three-form ρˆis given by J_ρ^∗ρ.

(b) The concrete values of Kρ, λ(ρ) and, if it is well-dened, of Jρ for the three-formsρ onR⁶ in Table 1.1 for the standard orientation ofR⁶ are given in the following table:

Table 2.1: Invariants for three-forms in six dimensions

ρ K_ρ λ(ρ) J_ρ

Q₁ 0 0

-Q₂ 0 0

-Q₃ −2e¹⊗e₆⊗e¹²³⁴⁵⁶ 0

-ρ₀ 2 e¹⊗e₄+e²⊗e₅+e³⊗e₆

⊗e¹²³⁴⁵⁶ 0

-ρ1 2J1⊗e¹²³⁴⁵⁶ 4 e¹²³⁴⁵⁶⊗2

J1

ρ−1 2J−1⊗e¹²³⁴⁵⁶ −4 e¹²³⁴⁵⁶⊗2

J−1

Proof. LetV be a 6-dimensional oriented real vector space andρ∈Λ³V^∗. Due to Propo-sition 1.45, ρ is equivalent to exactly one of the forms in Table 2.1 and ρ is an -complex structure if and only if it is equivalent to ρ in Table 2.1 and so by Theorem 1.35 if and only if it is stable. Hence, φ(ρ)6= 0if and only ifρis stable follows directly from (b) and (b) is a straightforward computation. For a proof that the dual three-form ρˆ of a stable three-formρ∈Λ³V^∗ is equal toJ_ρ^∗ρ we refer to [CLSS].

2.3 (Special) -Hermitian structures

In this section, we consider (special)-Hermitian structures on2m-dimensional real vector spaces. Special -Hermitian structures are -Hermitian structures together with an -complex volume form of certain length. We recall the exact denitions and some basic facts of these structures. Using the results of Section 2.2, we show how one can reconstruct a special-Hermitian structure via a pair of a non-degenerate two-formωand an-complex volume formΨfullling certain compatibility conditions. For oddmwe prove that for the reconstruction we only needωandRe(Ψ). Finally, we look at the casem= 3and show that then a special -Hermitian structure can be recovered from a pair (ω, ρ) ∈ Λ²V^∗×Λ³V^∗ of stable forms onV withω∧ρ= 0. Again, we closely follow [SHPhD].

We start with -Hermitian structures.

Denition 2.23. LetV be a real2m-dimensional vector space. An -Hermitian structure

2.3. (SPECIAL)-HERMITIAN STRUCTURES 35

(g, J) consists of a pseudo-Euclidean metric g on V and an -complex structure J such that J^∗g=−g. The fundamental two-form ω∈Λ²V^∗ (associated to (g, J)) is dened by ω(v, w) := g(v, J w). A 1-Hermitian structure is also called para-Hermitian structure, a (−1)-Hermitian structure is called pseudo-Hermitian structure or Hermitian structure in case g is positive denite.

Remark 2.24. • J^∗g = −g implies that the fundamental two-form ω(·,·) = g(·, J·) is, in fact, a two-form, which is non-degenerate sinceg is non-degenerate.

• IfV is a real2m-dimensional vector space andgis a pseudo-Euclidean metric, then a complex structureJ such that(g, J) is a pseudo-Hermitian structure is the same as a one-fold cross product on(V, g). In particular, the signature ofgis then(2p,2m−2p) for somep∈ {0, . . . , m} by Example 1.12 (c).

• If (g, J) is a para-Hermitian structure on the real 2m-dimensional vector space V, then g(J v, J v) =−g(v, v) for all v∈V. Hence, g has necessarily signature (m, m). The following examples of-Hermitian structures onR^2mwill be used as model tensors.

Example 2.25. For p ∈ {0, . . . , m}, (h·,·i_2p,2m−2p, J−1) is a pseudo-Hermitian struc-ture on R^2m, whereas (h·,·i_split, J₁) is a para-Hermitian structure on R^2m. Here, J ∈ End R^2m

, ∈ {−1,1}, is dened by Equation (2.3) and h·,·i2p,2m−2p = P2p

i=1eⁱ⊗eⁱ− P2m

j=2p+1e^j⊗e^j ∈S² R^2m∗

,h·,·i_split=P2m

i=1(−1)ⁱeⁱ⊗eⁱ∈S² R^2m∗

by our conventions.

The fundamental two-form is given by ωp,m−p=Pp

i=1e²ⁱ⁻¹∧e²ⁱ−Pm

j=p+1e^2j−1∧e^2j or ω0 =ωm,0 =Pm

i=1e²ⁱ⁻¹∧e²ⁱ, respectively.

Remark 2.26. • The common stabiliser of(h·,·i2p,2m−2p, J−1)isU(p, m−p)⊆GL(m, C)⊆GL(2m,R). To compute the common stabiliser of (h·,·i_split, J₁), note that both

R^2m

+ and R^2m

− are isotropic with respect toh·,·i_split. Thus, h·,·i_split induces a non-degenerate bilinear pairing of R^2m

+ and R^2m

−. For g∈End R^2m

+

we denote by g^t ∈ End

R^2m

−

the transpose with respect to the mentioned pairing.

Then the common stabiliser of (h·,·i_split, J1) is given by n

f ∈GL(2m,R)

f|_(R2m)₊ =f1, f|_(R2m)₋ =f₁^−t, f1 ∈GL

R^2m

+

o∼= GL(m,R).

We call this group the para-unitary group. To unify the treatment, we setU⁻¹(p, m−

p) := U(p, m−p) and denote, for arbitrary p ∈ {0, . . . , m}, the para-unitary group by U¹(p, m−p).

• Every pseudo-Hermitian structure (g, J) on a 2m-dimensional real vector space has the pair (h·,·i_2p,2m−2p, J−1) for some p ∈ {0, . . . , m} as model tensors and every para-Hermitian structure (g, J) has the pair (h·,·i_split, J₁) as model tensors. Hence,

Im Dokument Geometric structures on Lie algebras and the Hitchin flow (Seite 35-57)