Classification of 𝑘-forms on r𝑛 and the existence of associated geometry on manifolds2 Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 21. Выпуск 2.

УДК 512.64+514.745 DOI 10.22405/2226-8383-2020-21-2-362-382

Классификация &-форм на Мп и существование ассоциированной геометрии на многообразиях1

Хонг Ван Ле, И. Ванжура

Ле Хонг Ван — доктор наук, профессор, Институт математики Чешской академии наук, (г. Прага, Чехия). e-mail: [email protected]

Ванжура Иржи — доктор наук, профессор, Институт математики Чешской академии наук, (г. Прага, Чехия). vanzuraQmath. cas. cz

Аннотация

В этой статье мы рассмотрим методы и результаты классификации к-форм (соотв. fc-векторов на М"), понимаемых как описание пространства орбит стандартного GL(n, М)-действие на ЛкМ"* (соотв. на ЛкМ"). Мы обсудим существование связанной геометрии, определяемой дифференциальными формами на гладких многообразиях. Эта статья также содержит Приложение, написанное Михаил Боровым, о методах когомологии Галуа для нахождения вещественных форм комплексных орбит.

Ключевые слова: GL(n, М)-орбиты в ЛЙМ"*; 0-группа; геометрия, определяемая дифференциальными формами; когомологии Галуа

Библиография: 68 названий. Для цитирования:

Хонг Ван Ле, И. Ванжура. Классификация fc-форм на Мп и существование ассоциированной геометрии на многообразиях // Чебышевский сборник, 2019, т. 21, вып. 2, с. 362-382.

CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.

UDC 512.64+514.745 DOI 10.22405/2226-8383-2020-21-2-362-382

Classification of &-forms on Mn and the existence of associated

geometry on manifolds2

Hong Van Le, J. Vanzura

Le Hong Van — Doctor of Sciences, Professor, Institute of Mathematics of the Czech Academy of Sciences, (Praha, Czech Republic). e-mail: [email protected]

Vanzura Jifi — Doctor of Sciences, Professor, Institute of Mathematics of the Czech Academy of Sciences, (Praha, Czech Republic). vanzuraQmath. cas. cz

Исследование ХЕШ было поддержано GACR-project 18-00496S и RVO:67985840. The research of HVL was supported by the GACR-project 18-00496S and RVO:67985840.

Abstract

In this paper we survey methods and results of classification of fc-forms (resp. fc-vectors on R"), understood as description of the orbit space of the standard GL(n, Reaction on AfcR"* (resp. on AfcR"). We discuss the existence of related geometry defined by differential forms on smooth manifolds. This paper also contains an Appendix by Mikhail Borovoi on Galois cohomology methods for finding real forms of complex orbits.

Keywords: GL(n, R)-orbits in AkR"*; 0-group; geometry defined by differential forms; Galois cohomology

Bibliography: 68 titles. For citation:

Hong Van Le, J. Vanzura, 2019, "Classification of fc-forms on Rn and the existence of associated geometry on manifolds" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 362-382.

Preface

Hamiltonian systems were one of research topics of Hong Van Le in her undergraduate study and calibrated geometry was the topic of her Ph.D. Thesis under guidance of Professor Anatolv Timofeevich Fomenko. Hamiltonian systems are defined on svmplectic manifolds and calibrated geometry is defined by closed differential forms of comass one on Riemannian manifolds. Since that time she works frequently on geometry defined by differential forms, some of her papers were written in collaboration with Jin Vanzura, [38, 39, 40]. We dedicate this survey on algebra and geometry of fc-forms on Rn as well as on smooth manifolds to Anatolv Timofeevich Fomenko on the occasion of his 75th birthday and we wish him good health, happiness and much success for the coming years.

1. Introduction

Differential forms are excellent tools for the study of geometry and topology of manifolds and their submanifolds as well as dynamical systems on them. Kahler manifolds, and more generally, Riemannian manifolds (M,g) with non-trivial holonomv group admit parallel differential forms and hence calibrations on (M,g) [27], [55], [40], [17]. In the study of Riemannian manifolds with non-trivial holonomv groups these parallel differential forms are extremely important [7], [29]. In their seminal paper [27] Harvev-Lawson used calibrations as powerful tool for the study of geometry of calibrated submanifolds, which are volume minimizing. Their paper opened a new field of calibrated geometry [30] where one finds more and more tools for the study of calibrated submanifolds using differential forms, see e.g., [17]. In 2000 Hitchin initiated the study of geometry defined by a differential 3-form [25], and in a subsequent paper he analyzed beautiful geometry defined by differential forms in low dimensions [26]. One starts investigation of a differential form of degree k on a manifold Mn of dimension n by finding a normal form of at a point x £ Mn and, if possible, to find a normal form of ^>k up to certain order in a small neighborhood U(x) C Mn. Finding a normal form of at a point x £ Mn is the same as finding a canonical representative of the equivalence class of (pk(x) in Ak(TX*Mn), where two fc-forms on TxMn are equivalent if they are in the same orbit of the standard GL(n, Reaction on Ak(Tx*Mn) = AkRn*. We say that a manifold Mn is endowed by a differential form <p £ Q*(Mn) of type ip0 £ A*Rn*, if for all x £ Mn the equivalence class of f(x) £ A*T*Mn can be identified with the equivalent class of <^0 £ A*Rn* via a linear isomorphism TxMn = Rn. Instead of investigation of a normal form of a concrete form ^>k, we may be also interested in a classification of (equivalent) fc-forms on Rn, understood as a description of the moduli space of equivalent fc-forms on Rn, which could give us insight on a normal form of ^>k and could also suggest interesting candidates for the geometry defined by differential forms.

Classification of on Rra is a part of algebraic invariant theory. Recall that an invariant of

an equivalence relation on a set 5, e.g., defined by orbits of an action of a group G on S, is a mapping from S to another set Q that is constant on the equivalence classes. A system of invariants is called complete if it separates any two equivalent classes. If a complete system of invariants consists of one element, we call this invariant complete. In the classical algebraic invariant theory one deals mainly with actions of classical or algebraic groups on some space of tensors of a fixed type over a vector space over a field F [23], see [48] for a survey of modern invariant theory and source of algebraic invariant theory. From a geometric point of view, the most important invariants of a form pk on Rra are the rank of pk and the stabilizer of pk under the action of GL(n, R). Recall that the rank of pk, denoted by rk pk, is the dimension of the image of the linear operator L^k : Rra ^ Afc-1Rm, v ^ ivfk. We denote the stabilizer of by StGL(ra,R)(^>fc) and in general, we denote by Stc(^) the stabilizer of a point x in a set S where a group G acts. A form pk e AkR"* is called non-degenerate, or multisymplectic, if rk pk = n. Furthermore, it is important to study the topology of the orbit GL(n, R) ■ <pk = GL(n, R)/StGL(ra,R)(^fc), for example, the connectedness, see Proposition 2 below, the openness, the closure of the orbit GL(n, R) ■ <pk C AkRra*. It turns out that understanding these questions helps us to understand the structure of the orbit space of GL(n, Reaction on AfcRra*. These invariants of fc-forms shall be highlighted in our survey.

Let us outline the plan of our paper. In the first part of Section 2 we make several observations on the duality between GL(n, R)-orbits of fc-forms on R^d GL(n, R)-orbits of fc-vectors as well as the duality between GL+(n, R)-orbits of ^tas on R^d GL+(n, R)-orbits of (n — k)-forms on Rra. Then we recall the classification of 2-forms on Rra (Theorem 2) and present the Martinet's classification of (n — 2)-forms on Rra (Theorem 3).

In contrast to the classification of 2-forms on Rra, the classification of 3-forms on Rra depends on the dimension n. Since dimA3R"* ^ dimGL(n, R) + 1, if n ^ 9, there are infinite numbers of inequivalent 3-forms in Rra. Till now there is no classification of the GL(n, R^^^^ton on A3Rra*, if n ^ 10.

In the dimension n = 9 the classification of the SL(9, C)-orbits on A3C9 has been obtained by

Vinberg-Elashvili [65]. In the second part of Section 2 we survey Vinberg-Elashvili's result and some

further developments by Le [34] and Dietrich-Facin-de Graaf [12], which give partial information

on GL(9, R)-orbits on A3R9. Then we review Djokovic' classification of 3-vectors in R8 and present

a classification of 5-forms on R8 (Corollary 1). Djokovic's classification method combines some

ideas from Vinberg-Elashvili's work and Galois cohomologv method for classifying real forms of a

R8

R8 (Proposition 1) as well as the classifications of 3-forms in R^ for n ^ 7 (Theorem 1, Remark 5).

Then we review a classification of GL(8, C)-action on A4C8 by Antonvan [1], which is important

R8

R8

R8

In Section 3, for k = 2, 3, 4, we compile known results and discuss some open problems on necessary and sufficient topological conditions for the existence of a differential fc-form tp of given type StoL(n,R)(^(®)) on manifoIds Mn (in these cases the equivalence class of p(x) is defined uniquely by the type of the stablizer of <p(x), i.e., the conjugation class of StoL(ra,R)(^(^)) in GL(n, R)). In dimension n = 8 (and hence also for n = 6, 7) we observe that the stabilizer StGL(ra,R) (p) of a 3-form p e A3Rra* forms a ^^^^^^^e system of invariants of the action of GL(n, R) on Rra (Remark 6).

We include two appendices in this paper. The first appendix contains a result due to Hong Van Le concerning the existence of 3-form of type G2 on a smooth 7-manifold, which has been posted in arxiv in 2007 [33]. The second appendix outlines the Galois cohomologv method for classification of real forms of a complex orbit. This appendix is taken from a private note by Mikhail Borovoi

with his kind permission.

Finally we would like to emphasize that our paper is not a bibliographical survey. Some important papers may have been missed if they are not directly related to the main lines of our narrative. We also don't mention in this survey the relations of geometry defined by differential forms to physics and instead refer the reader to [30], [15], [14], [60].

2. Classification of GL(n, R)-orbits of &-forms on Rn

2.1. General theorems

We begin the classification of GL(n, R)-orbits on AkRn* with the following observation that the orbit of the standard action of GL(n, R) on AkRn can be identified with the orbit of the standard action of GL(n, R) on AkRn* by using an isomorphism ^ £ Hom(Rn, Rn*) = Rn* ® Rn* D S2Rn*. Note that there are several papers and books devoted to the classification of fc-vectors on Rn [23, Chapter VII] 3, [11], [65]. Hence we have the following well-known fact, see e.g., [45],

Proposition 1. There exists a bijection between the GL(n, R)-orbits in AkRn and GL(n, R)-orbits in AkRn*.

Next we shall compare GL+(n, R)-orbits on AkRn with GL+(n, R)-orbits on An-kRn*. We take a volume form Q £ AnRn*\{0} and define the Poincare isomorphism Pq : AkRn ^ An-kRn*, £ ^ Q. Since GL+(n, R) is a direct product of its center Z(GL+ (n, R)) = R+ with its semisimple subgroup SL(n, R), fa any a £ R the group GL+(n, R) admits a a-twisted action on AkRn* defined as follows: 9[\](^) := (detg)x ■ g(<p) for g £ GL+(n,R), <p £ AkRn*, where g(<p) denotes the standard action of g on <p.

Denote also by ^ the isomorphism AkRn ^ AkRn* induced from a scalar product ^ on Rn.

Lemma 1. The composition Pq o : AkRn* ^ An-kRn* is a GL+(n, R)-equivariant map where GL+(n, R) acts on AkRn* by the standard action and on An-kRn* by the (—1)-twisted action.

Proof. Let tp = y(X) £ AkRn* and g £ GL+(n, R). Then

Pq o n-l(g*^) = Pn(g-1 o = ^-i^-i^Q

= (detg)-1 ■ = 9[-I](PQ o V■-1((P)),

which proves the first assertion of Lemma 1. □

Proposition 2. (1) There is a 1-1 correspondence between GL+(n, R)-or&fte of k-forms on Rn and GL+(n, R)-or&fte of (n — k)-forms on Rn. This correspondence preserves the openness of GL+(n, R)-orbits (and hence the openness of GL(n, R)-orbits).

(2) The GL(n, R)-or&fi of £ AkRn* has two connected components if and only if StoL(n,R)(^k) C GL+(n, R). In other cases the GL(n, R)-orfo'i of (pk is connected.

(3) Assume that ^>k £ AkRn* is degenerate. Then the GL(n, R)-orfo'i of (pk is connected.

proof. 1. The first assertion of Proposition 2 is a consequence of Lemma 1.

2. The second assertion of Proposition 2 follows from the fact that GL(n, R) has two connected components.

3. Assume that <p is degenerate. Then W := ker L^ is non-empty. Let W± be any complement to W in Rn i.e., Rn = W © W\ Then GL(^) © Idw± is a subgroup of St(ip). Since this subgroup

has non-trivial intersection with GL-(n, R), this implies the last assertion of Proposition 2 follows

□

3under "polyvectors"Gurevich meant both covariant and contravariant polyvectors

The following theorem due to Vinberg-Elashvili reduces a classification of (degenerate) fc-forms of rank r in Rra to a classification of ^^s on Rr. (Vinberg-Elashvili considered only the case k = 3 and the SL(n, C)-action on A3Cra but their argument works for any k and for GL(n, R)-action on

Theorem 1. (cf. [65, §441, [53, Lemma 3.2]) There is a 1-1 correspondence between GL(n, R)-orbits of k-form,s of rank less or equal to r on Rra and GL(r, R)-or&fte of k-form,s on Rr.

2.2. Classification of 2-forms and (n — 2)-forms on Rra

From Proposition 2 we obtain immediately the following known theorem [10], cf. [23, Theorem 34.9].

Theorem 2. (1) The rank of a 2-form p e A2R"* is a complete invariant of the standard GL(n, R)-acfe'on on A2Rra*. Hence A2Rra* decomposes into [n/2] + 1 GL(n, R)-orbits.

(2) The GL(n, R)-orbit of a 2-form p e A2Rra* has two connected components if and only if n = 2k and p has maximal rank.

(3) If p is of maximal rank, then the GL(n, R)-orbit of p is open and its closure contains the GL(n, R)-orbit of any degenerate 2-form on Rra.

The classification of (n — on Rra has been done by Martinet [41]. Martinet used the

inverse Poincare isomorphism P-1 : Ara-2Rra* ^ A2Rra to define the length of p e Ara-2Rra, denoted by l(p), to be the half of the rank of the bi-vector P-1(p) 4. By Proposition 2 and Theorem 2 the map P-1 induces an isomorphism bet ween the GL(n, R)-orbits of degenerate (n — 2)-form s p on Rra and degenerate bivectors P- ^^ on Rra.

If 2l(p) < n then p has the following canonical form

K<p)

p = ^2 a1 A"' a2i-2 A a2i+1 A •••A an. (1)

i=1

Bv Theorem 2 (2) the orbit GL(n, R) ■ P-1 (p) is connected, and hence by Proposition 2 the orbit GL(n, R) ■ p is connected.

• If 2l(p) = ^^d l(p) is odd, then using Lemma 1 and Theorem 2(2) we conclude that the set of (n — 2)-forms of length I consists of two open connected GL(n, R)-orbits that correspond to the sign of A = An (p) where

P-1(p) = e1 A e2 +-----+ e2fc-1 A e2k,

Q = Xa1 A ■ ■ ■ A an,

i(<p)

p = A ^ a1 A ■ ■ ■ a2i-2 A a2i+1 A ■ ■ ■ A an and A = ±1. (2)

i=1

• If 2l(p) = n and l(p) is even, using the same argument as in the previous case, we conclude that the set of (n-2)-forms of length I consists of one open GL(n, R)-orbit, which has two connected components.

To summarize Martinet's result, we assign the sign sq(p) of a (n-2)-form p e Ara-2Rra to be the number An (p)l(tfi) if 2l(p) = n, and to be 1, if 2l(p) < n.

Theorem 3. (cf. [41, §5]) (1) The length l(p) and the sign sn (p) of a (n-2)-form p e Ara-2Rm form, a complete system of invariants of the standard GL(n, R)-acfe'on on Ara-2Rra*.

(2) The GL(n, R)-orbit of a (n-2)-form p e A2R"* has two connected components if and only if n = 2k, l(p) = n/2 and I is even.

4the rank of a fc-vector is defined similarly as the rank of a fc-form.

2.3. Classification of 3-forms and 6-forms on R9

We observe that the vector space AkRn* is a real form of the complex vector space AkCn*. Hence, for any <p £ Ak Rn* the orb it GL(n, R) ■ ^ lies in the orbit GL(n, C) ■ (p. We shall say that GL(n, R) ■ (f is a real form of the complex orbit GL(n, C) ■ (f. It is known that every complex orbit has only finitely many real forms [3, Proposition 2.3]. Thus, the problem of classifying of the GL(n, R)-orbits in AkRn can be reduced to the problem of classifying the real forms of the GL(n, C)-orbits on AkCn. The classification of GL(n, C)-orbits on A3Cn is trivial, if n ^ 5, cf. Proposition 2. For n = 6 it was solved by W. Reichel [50]; for n = 7 it was solved by J. A. Schouten [57]; for n = 8 it was solved by Gurevich in 1935, see also [23]; and for n = 9 it was solved by Vinberg-Elashvili

[65]. In fact Vinberg-Elashvili classified SL(9,C)-orbits on A3C9, which are in 1-1 correspondence with SL(9,C)-orbits in A3C9* and SL(9, C)-orbits on A6C9*. Since the center of GL(9, C) acts on A3C9 \ {0} with the kernel Z3, it is not hard to obtain a classification of GL(9, C)-orbits on A3C9,

A3C9* A6C9* SL(9, C) A3C9

As we have remarked before, there are infinitely many GL(n, C)-orbits on A3C9, and to solve this complicated classification problem Vinberg-Elashvili made an important observation that the standard SL(9, C)-action on A3C9 is equivalent to the action of the adjoint group GC (also called the 0-group) of the Z3-graded complex simple Lie algebra

es = fl-1 © flC © flC (3)

where g| = sl(9, C), g| = A3C3, g-1 = A3C9* and = SL(9, C)/Z3 is the connected subgroup, corresponding to the Lie subalgebra gC, of the simply connected Lie group EC whose Lie algebra es

Remark 1. Let gC be a complex Lie algebra. Any Zm-grading gC := ©¿ezmgC on gC defines an automorphism a £ Aut(gC) of order m by setting a(x) := e%x where e = exp(^y—ln/m) and x £ gC. Conversely, any a £ Aut(gC) of ord er m defines a Zm-gradin g gC := ©¿ezm gC by setting gC := {x £ gC| a(x) = elx}.

In [65, §2.2] Vinberg and Elashvili considered the automorphism dC of order 3 on es associated to the Z3-gradation in (6) 5. To describe dC we recall the root system £ of es:

9

£ = {£i — £j, ±(e% + £j + £k)}, (i,j, k distinct), = 0}.

¿=1

Remark 2. Given a complex semisimple Lie algebra gC let us choose a Cartan subalgebra hC of gC. Let £ be the root system of g^. Denote by {Ha,Eal a £ £} the Chevalley system in gC i.e., Ha £ hC and Ea is the root vector corresponding to a such that for any H £ hC we have [H,Ea] = a(H)Ea, [Ha,Ea] = 2Ea and [Ea,E-a] = Ha [28, §32.2]. Then

gC = ©«es+ (h«)c ©«es+ ©«es+ (^-«)c (4)

where £+ C £ denote the system of positive roots, and £+ - the subset of simple roots. The automorphism dC of order 3 on es is defined as follows

)c = Id,

0CEa,«=(£i+£j +£fc))C = exP(^2^/3) ■ Id,

_+£fc))C = eXP(—i2*/3) ■ Id.

5 Automorphisms of finite order of semisimple Lie algebras have been classified earlier independently by Wolf-Gray

[66] and Kac [31].

Remark 3. Let {Ha, Ea \ a e S} he the Chevalley system, of a complex semisimple Lie algebra gC. Then {Hp,Ea\ a e S, ft e S+} is a basis of the normal form g also called split real form, of gC. The normal form of the complex simple Lie algebra e8 is denoted by e8(8); and the normal form, of sl(n, C) is the real simple Lie algebra sl(n, R). Clearly the Lie subaIgebra e8(8) has the induced Z3-grading from the one on e8 defined in (3) (note that e8(8) is not invariant under dC), i.e., we have

e8(8) = fl-1 © So © fl1 (5)

where Qi = e8(8) R gC is a real form of gC for i e {—1, 0,1}. Hence there is a 1-1 correspondence between SL(9, R)-orfo'is on A3R9* and the adjoint action of the subgroup G0, corresponding to the Lie subalgebra q0, of the Lie group GC.

Now let F be the field R or C. Based on (5), (3), Remark 3, and following [65, §1], [34, Lemma 2.5], we shall call a nonzero element x e A3F9 semisimple, if its orbit SL(9, F) ■ x is closed in A3F9, and nilpotent, if the closure of its orbit SL(9, F) ■ x contains the zero 3-vector. Our notion of semisimple and nilpotent elements agrees with the notion of semisimple and nilpotent elements in semisimple Lie algebras [65], [34], see also [11] for an equivalent definition of semisimple and nilpotent elements in homogeneous components of graded semisimple Lie algebras.

Example 12. ([65, §4-4]) Let x e A3F9 be a degenerate vector of rank r ^ 8, where F = R or C. (The definition of the rank of a k-vector can be defined in the same way as the definition of the rank of a k-form). Then for any A e R there exists an element g e SL(9, F) such that g ■ x = A ■ x. Hence the closure of the orbit SL(9, F) ■ x contains 0 e A3F9 and therefore x is a nilpotent element.

Proposition 3. Every nonzero 3-vector x in A3F9 can be uniquely written as x = p + e, where p is a semisimple 3-vector, e - a nilpotent 3-vector, and p A e = 0.

F=C

F = R

ZTO-graded Lie semisimple algebra and a version of the Jacobson-Morozov-Vinberg theorem for real graded semisimple Lie algebras [34, Theorem 2.1].

Using Proposition 3, Vinberg-Elashvili proposed the following scheme for their classification of 3-vectors on C9. First they classified semisimple 3-vectors p. The SL(9, C)-equivalence class of semisimple 3-vectors p has dimension 4 - the dimension of a maximal subspace consisting of commuting semisimple elements in g^. Then the equivalence classes of semisimpie elements p are divided into seven types according to the type of the stabilizer subgroup St(p) and the subspace E(p) := {x e A3C9\p Ax = 0} We assign a 3-vector on F9 to the same family as its semisimple part. Then Vinberg-Elashvili described all possible nilpotent parts for each family of 3-vectors. WThen the semisimple part is p, the latter are all the nilpotent 3-vectors e of the space E(p). The classification is made modulo the action of StSL(9,C)(p). Note that there is only finite number of nilpotent orbits in E(p) for any semisimple 3-vector p. Therefore the dimension of the orbit space A3C9/SL(9, C) is 4, which is the dimension of the space of all semisimple 3-vectors.

To classify semisimple elements p e A3C9 and nilpotent elements in E(p) Vinberg-Elashvili developed further the general method invented by Vinberg [61, 62, 63, 64] for the study of the

orbits of the adjoint action of the 0-group on ZTO-graded semisimple complex Lie algebras.

C8

we shall describe in more detail in Subsection 2.5, by Le [34] and Dietrich-Faccin-de Graaf [12] for real graded semisimple Lie algebras. As a result, we have partial results concerning the orbit space of the standard SL(9, R)-action on A3R9* (as well as partial results concerning the orbit space of

SL(8, R) A4R8*

SL(9, R) A3R9

to the classification of semisimple elements p in A3R9, which is the same as the classification of real

forms of SL(9, C)-orbits of semisimple elements p in A3C9 (the classification of the SL(9, C)-orbits

has been given in [65]) and the classification of nilpotent elements e £ A3R9 such that e A p = 0.

Note that e is a nilpotent element in the semisimple component Z(p)' of the zentralizer Z(p) of

the semisimple element p. Thus the latter problem is reduced to the classification of real forms of

complex nilpotent orbits in Z(p)^C, and the classification of the latter orbits has been done in [65].

Le's method [34] and Dietrich-Faccin-de Graaf's method of classification of nilpotent orbits of real

graded Lie algebras [12] give partial information on the real forms of these nilpotent orbits. We shall

Rs

R9

include an appendix outlining the Galois cohomologv method in this paper. 2.4. Classification of 3-forms and 5-forms on Rs

3 3 Rs

Similar to [65], see (3), Djokovic made an important observation that for F = R (resp. for F = C) the standard GL(8, Fraction on A3Fs is equivalent to the action of the adjoint group Ad Go of the Z-graded Lie algebra g = eS(S) (resp. g = es) on the homogeneous component g1 of degree 1, where

Here AdGQ = GL(8,F)/Z3 [11, Proposition 3.2], g_3 = F8*, g_2 = Л^8, g_i = Л^8*, go = gl(8,F), gi = Л^8, g2 = Л^8*, g3 = F8.

Since there is only finite number of GL(n,F)-orbits in g1; any element in g1 is nilpotent. To

g1 = A3Rs

A3C9

each nilpotent element e £ g1 a semisimple element h(e) £ go and a nilpotent element f £ g-1 that satisfy the following condition [11, Lemma 6.1]

Element h is defined by e uniquely up to conjugation and h = h(e) is called a characteristic of e [11, Lemma 6.2], see also [34, Theorem 2.1] for a general statement. Given e and h, element f is defined uniquely. A triple (h, e, f) in (7) is called an sl2-triple, which we shall denote by sl2(e). WTith help of s[2(e)-triples Djokovic classified real forms of nilpotent orbits GL(8, C) ■ e, where e £ g1 = A3Cs, as follows. Denote by ZGL(S,C)(s[2(e)) the centralizer of sl2(e) in GL(8, C). Let $ = Z2 be the Galois

CR

Galois cohomologv ($, ^GL(S,C)(s[2(e))) to the set of GL(8, R)-orbits contained in GL(8, C) ■ e [11, Theorem 8.2]. A similar argument has been first used by Revov [51] and later by Midoune and Noui for classification of alternating forms in dimension 8 over a finite field [43]. Recall that classification GL(8, C)

Vinberg-Elashvili in their classification of 3-vectors on C9. There are altogether 23 GL(8, C)-orbits A3Cs Z

es

GL(8, R)

A3Rs. The space A3Rs decomposes into 35 GL(8, R)-orbits.

GL(8, R)

A3Rs*

exists

<p £ A3Rs* such that the orbit GL(8, R) ■ ip is open in A3R8*. Such a 3-form <p is called stable. Clearly any stable 3-form (p is nondegenerate, i.e., rk^ = 8. In general, a k-form, (p on Rn is called stable, if the orbit GL(n, R) ■ <p is open in AkRn. Clearly any symplectic form is stable. It is not hard to see that if <p £ AkRn is open, and fc ^ 2, then either k = 3 and n = 5, 6, 7, 8, or fc = 4 and

g = g_3 ф g_2 Ф g_1 Ф g0 Ф gi Ф g2 Ф g3.

(6)

[h,e]=2e, [h,f ] = -2/, [e, f ] = h.

(7)

n = 6, 7, or k = 5 and n = 8. Stable forms on R8 have been studied in deep by Hitchin [26], Witt [68] and later by Le-Panak- Vanzura in [38], where they classified all stable forms on Rra (they proved that stable k-forms exist on Rn only in dimensions n = 6, 7, 8 if 3 ^ k ^ n — k), and determined their stabilizer groups [38, Theorem 4-1]-

R8

on R6 and the classification of 3-vectors on R7 by Theorem 1. The classification of 3-forms on

R7 has been first obtained by Westwick [67] by adhoc method. There are 8 equivalence classes of 3 R7

C7 3 R6

3 C6 3 R6

3 R7 3

F7

F

3

R8

are the

3 C8

the stabilizer groups StGL(8,R)(^) °f each multi-symplectic 3-form p on R8 has not been obtained till now according to our knowledge. The stabilizer StGL(8,C)(^) has been obtained by Midoune in

GL(8, R)

in

A3R8

and the centralizer ^gl(8,r)(sl2(e)) for each nilpotent element e e e8(8). It follows that the stabilizer algebra Zsl(8R)(p) of 3-forms p e A3R8 forms a complete system, of invariants of the GL(8, R) A3R8

3-form, p on R8 is not connected.

Proposition 4. For any multisymplectic 3-form p e A3R8* we have StGL(8,R)(^)RGL (8, R) = = 0. Hence the GL(8, R)-orbit of any 3-form, on R8 is connected.

Proof. For each equivalence class of a 3-form p of rank 8 we choose a canonical element p0 in the Djokovic's list [11, p. 36-37]. Then we find an element g e StGL(8,R)(^0) R GL-(8, R). Hence the GL(n, R)-orbit of each multisymplectic 3-form on R8 is connected. If p is not multisymplectic, the orbit GL(8, R) ■ p is connected by Proposition 2. This completes the proof of Proposition 4. □ Proposition 4 and Proposition 2 imply immediately the following

Corollary 1. (cf. [53, Proposition 4-1]) The Poincare map Pq induces an isomorphism between GL(8, R)-or&*'te on A3R8 and GL(8, R)-or&*'i on A5R8*. Each GL(8, R^^^i on A5R8 is connected.

R8

C8 SL(8, C)

has been given by Antonvan [1], following the scheme proposed by Vinberg-Elashvili for the

C9 R8

application of her study of the adjoint orbits in ZTO-graded real semisimple Lie algebras. In this subsection we outline Antonvan's method and Le's method.

Let F = C (resp. R). Denote by g the exceptional complex simple Lie algebra e7 (rep. e7(7)

e7

C8 R8

GL(8, F) A4F8

0-group of the Z2-graded simple Lie algebra

Q = Q0 © Q1

(8)

on its homogeneous component g1; which is isomorphie to A4Fs. Here go = sl(8,F).

go g1 F = C

of e7. Recall that e7 has the following root system:

s

£ = {£% — £j,£v + £q + £r + £s, | i = j, (p, Q, r, ^inct), ^ £i = 0}.

j=1

By Remark 1, the Z2-grading on e7 is defined uniquely bv an in volution dC of e7. In terms of the Chevallev system of e7, see Remark 2, the involution dC is defined as follows:

0C\ho =

0C(Ea) = Ea,i£a = ^ — e3,

dC(Ea) = —Ea,iia = £i + £j + £k + £i.

Note that d := #C=e7(7) 311 involution of e7(7) and it defines the induced Z2-gradation from e7 on e7(7) •

C9

SL(8, C) Cs

Cs

semisimple algebraic groups [61], which has been employed by Vinberg-Elashvili for the classification of semisimple 3-vectors as we mentioned above. Next we include each semisimple element x £ g1 of Z2 e7 g1

g1

real or complex Zm-graded semisimple Lie algebras g). If g is a complex ZTO-graded sesmisimple

g1

adjoint group GCf. To reduce the classification of semisimple elements in g1 further we introduce the notion of the Wevl group W (g, C) of a ramp lex Zm-graded semisimple Lie algebra g w.r.t. to a Cartan subalgebra C C g1 as follows. Let GC be the connected semisimple Lie algebra having the Lie algebra g and G^ the Lie subgroup of the GC having the Lie algebra go. We define

No (C) := {g £ GolVx £C g(x) £C},

Zo(C) := {g £ GolVx £C g(x) = x}.

Then W(g,C) := No(C)/Zo(C). The Weyl group W(g,C) is finite, moreover W(g,C) is generated by complex reflections, which implies that the algebra of W(g, C)-invariants on C is free [61]. Furthermore, two semisimple elements in C belong to the same GjCf-orbit if and only if they are in the same orbit of the W(g, C^^^^ton on C. showed that the Wevl group W(e7, C) has

order 2903040 and the generic semisimple element has trivial stabilizer. He also found a basis of a Cartan algebra C C g1; which is also a Cartan subalgebra of the Lie algebra e7. Thus the set of SL(8, C) Cs

depending on the type of the stabilizer of the action of the Wevl group W(e7, C) on the Cartan algebra C. For the classification of nilpotent elements and mixed 4-vectors on Cs Antonvan used the Vinberg method of support [64].

Le suggested the following scheme of classification of the SL(8, R)-orbits on A4Rs [34]. Observe

A4Rs

element and a nilpotent element [34, Theorem 2.1], as in Proposition 3. First, we classify semisimple elements, using the fact that every Cartan subspace C C g1 is conjugated to a standard Cartan subspace Co that is invariant under the action of a Cart an involution ru of th e Z2-graded Lie algebra e7(7) [47]. The set of all standard Cartan subspaces Co C g1 C g = e7(7), and more generally, the set

of all standard Cartan subspaces C C g1 in any Z2-graded real semisimple Lie algebra g, has been

classified by Matsuki and Oshima in [47]. Le decomposed each semisimple element into a sum of an

elliptic semisimple element, i.e., a semisimple element whose adjoint action on g^C = e7 has purely

imaginary eigenvalues, and a real semisimple element, i.e., a semisimple element whose adjoint

action on g^C = e7 has real eigenvalues, cf. [52] for a similar decomposition of semisimple elements

in a real sesimsimple Lie algebra. The classification of real semisimple elements and commuting

elliptic semisimple elements in C0 C g1 is then reduced to the classification of the orbits of the Wevl

groups of associated Z2-graded symmetric Lie algebras on their Cartan subalgebras [34, Corollary

R8

their semisimple parts and the corresponding nilpotent parts. The classification of nilpotent parts

can be done using algorithms in real algebraic geometry based on Le's theory of nilpotent orbits in

graded semisimple Lie algebras [34], that develops further Vinberg's method of support also called

carrier algebra. In [12] Dietrich-Faccin-de Graaf developed Vinberg's method further and applied

their method to classification of the orbits of homogeneous nilpotent elements in certain graded

real semisimple Lie algebras. In particular, they have a new proof for Djokovic's classification of R8

Remark 7. (1) The method of 9-group has been extended by Antonyan and Elashvili for classifications of spinors in dimension 16 [2].

(2) Many results of classifications of k-vectors over the fields R and C have their analogues over other fields F and their closures F [4-3]. Over the field F = Z2 the classification of 3-vectors in Fra is related to some open problems in the theory of self-dual codes [49]. Till now there is no classification of 3-vectors in Fn if n ^ 9 and F = C.

3. Geometry defined by differential forms

In this section we briefly discuss several results and open questions on the existence of differential fc-forms of given type on a smooth manifold, where k = 2, 3, 4.

• Assume that k = 2 and p is a closed 2-form with constant rank on Mn, then p is called a pre-symplectic form [60]. Till now there is no general necessary and sufficient condition for the existence of a pre-svmplectic form p on a manifold Mn except the case that p is a svmplectic form. Necessary conditions for the existence of a svmplectic form p on M2n are the existence of an almost complex structure on M2n and if M2n is closed, the existence of a cohomologv class a e H2(M2n; R) with an > 0. If M2n is open, a theorem of Gromov [18, 19] asserts that the existence of an almost complex structure is also sufficient, his argument has been generalized in [13] and used in the proof of Theorem 4(2) below. Taubes using Seiberg-Witten theory proved that there exist a closed 4-manifold M4 admitting an almost complex structure and a e H2(M, R) such that a2 = 0 but M4 has no svmplectic structure [59]. Note that for any svmplectic form w on M2n there exists uniquely up to homotopv an almost complex structure J on M2n that is compatible with w, i.e., g(X, Y) := u(X, JY) is a Riemannian metric on M2n. Connollv-Le-Ono using the Seiberg-Witten theory showed that a half of all homotopv classes of almost complex structures on a certain class of oriented compact 4-manifolds is not compatible with any svmplectic structure [9].

• Manifolds M2n endowed with a nondegenerate conformallv closed 2-form w, i.e., dw = d Aw for some closed 1-form 0 on M2n, are called conformally symplectic manifolds. A necessary condition for the existence of nondegenerate 2-form w on M2n is the existence of an almost complex structure on TM2n, which is equivalent to the existence of a section J of the associated bundle SO(2n)/U(n), see [56] where a necessary condition for the existence of a section J has been determined in terms of the WThitnev-Stiefel characteristic classes. We don't have necessary and sufficient conditions for the existence of a general conformallv svmplectic form on M2n, except the existence of an

almost complex structure on M2n. In [39] Le-Vanzura proposed new cohomologv theories of locally conformal svmplectic manifolds.

• Assume that k = 3 and p is a stable 3-form on Ms. In [46] Noui and Revov proved that the Lie algebra of the stabilizer of <p is a real form of the Lie algebra sl(3, C). Hence stable 3-forms on Rs are equivalent to the Cartan 3-forms on the real forms sl(3, R), su(1, 2^d su(3) of the complex Lie algebra sl(3, C). Later in [38] Le-Panak-Vanzura reproved the Noui-Revov result by

Rs GL(8, R)

A3Rs*

form p £ A3Rs* and found a necessary and sufficient condition for a closed orientable manifold Ms to admit a stable 3-form [38, Proposition 7.1]. In [36] Le initiated the study of geometry and topology of manifolds admitting a Cartan 3-form associated with a simple compact Lie algebra.

• Necessary and sufficient conditions for a closed connected 7-manifold M7 to admit a multisvmplectic 3-form has been determined in [54], see also Appendix 4 below. There are two equivalence classes of stable 3-forms on R7 with the stablizer groups G2 and G2 respectively. Since G2 and G2 are exceptional Riemannian and pseudo Riemannian holonomv groups, manifolds M7 Emitting stable 3-form of G2-tvpe (resp. of G2-tvpe) are in focus of research in Riemannian geometry (respectively in pseudo Riemannian geometry) [30], [35], [32]. As we have mentioned, the

study of geometries of stable forms in dimension 6,7, 8 have been initiated by Hitchin [25, 26].

•

generated by the quaternionic 4-form, the algebra of parallel forms on a Spin(7)-manifold is generated by the self-dual Cavlev 4-form. Riemannian manifolds admitting parallel 2-forms of maximal rank are Kahler manifolds, which are the most studied subjects in geometry, in particular in the theory of minimal submanifolds, see e.g., [37].

4. Manifolds admitting a ¿^-structure

In 2000 Hitchin initiated the study of geometries defined by differential forms [25], and subsequently in [26] he initiated the study of geometries defined by stable forms. The latter geometries have been investigated further in [68], [38]. A necessary and sufficient condition for a manifold M to admit a stable form p of G2-tvpe, i.e., the stabilizer of p is isomorphic to the group G2, has been found by Gray [20]. In this Appendix we state and prove a necessary and sufficient condition for a manifold M to admit a stable form p of G2-tvpe. We recall that a 3-form p on R7 is called of G2-tvpe, if it lies on the GL(R7)-orbit of a 3-form

Po = &1 A 92 A O3 + «1 A O1 + (12 A 62 + «3 a O3.

Here a1, a2 are 2-forms on R7 which can be written as

«1 = y1 A y2 + y3 A y4, 0,2 = y1 A y3 — y2 A y4, a.3 = y1 A y4 + y2 A y3

and (d1,d2,03, y1, y2, y3, y4) is an oriented basis of R7*.

Bryant showed that StGL(7,R)(^o) = G2 [7]. He also proved that G2 coincides with the automorphism group of the split octonians [7].

Theorem 4. (1) Suppose that M7 is a compact 7-manifold. Then M7 admits a 3-form of G2-type, if and only if M7 is orientable and spinnable. Equivalently the first and second Stiefel-Whitney classes of M7 vanish.

(2) Suppose that M7 is an open manifold which admits an embedding to a compact orientable and spinnable 7-manifold. Then M7 admits a closed 3-form p of G2-type.

proof. First we recall that the maximal compact Lie subgroup of <52 is SO(4). This follows from the Cartan theory on symmetric spaces. We refer to [27, p. 115] for an explicit embedding of SO(4) into G2. The reader can also check that the mage of this group is also a subgroup of G2 С GL(R7). We shall denote this image bv SO(4)3,4.

Now assume that a smooth manifold M7 admits a G2-structure. Then it must be orientable and spinnable, since the maximal compact Lie subgroup SO(4)3,4 of G2 is also a compact subgroup of the group G2.

Lemma 2. Assume that M7 is compact, orientable and spinnable. Then M7 admits a G2-structure.

Proof. Since M7 is compact, orientable and spinable, M7 admits a SU(2)-structure [16]. Since

SU(2) is a subgroup of SO(4)3,4, M7 admits a 50(4)3,4-structure. Hence M7 admits a <52-structure. □

This completes the proof of the first assertion of Theorem 4.

Let us prove the last statement of Theorem 4. Assume that M7 is a smooth open manifold which admits an embedding into a compact orientable and spinnable 7-manifold. Taking into account the first assertion of Theorem 4, there exists a 3-form p on M7 of G2-tvpe. We shall use the following theorem due to Eliashberg-Mishachev to deform the 3-form p to a closed 3-form p of G2-tvpe on M 7.

Let M be a smooth manifold and a G HP(M, R). For a subspace ft С ЛРМ we denote by Cloaft the subspace of the space Г(М, ft.) of smooth sect ions M ^ ft that consists of closed p-forms и G Г(М, ft) С Qp(M) such that [w] = a G Hp(M, R). Denote bv Diff(M) the diffeomorphism group of M.

Proposition 5 (Eliashberg-Mishashev Theorem). ([13, 10.2.1]) Let M be an open manifold, a G HP(M, R) and ft С ЛРМ an open Diff(M)-invariant subset. Then the inclusion

ClOaft ^ Г(М, ft)

is a homotopy equivalence. In particular,

- any p-form ш G Г(М, ft) is homotopiс in ft to a closed form ш;

- any homotopy wt G Г(М, ft) of p-forms which connects two closed forms ш0,ш1 such that [w0] = [wi] = a G HP(M, R) can be deformed in ft into a homotopy of closed forms ujt connecting Шо and wi such that [wt] = a for all t.

Let ft be the space of all 3-forms of G2-t^e оn M7. Clearly this фасе is an open Diff(M7)-invariant subset of Л3М7. Now we apply the Eliashberg-Mishashev theorem to the 3-form p3 of (52-type whose existence has been proved above. This completes the proof of Theorem 4. □

5. Classification of orbits over a nonclosed field of characteristic 0

by Mikhail Borovoi

We consider a linear algebraic group G with group of fc-points G(k) over an algebraically closed field к of characteristic 0. Assume that G acts on a fc-variety X with set of fc-points X(k), and assume that we know the classification of G(fc)-orbits in X(к), e.g., к = C, G = GL(9, C) X = Л3С9. Let fc0 ^e a subfield of к such that к is ад algebraic closure of fc0. We write Г = Gal(fc/fc0) for the Galois group of the extension к over к0. И k0 = R, then Г = Gal(C/R) = {1,7} where 7 is the complex

conjugation. Assume that we have compatible fc0-forms G0 of G and X0 of X. We wish to classify G0(fc0)-orbits in X0(k0). We start with one G-orbit У in X. We check whether У is Г-stable. If not, then У has no fc0-points. Assume that У is Г-stable. Then the Г-action on У defines a fc0-model У0 of У. Now G0 acts on У0 over fc0. We say that У0 is (a twisted form of) a homogeneous space of G0. We ask

(1) whether У0 has fc0-points;

(2) if the answer to (1) is positive, we wish to classify G(fc0)-orbits in У0(к0).

У

Let y0 £ У0(к0), and let H0 = St^0(y0)- Then we may write У0 = G0/H0. The Galois group Г = Gal(ft/ft0) acts compatibly on G0(k) = G(k), H0(k) = H (fc), and y(fc) = У (к) = G(k)/H (к).

Theorem 5 ([58], Section 1.5.4, Corollary 1 of Proposition 36). There is a canonical bijection between the set of orbits У0(к0)/G0(k0) and the kernel ker[# 1(k0,H0) ^ H 1(k0,G0)].

Here H 1(к0,Щ) := H 1(Г,Я0(к)).

6. Acknowledgement

The authors would like to thank Professor Alexander Elashvili and Professor Andrea Santi for their interest in this subjects and for their suggestions of references, Professor Lemnouar Noui for sending us a copy of the PhD Thesis of Midoune [42] and Professor Mahir Can for his helpful comments on a preliminary version of this paper. We are grateful to Professor Mikhail Borovoi for his help in literature and for his writing up an explanation of the Galois cohomologv method for finding real forms of complex orbits, which we put as an Appendix to this paper.

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