Classification of k-forms on R n and the existence of associated geometry on manifolds 2

In this paper we survey methods and results of classification of k-forms (resp. k-vectors on R), understood as description of the orbit space of the standard GL(n,R)-action on ΛkRn* (resp. on ΛR). 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.


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. Kähler manifolds, and more generally, Riemannian manifolds ( , ) with non-trivial holonomy group admit parallel differential forms and hence calibrations on ( , ) [27], [55], [40], [17]. In the study of Riemannian manifolds with non-trivial holonomy groups these parallel differential forms are extremely important [7], [29]. In their seminal paper [27] Harvey-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 on a manifold of dimension by finding a normal form of at a point ∈ and, if possible, to find a normal form of up to certain order in a small neighborhood ( ) ⊂ . Finding a normal form of at a point ∈ is the same as finding a canonical representative of the equivalence class of ( ) in Λ ( * ), where two -forms on are equivalent if they are in the same orbit of the standard GL( , R)-action on Λ ( * ) = Λ R * . We say that a manifold is endowed by a differential form ∈ Ω * ( ) of type 0 ∈ Λ * R * , if for all ∈ the equivalence class of ( ) ∈ Λ * * can be identified with the equivalent class of 0 ∈ Λ * R * via a linear isomorphism = R . Instead of investigation of a normal form of a concrete form , we may be also interested in a classification of (equivalent) -forms on R , understood as a description of the moduli space of equivalent -forms on R , which could give us insight on a normal form of and could also suggest interesting candidates for the geometry defined by differential forms.
Хонг Ван Ле, И. Ванжура Classification of -forms on R is a part of algebraic invariant theory. Recall that an invariant of an equivalence relation on a set , e.g., defined by orbits of an action of a group on , is a mapping from to another set 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 on R are the rank of and the stabilizer of under the action of GL( , R). Recall that the rank of , denoted by rk , is the dimension of the image of the linear operator : R → Λ −1 R * , ↦ → . We denote the stabilizer of by St GL( ,R) ( ), and in general, we denote by St ( ) the stabilizer of a point in a set where a group acts. A form ∈ Λ R * is called non-degenerate, or multisymplectic, if rk = . Furthermore, it is important to study the topology of the orbit GL( , R) · = GL( , R)/St GL( ,R) ( ), for example, the connectedness, see Proposition 2 below, the openness, the closure of the orbit GL( , R) · ⊂ Λ R * . It turns out that understanding these questions helps us to understand the structure of the orbit space of GL( , R)-action on Λ R * . These invariants of -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( , R)-orbits of -forms on R and GL( , R)-orbits of -vectors as well as the duality between GL + ( , R)-orbits of -forms on R and GL + ( , R)-orbits of ( − )-forms on R . Then we recall the classification of 2-forms on R (Theorem 2) and present the Martinet's classification of ( − 2)-forms on R (Theorem 3).
In contrast to the classification of 2-forms on R , the classification of 3-forms on R depends on the dimension . Since dim Λ 3 R * dim GL( , R) + 1, if 9, there are infinite numbers of inequivalent 3-forms in R . Till now there is no classification of the GL( , R)-action on Λ 3 R * , if 10.
In the dimension = 9 the classification of the SL(9, C)-orbits on Λ 3 C 9 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 Λ 3 R 9 . Then we review Djokovic' classification of 3-vectors in R 8 and present a classification of 5-forms on R 8 (Corollary 1). Djokovic's classification method combines some ideas from Vinberg-Elashvili's work and Galois cohomology method for classifying real forms of a complex orbit. Note that the classification of 3-vectors in R 8 implies the classification of 3-forms in R 8 (Proposition 1) as well as the classifications of 3-forms in R for 7 (Theorem 1, Remark 5). Then we review a classification of GL(8, C)-action on Λ 4 C 8 by Antonyan [1], which is important for classification of 4-forms on R 8 . At the end of Section 2 we review a scheme of classification of 4-forms on R 8 proposed by Lê in 2011 [34] and Dietrich-Facin-de Graaf's method of classification of 3-forms on R 8 in [12].
In Section 3, for = 2, 3, 4, we compile known results and discuss some open problems on necessary and sufficient topological conditions for the existence of a differential -form of given type St GL( ,R) ( ( )) on manifolds (in these cases the equivalence class of ( ) is defined uniquely by the type of the stablizer of ( ), i.e., the conjugation class of St GL( ,R) ( ( )) in GL( , R)). In dimension = 8 (and hence also for = 6, 7) we observe that the stabilizer St GL( ,R) ( ) of a 3-form ∈ Λ 3 R * forms a complete system of invariants of the action of GL( , R) on R (Remark 6).
We include two appendices in this paper. The first appendix contains a result due to Hông Vân Lê concerning the existence of 3-form of type˜2 on a smooth 7-manifold, which has been posted in arxiv in 2007 [33]. The second appendix outlines the Galois cohomology 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( , R)-orbits of -forms on R

General theorems
We begin the classification of GL( , R)-orbits on Λ R * with the following observation that the orbit of the standard action of GL( , R) on Λ R can be identified with the orbit of the standard action of GL( , R) on Λ R * by using an isomorphism ∈ (R , R * ) = R * ⊗ R * ⊃ 2 R * . Note that there are several papers and books devoted to the classification of -vectors on R [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( , R)-orbits in Λ R and GL( , R)orbits in Λ R * .
(2) The GL( , R)-orbit of ∈ Λ R * has two connected components if and only if St GL( ,R) ( ) ⊂ GL + ( , R). In other cases the GL( , R)-orbit of is connected. (3) Assume that ∈ Λ R * is degenerate. Then the GL( , R)-orbit of 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( , R) has two connected components.
3. Assume that is degenerate. Then := ker is non-empty. Let ⊥ be any complement to in R i.e., R = ⊕ ⊥ . Then GL( ) ⊕ ⊥ is a subgroup of ( ). Since this subgroup has non-trivial intersection with GL − ( , R), this implies the last assertion of Proposition 2 follows from the second one. This completes the proof of Proposition 2. 2 Хонг Ван Ле, И. Ванжура The following theorem due to Vinberg-Elashvili reduces a classification of (degenerate) -forms of rank in R to a classification of -forms on R . (Vinberg-Elashvili considered only the case = 3 and the SL( , C)-action on Λ 3 C but their argument works for any and for GL( , R)-action on Λ R * .) Theorem 1. (cf. [65, §4.4], [53, Lemma 3.2]) There is a 1-1 correspondence between GL( , R)orbits of -forms of rank less or equal to on R and GL( , R)-orbits of -forms on R .
(2) The GL( , R)-orbit of a 2-form ∈ Λ 2 R * has two connected components if and only if = 2 and has maximal rank.
(3) If is of maximal rank, then the GL( , R)-orbit of is open and its closure contains the GL( , R)-orbit of any degenerate 2-form on R .
• If 2 ( ) = , and ( ) is odd, then using Lemma 1 and Theorem 2(2) we conclude that the set of ( − 2)-forms of length consists of two open connected GL( , R)-orbits that correspond to the sign of = Ω ( ) where −1 • If 2 ( ) = and ( ) is even, using the same argument as in the previous case, we conclude that the set of (n-2)-forms of length consists of one open GL( , R)-orbit, which has two connected components.

Classification of 3-forms and 6-forms on R 9
We observe that the vector space Λ R * is a real form of the complex vector space Λ C * . Hence, for any ∈ Λ R * the orbit GL( , R) · lies in the orbit GL( , C) · . We shall say that GL( , R)· is a real form of the complex orbit GL( , C)· . 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( , R)orbits in Λ R can be reduced to the problem of classifying the real forms of the GL( , C)-orbits on Λ C . The classification of GL( , C)-orbits on Λ 3 C is trivial, if 5, cf. Proposition 2. For = 6 it was solved by W. Reichel [50]; for = 7 it was solved by J. A. Schouten [57]; for = 8 it was solved by Gurevich in 1935, see also [23]; and for = 9 it was solved by Vinberg-Elashvili [65]. In fact Vinberg-Elashvili classified SL(9, C)-orbits on Λ 3 C 9 , which are in 1-1 correspondence with SL(9, C)-orbits in Λ 3 C 9* and SL(9, C)-orbits on Λ 6 C 9* . Since the center of GL(9, C) acts on Λ 3 C 9 ∖ {0} with the kernel Z 3 , it is not hard to obtain a classification of GL(9, C)-orbits on Λ 3 C 9 , and hence on Λ 3 C 9* and on Λ 6 C 9* from the classification of the SL(9, C)-orbits on Λ 3 C 9 .
As we have remarked before, there are infinitely many GL( , C)-orbits on Λ 3 C 9 , and to solve this complicated classification problem Vinberg-Elashvili made an important observation that the standard SL(9, C)-action on Λ 3 C 9 is equivalent to the action of the adjoint group C 0 (also called the -group) of the Z 3 -graded complex simple Lie algebra = Λ 3 C 9* and C 0 = SL(9, C)/Z 3 is the connected subgroup, corresponding to the Lie subalgebra g C 0 , of the simply connected Lie group C 8 whose Lie algebra is e 8 . Remark 1. Let g C be a complex Lie algebra. Any Z -grading g C := ⊕ ∈Z g C on g C defines an automorphism ∈ Aut(g C ) of order by setting ( ) := where = exp(2 √ −1 / ) and ∈ g C . Conversely, any ∈ Aut(g C ) of order defines a Z -grading g C := ⊕ ∈Z g C by setting In [65, §2.2] Vinberg and Elashvili considered the automorphism C of order 3 on e 8 associated to the Z 3 -gradation in (6) 5 . To describe C we recall the root system Σ of e 8 : Remark 2. Given a complex semisimple Lie algebra g C let us choose a Cartan subalgebra h C 0 of g C . Let Σ be the root system of g C . Denote by { , | ∈ Σ} the Chevalley system in g C i.e., ∈ h C 0 and is the root vector corresponding to such that for any Then where Σ + ⊂ Σ denote the system of positive roots, and Σ + -the subset of simple roots.
The automorphism C of order 3 on e 8 is defined as follows 5 Automorphisms of finite order of semisimple Lie algebras have been classified earlier independently by Wolf-Gray [66] and Kac [31]. Remark 3. Let { , | ∈ Σ} be the Chevalley system of a complex semisimple Lie algebra g C . Then { , | ∈ Σ, ∈ Σ + } is a basis of the normal form g, also called split real form, of g C . The normal form of the complex simple Lie algebra e 8 is denoted by e 8 (8) , and the normal form of sl( , C) is the real simple Lie algebra sl( , R). Clearly the Lie subalgebra e 8(8) has the induced Z 3 -grading from the one on e 8 defined in (3) Hence there is a 1-1 correspondence between SL(9, R)-orbits on Λ 3 R 9* and the adjoint action of the subgroup 0 , corresponding to the Lie subalgebra g 0 , of the Lie group C 0 . 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 ∈ Λ 3 F 9 semisimple, if its orbit SL(9, F) · is closed in Λ 3 F 9 , and nilpotent, if the closure of its orbit SL(9, F) · 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.
(The definition of the rank of a -vector can be defined in the same way as the definition of the rank of a -form). Then for any ∈ R there exists an element ∈ SL(9, F) such that · = · . Hence the closure of the orbit SL(9, F) · contains 0 ∈ Λ 3 F 9 and therefore is a nilpotent element. Proposition 3. Every nonzero 3-vector in Λ 3 F 9 can be uniquely written as = + , where is a semisimple 3-vector, -a nilpotent 3-vector, and ∧ = 0. Proposition 3 has been obtained by Vinberg-Elashvili in [65] for the case F = C. To prove Proposition 3 for F = R, we use the Jordan decomposition of a homogeneous element in a real Z -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 C 9 . First they classified semisimple 3-vectors . The SL(9, C)-equivalence class of semisimple 3-vectors has dimension 4 -the dimension of a maximal subspace consisting of commuting semisimple elements in g 1 . Then the equivalence classes of semisimple elements are divided into seven types according to the type of the stabilizer subgroup St( ) and the subspace ( ) := { ∈ Λ 3 C 9 | ∧ = 0}. We assign a 3-vector on F 9 to the same family as its semisimple part. Then Vinberg-Elashvili described all possible nilpotent parts for each family of 3-vectors. When the semisimple part is , the latter are all the nilpotent 3-vectors of the space ( ). The classification is made modulo the action of St SL(9,C) ( ). Note that there is only finite number of nilpotent orbits in ( ) for any semisimple 3-vector . Therefore the dimension of the orbit space Λ 3 C 9 /SL(9, C) is 4, which is the dimension of the space of all semisimple 3-vectors.
To classify semisimple elements ∈ Λ 3 C 9 and nilpotent elements in ( ) 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 -group on Z -graded semisimple complex Lie algebras.
Vinberg's method has been developed by Antonyan for classification of 4-forms in C 8 , which we shall describe in more detail in Subsection 2.5, by Lê [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 Λ 3 R 9* (as well as partial results concerning the orbit space of the standard action of SL(8, R) on Λ 4 R 8* we mentioned above). By Proposition 3, and following Vinberg-Elashvili scheme, the classification of the orbits of SL(9, R)-action on Λ 3 R 9 can be reduced to the classification of semisimple elements in Λ 3 R 9 , which is the same as the classification of real forms of SL(9, C)-orbits of semisimple elements in Λ 3 C 9 (the classification of the SL(9, C)-orbits has been given in [65]) and the classification of nilpotent elements ∈ Λ 3 R 9 such that ∧ = 0. Note that is a nilpotent element in the semisimple component ( ) ′ of the zentralizer ( ) of the semisimple element . Thus the latter problem is reduced to the classification of real forms of complex nilpotent orbits in Z( ) ′ ⊗C , and the classification of the latter orbits has been done in [65]. Lê'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 discuss a similar scheme of classification of 4-forms on R 8 in Subsection 2.5. Currently we consider the Galois cohomology method for classification of 3-forms on R 9 promising [4], and therefore we include an appendix outlining the Galois cohomology method in this paper.

Classification of 3-forms and 5-forms on R 8
The classification of 3-vectors (and hence 3-forms) on R 8 has been given by Djokovic in [11]. Similar to [65], see (3), Djokovic made an important observation that for F = R (resp. for F = C) the standard GL(8, F)-action on Λ 3 F 8 is equivalent to the action of the adjoint group Ad 0 of the Z-graded Lie algebra g = e 8(8) (resp. g = e 8 ) on the homogeneous component g 1 of degree 1, where Here Since there is only finite number of GL( , F)-orbits in g 1 , any element in g 1 is nilpotent. To study nilpotent elements in g 1 = Λ 3 R 8 , as Vinberg-Elashvili did for complex nilpotent 3-vectors on Λ 3 C 9 , Djokovic used a real version of Jacobson-Morozov-Vinberg's theorem that associates with each nilpotent element ∈ g 1 a semisimple element ℎ( ) ∈ g 0 and a nilpotent element ∈ g −1 that satisfy the following condition [11, Lemma 6.1] Element ℎ is defined by uniquely up to conjugation and ℎ = ℎ( ) is called a characteristic of [11, Lemma 6.2], see also [34, Theorem 2.1] for a general statement. Given and ℎ, element is defined uniquely. A triple (ℎ, , ) in (7) is called an sl 2 -triple, which we shall denote by sl 2 ( ). With help of sl 2 ( )-triples Djokovic classified real forms of nilpotent orbits GL(8, C) · , where ∈ g 1 = Λ 3 C 8 , as follows. Denote by GL(8,C) (sl 2 ( )) the centralizer of sl 2 ( ) in GL(8, C). Let Φ = Z 2 be the Galois group of the field extension of C over R. Then Djokovic proved that there is a bijection from the Galois cohomology (Φ, GL(8,C) (sl 2 ( ))) to the set of GL(8, R)-orbits contained in GL(8, C) · [11, Theorem 8.2]. A similar argument has been first used by Revoy [51] and later by Midoune and Noui for classification of alternating forms in dimension 8 over a finite field [43]. Recall that classification of GL(8, C)-orbits has been obtained by Gurevich and later this classification is also re-obtained by Vinberg-Elashvili in their classification of 3-vectors on C 9 . There are altogether 23 GL(8, C)-orbits on Λ 3 C 8 . In [11] Djokovic gave another proof of this classification using the Z-graded Lie algebra e 8 in (6). Finally Djokovic computed the related Galois cohomology to obtain the number of real forms of each complex orbit and he also found a canonical representation of each GL(8, R)-orbit on Λ 3 R 8 . The space Λ 3 R 8 decomposes into 35 GL(8, R)-orbits.

Remark 4.
Since there is only finite number of GL(8, R)-orbits on Λ 3 R 8* , there exists ∈ Λ 3 R 8* such that the orbit GL(8, R) · is open in Λ 3 R 8* . Such a 3-form is called stable. Clearly any stable 3-form is nondegenerate, i.e., rk = 8. In general, a -form on R is called stable, if the orbit GL( , R) · is open in Λ R . Clearly any symplectic form is stable. It is not hard to see that if ∈ Λ R is open, and 2, then either = 3 and = 5, 6, 7, 8, or = 4 and = 6, 7, or = 5 and = 8. Stable forms on R 8 have been studied in deep by Hitchin [26], Witt [68] and later by Lê-Panak-Vanžura in [38], where they classified all stable forms on R (they proved that stable -forms exist on R only in dimensions = 6, 7, 8 if 3 − ), and determined their stabilizer groups [38, Theorem 4.1]. Remark 5. Djokovic's classification of 3-vectors on R 8 contains the classification of 3-vectors on R 6 and the classification of 3-vectors on R 7 by Theorem 1. The classification of 3-forms on R 7 has been first obtained by Westwick [67] by adhoc method. There are 8 equivalence classes of multisymplectic 3-forms on R 7 , which are the real forms of 5 equivalent classes of multisymplectic 3-forms on C 7 , and there are 6 equivalence classes of 3-forms on R 6 , which are the real forms of 5 equivalence classes of 3-forms on C 6 . The stabilizer of 3-forms in R 6 has been determined in [25] and the stabilizer of multisymplectic 3-forms in R 7 has been defined in [6]. The stabilizer of 3-forms on F 7 has been described by Cohen-Helminck in [8, Theorem 2.1] for any algebraically closed field F. Remark 6. There are 21 equivalence classes of multisymplectic 3-forms on R 8 which are the real forms of 13 equivalence classes of multisymplectic 3-forms on C 8 [11, §9]. A complete list of the stabilizer groups St GL(8,R) ( ) of each multi-symplectic 3-form on R 8 has not been obtained till now according to our knowledge. The stabilizer St GL(8,C) ( ) has been obtained by Midoune in his PhD Thesis [42], see also [43]. In [11] Djokovic computed the dimension of each GL(8, R)-orbit in Λ 3 R 8 and the centralizer GL(8,R) (sl 2 ( )) for each nilpotent element ∈ e 8 (8) . It follows that the stabilizer algebra gl(8,R) ( ) of 3-forms ∈ Λ 3 R 8 forms a complete system of invariants of the GL(8, R)-action on Λ 3 R 8 . In Proposition 4 below we show that the stabilizer of any multisymplectic 3-form on R 8 is not connected.

Classification of 4-forms on R 8
Classification of 4-forms on C 8 , whose equivalence is defined via the standard action of SL(8, C), has been given by Antonyan [1], following the scheme proposed by Vinberg-Elashvili for the classification of 3-vectors on C 9 . In [34] Lê proposed a scheme of classification of 4-forms on R 8 as application of her study of the adjoint orbits in Z -graded real semisimple Lie algebras. In this subsection we outline Antonyan's method and Lê's method.
Let F = C (resp. R). Denote by g the exceptional complex simple Lie algebra e 7 (rep. e 7(7) -the split form of e 7 ). The starting point of Antonyan's work on the classification on 4-vectors on C 8 (resp. the starting point of Lê's scheme of classification of 4-forms on R 8 ) is the following observation, cf. (3), (5). The standard GL(8, F)-action on Λ 4 F 8 is equivalent to the action of the -group of the Z 2 -graded simple Lie algebra on its homogeneous component g 1 , which is isomorphic to Λ 4 F 8 . Here g 0 = sl(8, F). Let us describe the components g 0 and g 1 in (8) for the case F = C using the root decomposition of e 7 . Recall that e 7 has the following root system: By Remark 1, the Z 2 -grading on e 7 is defined uniquely by an involution C of e 7 . In terms of the Chevalley system of e 7 , see Remark 2, the involution C is defined as follows: Note that := C |g=e 7 (7) is an involution of 7 (7) and it defines the induced Z 2 -gradation from e 7 on e 7 (7) .
Following the Vinberg-Eliashivili scheme of the classification of 3-vectors on C 9 , Antonyan classified SL(8, C)-equivalent 4-vectors on C 8 by using the Jordan decomposition (Proposition 3). First he classified all semisimple 4-vectors on C 8 using Vinberg's theory on finite automorphisms of 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 ∈ g 1 of the Z 2 -graded complex Lie algebra e 7 into a Cartan subalgebra of g 1 , which is defined as a maximal subspace in g 1 consisting of commuting semisimple elements [63] (this definition is also applied to real or complex Z -graded semisimple Lie algebras g). If g is a complex Z -graded sesmisimple Lie algebra, then all the (complex) Cartan subalgebras in g 1 are conjugate under the action of the adjoint group C 0 . To reduce the classification of semisimple elements in g 1 further we introduce the notion of the Weyl group (g, ) of a complex Z -graded semisimple Lie algebra g w.r.t. to a Cartan subalgebra ⊂ g 1 as follows. Let C be the connected semisimple Lie algebra having the Lie algebra g and C 0 the Lie subgroup of the C having the Lie algebra g 0 . We define Then (g, ) := 0 ( )/ 0 ( ). The Weyl group (g, ) is finite, moreover (g, ) is generated by complex reflections, which implies that the algebra of (g, )-invariants on is free [61]. Furthermore, two semisimple elements in belong to the same C 0 -orbit if and only if they are in the same orbit of the (g, )-action on . Antonyan showed that the Weyl group (e 7 , ) has order 2903040 and the generic semisimple element has trivial stabilizer. He also found a basis of a Cartan algebra ⊂ g 1 , which is also a Cartan subalgebra of the Lie algebra e 7 . Thus the set of SL(8, C)-equivalent semisimple 4-vectors on C 8 has dimension 7. This set is divided into 32 families depending on the type of the stabilizer of the action of the Weyl group (e 7 , ) on the Cartan algebra . For the classification of nilpotent elements and mixed 4-vectors on C 8 Antonyan used the Vinberg method of support [64].
Lê suggested the following scheme of classification of the SL(8, R)-orbits on Λ 4 R 8 [34]. Observe that we also have the Jordan decomposition of each element in Λ 4 R 8 into a sum of a semisimple 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 ⊂ g 1 is conjugated to a standard Cartan subspace 0 that is invariant under the action of a Cartan involution u of the Z 2 -graded Lie algebra e 7(7) [47]. The set of all standard Cartan subspaces 0 ⊂ g 1 ⊂ g = e 7(7) , and more generally, the set Хонг Ван Ле, И. Ванжура of all standard Cartan subspaces ⊂ g 1 in any Z 2 -graded real semisimple Lie algebra g, has been classified by Matsuki and Oshima in [47]. Lê decomposed each semisimple element into a sum of an elliptic semisimple element, i.e., a semisimple element whose adjoint action on g ⊗C = e 7 has purely imaginary eigenvalues, and a real semisimple element, i.e., a semisimple element whose adjoint action on g ⊗C = e 7 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 0 ⊂ g 1 is then reduced to the classification of the orbits of the Weyl groups of associated Z 2 -graded symmetric Lie algebras on their Cartan subalgebras [34,Corollary 5.3]. As in [65] and [1], the classification of mixed 4-vectors on R 8 is reduced to the classification of their semisimple parts and the corresponding nilpotent parts. The classification of nilpotent parts can be done using algorithms in real algebraic geometry based on Lê'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 3-vectors on R 8 . Remark 7. (1) The method of -group has been extended by Antonyan and Elashvili for classifications of spinors in dimension 16 [2].
(2) Many results of classifications of -vectors over the fields R and C have their analogues over other fields F and their closures F [43]. Over the field F = Z 2 the classification of 3-vectors in F is related to some open problems in the theory of self-dual codes [49]. Till now there is no classification of 3-vectors in F if 9 and F ̸ = C.

Geometry defined by differential forms
In this section we briefly discuss several results and open questions on the existence of differential -forms of given type on a smooth manifold, where = 2, 3, 4.
• Assume that = 2 and is a closed 2-form with constant rank on , then is called a pre-symplectic form [60]. Till now there is no general necessary and sufficient condition for the existence of a pre-symplectic form on a manifold except the case that is a symplectic form. Necessary conditions for the existence of a symplectic form on 2 are the existence of an almost complex structure on 2 and if 2 is closed, the existence of a cohomology class ∈ 2 ( 2 ; R) with > 0. If 2 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 4 admitting an almost complex structure and ∈ 2 ( , R) such that 2 ̸ = 0 but 4 has no symplectic structure [59]. Note that for any symplectic form on 2 there exists uniquely up to homotopy an almost complex structure on 2 that is compatible with , i.e., ( , ) := ( , ) is a Riemannian metric on 2 . Connolly-Lê-Ono using the Seiberg-Witten theory showed that a half of all homotopy classes of almost complex structures on a certain class of oriented compact 4-manifolds is not compatible with any symplectic structure [9].
• Manifolds 2 endowed with a nondegenerate conformally closed 2-form , i.e., = ∧ for some closed 1-form on 2 , are called conformally symplectic manifolds. A necessary condition for the existence of nondegenerate 2-form on 2 is the existence of an almost complex structure on 2 , which is equivalent to the existence of a section of the associated bundle SO(2 )/U( ), see [56] where a necessary condition for the existence of a section has been determined in terms of the Whitney-Stiefel characteristic classes. We don't have necessary and sufficient conditions for the existence of a general conformally symplectic form on 2 , except the existence of an almost complex structure on 2 . In [39] Lê-Vanžura proposed new cohomology theories of locally conformal symplectic manifolds.
• Assume that = 3 and is a stable 3-form on 8 . In [46] Noui and Revoy proved that the Lie algebra of the stabilizer of is a real form of the Lie algebra sl(3, C). Hence stable 3-forms on R 8 are equivalent to the Cartan 3-forms on the real forms sl(3, R), s (1, 2) and s (3) of the complex Lie algebra sl (3, C). Later in [38] Lê-Panak-Vanžura reproved the Noui-Revoy result by associating to each 3-form on R 8 various bilinear forms, which are invariants of the GL(8, R)-action on Λ 3 R 8* , and studied properties of these forms. They computed the stabilizer group of a stable form ∈ Λ 3 R 8* and found a necessary and sufficient condition for a closed orientable manifold 8 to admit a stable 3-form [38,Proposition 7.1]. In [36] Lê 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 7 to admit a multisymplectic 3-form has been determined in [54], see also Appendix 4 below. There are two equivalence classes of stable 3-forms on R 7 with the stablizer groups 2 and˜2 respectively. Since 2 and˜2 are exceptional Riemannian and pseudo Riemannian holonomy groups, manifolds 7 admitting stable 3-form of 2 -type (resp. of˜2-type) 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].
• It is worth noting that the algebra of parallel forms on a quaternion Kähler manifold is generated by the quaternionic 4-form, the algebra of parallel forms on a Spin(7)-manifold is generated by the self-dual Cayley 4-form. Riemannian manifolds admitting parallel 2-forms of maximal rank are Kähler manifolds, which are the most studied subjects in geometry, in particular in the theory of minimal submanifolds, see e.g., [37].

Manifolds admitting a˜2-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 to admit a stable form of 2 -type, i.e., the stabilizer of is isomorphic to the group 2 , has been found by Gray [20]. In this Appendix we state and prove a necessary and sufficient condition for a manifold to admit a stable form of˜2-type. We recall that a 3-form on R 7 is called of˜2-type, if it lies on the GL(R 7 )-orbit of a 3-form Here 1 , 2 are 2-forms on R 7 which can be written as Bryant showed that St GL(7,R) ( 0 ) =˜2 [7]. He also proved that˜2 coincides with the automorphism group of the split octonians [7]. Theorem 4. (1) Suppose that 7 is a compact 7-manifold. Then 7 admits a 3-form of˜2type, if and only if 7 is orientable and spinnable. Equivalently the first and second Stiefel-Whitney classes of 7 vanish.
(2) Suppose that 7 is an open manifold which admits an embedding to a compact orientable and spinnable 7-manifold. Then 7 admits a closed 3-form of˜2-type.
Proof. First we recall that the maximal compact Lie subgroup of˜2 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 2 . The reader can also check that the image of this group is also a subgroup of˜2 ⊂ GL(R 7 ). We shall denote this image by (4) 3,4 . Now assume that a smooth manifold 7 admits a˜2-structure. Then it must be orientable and spinnable, since the maximal compact Lie subgroup SO(4) 3,4 of 2 is also a compact subgroup of the group 2 . Lemma 2. Assume that 7 is compact, orientable and spinnable. Then 7 admits a˜2structure.
Let us prove the last statement of Theorem 4. Assume that 7 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 on 7 of˜2-type. We shall use the following theorem due to Eliashberg-Mishachev to deform the 3-form to a closed 3-form¯of˜2-type on 7 .

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 cohomology method for finding real forms of complex orbits, which we put as an Appendix to this paper.