Coadjoint orbits of Lie algebras and Cartan class

We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $\mathcal{O}(\alpha)$ at the point $\alpha$ corresponds to the characteristic space associated to the left invariant form;$\alpha$ and its dimension is the even part of the Cartan class of $\alpha$. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension $2n$ or $2n+1$ having an orbit of dimension $2n$.


Introduction
Let G be a connected Lie group, g its Lie algebra and g * the dual vector space of g. We identify g with the Lie algebra of left invariant vector fields on G and g * with the vector space of left invariant Pfaffian forms on G. The Lie group G acts on g * by the coadjoint action. If α belongs to g * , its coadjoint orbit O(α) associated with this action is reduced to a point if α is closed for the adjoint cohomology of g. If not, the coadjoint orbit is an evendimensional manifold provided with a symplectic structure. Such manifolds are interesting because any symplectic homogeneous manifold is of type O(α) for some Lie group G and α ∈ g * . From the Kirillov theory, if G is a connected and simply connected nilpotent Lie group, the set of coadjoint orbits coincides with the set of equivalent classes of unitary representations of this Lie group. In this work, we establish a link between the dimension of the coadjoint orbit of the form α and cl(α) its class in Elie Cartan's sense. More precisely dim O(α) = 2 cl(α) 2 . Recall that the Cartan class of α corresponds to the number of independent Pfaffian forms needed to define α and its differential dα and it is equal to the codimension of the characteristic space. The dimension of O(α) results in a natural relation between this characteric space and the tangent space at the point α to the orbit O(α).
As applications, we describe classes of Lie algebras with additional properties related to its coadjoint orbits. For example, we determine all Lie algebras whose nonsingular orbits are all of dimension 2 or 4 and also the Lie algebras of dimension 2p or 2p + 1 admitting a maximal orbit of dimension 2p that is admitting α ∈ g * such that cl(α) ≥ 2p.

Dimension of coadjoint orbits and Cartan class
2.1. Cartan class of a Pfaffian form. Let M be a n-dimensional differentiable manifold and α a Pfaffian form on M, that is a differential form of degree 1. The characteristic space of α at a point x ∈ M is the linear subspace C x (α) of the tangent space T x (M) of M at the point x defined by where A(α(x)) = {X x ∈ T x (M), α(x)(X x ) = 0} is the associated subspace of α(x), is the associated subspace of dα(x) and i(X x )dα(x)(Y x ) = dα(x)(X x , Y x ). Definition 1. Let α be a Pfaffian form on the differential manifold M. The Cartan class of α at the point x ∈ M is the codimension of the characteristic space C x (α) in the tangent space T x (M) to M at the point x. We denote it by cl(α)(x).
The function x → cl(α)(x) is with positive integer values and is semi-continuous, that is, if x 1 ∈ M is in a neighborhood of x, then cl(α)(x 1 ) ≥ cl(α)(x).
If the function cl(α)(x) is constant, that is, cl(α)(x) = cl(α)(y) for any x, y ∈ M, we say that the Pfaffian form α is of constant class and we denote by cl(α) this constant. The distribution x → C x (α) is then regular and it is an integrable distribution of dimension n − cl(α), called the characteristic distribution of α. It is equivalent to say that the Pfaffian system If M = G is a connected Lie group, we identify its Lie algebra g with the space of left invariant vector fields and its dual g * with the space of left invariant Pfaffian forms. Then if α ∈ g * , the differential dα is the 2-differential left invariant form belonging to Λ 2 (g * ) and defined by for any X, Y ∈ g. It is obvious that any left invariant form α ∈ g * is of constant class and we will speak on the Cartan class cl(α) of a linear form α ∈ g * . We have • cl(α) = 2p + 1 if and only if α ∧ (dα) p = 0 and (dα) p+1 = 0, • cl(α) = 2p if and only if (dα) p = 0 and α ∧ (dα) p = 0.
Definition 3. Let g be an n-dimensional Lie algebra. If α ∈ g * is neither a contact nor a frobeniusian form, the characteristic space C(α) = C e (α) at the unit e of G is not trivial and the characteristic distribution on G given by C x (α) with x ∈ G has a constant non zero dimension. As it is integrable, the subspace C(α) is a Lie subalgebra of g.
Then C(α) is an abelian subalgebra of g.
Recall some properties of the class of a linear form on a Lie algebra. The proofs of these statements are given in [19,14,16] • If g is a finite dimensional nilpotent Lie algebra, then the class of any non zero α ∈ g * is always odd.
• A real or complex finite dimensional nilpotent Lie algebra is never a Frobenius Lie algebra. More generally, an unimodular Lie algebra is non frobeniusian [10].
• Let g be a real compact Lie algebra. Any non trivial α ∈ g * has an odd Cartan class.
• Let g be a real or complex semi-simple Lie algebra of rank r. Then any α ∈ g * satisfies cl(α) ≤ n − r + 1. In particular, a semi-simple Lie algebra is never a Frobenius algebra. A semi-simple Lie algebra is a contact Lie algebra if its rank is 1, that is, g is isomorphic to sl(2, R) or so(3).
• The Cartan class of any linear non trivial form on a simple non exceptional Lie algebra of rank r satisfies cl(α) ≥ 2r. Moreover, if g is isomorphic of type A r , there exists a linear form of class 2r which reaches the lower bound.
• Any 2p + 1-dimensional contact real Lie algebra such that any non trivial linear form is a contact form is isomorphic to so(3).

2.2.
Cartan class and the index of a Lie algebra. For any α ∈ g * , we consider the stabilizer g α = {X ∈ g, α • adX = 0} and d the minimal dimension of g α when α lies in g * . It is an invariant of g which is called the index of g. If α satisfies dim g α = d then g α is an abelian subalgebra of g. Considering the Cartan class of α, g α is the associated subspace of dα: g α = A(dα), so the minimality is realized by a form of maximal class and we have d = n − cl(α) + 1 if the Cartan class cl(α) is odd or d = n − cl(α) if cl(α) is even. In particular (1) If g is a 2p-dimensional Frobeniusian Lie algebra, then the maximal class is 2p and d = 0. (2) If g is (2p + 1)-dimensional contact Lie algebra, then d = n − n + 1 = 1.
2.3. The coadjoint representation. Let G be a connected Lie group an g its Lie algebra. The adjoint representation of G on g is the homomorphism of groups : Ad : G → Aut(g) defined as follows. For every x ∈ G, let A(x) be the automorphism of G given by A(x)(y) = xyx −1 . This map is differentiable and the tangent map to the identity e of G is an automorphism of g. We denote it by Ad(x). Definition 6. The coadjoint representation of G on the dual g * of g is the homomorphism of groups: for any α ∈ g * and X ∈ g.
The coadjoint representation is sometimes called the K-representation. For α ∈ g * , we denote by O(α) its orbit, called the coadjoint orbit, for the coadjoint representation. The following result is classical [22]: any coadjoint orbit is an even dimensional differentiable manifold endowed with a symplectic form.
Let us compute the tangent space to this manifold O(α) at the point α.
Proposition 7. Consider a non zero α ∈ g * . The tangent space to O(α) at the point α is isomorphic to the dual space A * (dα) of the associated space A(dα) of dα. An immediate application is Proposition 9. Any (2p + 1)-dimensional Lie algebra with all coadjoint orbits O(α) are of maximal dimension 2p for α = 0, is isomorphic to so(3) or sl(2, R) and then of dimension 3.
Proof. From [19,14] any contact Lie algebra such that any non trivial linear form is a contact form is isomorphic to so (3). Assume now that any non trivial form on g is of Cartan class equal to 2p or 2p + 1. With similar arguments developed in [19,14], we prove that such a Lie algebra is semi-simple. But if g is simple of rank r, we have 2r ≤ cl(α) ≤ 2p + 2 − r. Then r = 1 and g is isomorphic to sl(2, R).
Remark. Assume that dim g = 2p and all the nonsingular coadjoint orbits are also of dimension 2p. Then for any non trivial α ∈ g * , cl(α) = 2p. If I is a non trivial abelian ideal of g, there exists ω ∈ g * , ω = 0 such that ω(X) = 0 for any X ∈ I. The Cartan class of this form ω is smaller than 2p. Then g is semi-simple. But the behavior of the Cartan class on simple Lie algebra leads to a contradiction.
We deduce also from the previous corollary: (1) If g is isomorphic to the (2p + 1)-dimensional Heisenberg algebra, then any nontrivial coadjoint orbit is of dimension 2p.
(2) If g is isomorphic to the graded filiform algebra L n , then any non trivial coadjoint orbit is of dimension 2.
(3) If g is isomorphic to the graded filiform algebra Q n , then any non trivial coadjoint orbit is of dimension 2 or n − 2.
(5) If g is a complex classical simple Lie algebra of rank r, then the maximal dimension of the coadjoint orbits is equal to n − r is this number is even, if not to n − r − 1 (see [14]).
3. Lie algebras whose coadjoint orbits are of dimension 2 or 0 In this section, we determine all Lie algebras whose coadjoint orbits are of dimension 2 or 0. This problem was initiated in [4,5]. This is equivalent to say that the Cartan class of any linear form is smaller or equal to 3. If g is a Lie algebra having this property, any direct product g 1 = g I of g by an abelian ideal I satisfies also this property. We shall describe these Lie algebras up to an abelian direct factor, that is indecomposable Lie algebras. It is obvious to see that for any Lie algebra of dimension 2 or 3, the dimensions of the coadjoint orbits are equal to 2 or 0. We have also seen: Proposition 11. Let g be a simple Lie algebra of rank 1. Then for any α = 0 ∈ g * , dim O(α) = 2. Conversely, if g is a Lie algebra such that dim O(α) = 2 for any α = 0 ∈ g * then g is simple of rank 1. Now we examine the general case. Assume that g is a Lie algebra of dimension greater or equal to 4 such that for any nonzero α ∈ g * we have cl(α) = 3, 2 or 1.
Since g is indecomposable, for any X ∈ A(dω 3 ) and X / ∈ D(g), there exists X 12 ∈ R{X 1 , X 2 } such that [X 12 , X] = 0. We deduce Proposition 12. Let g an indecomposable Lie algebra such that the dimension of the nonsingular coadjoint orbits is 2. We suppose that there exists ω ∈ g * such that cl(ω) = 3. If n ≥ 7 then g = t ⊕ I n−1 where I n−1 is an abelian ideal of codimension 1.
It remain to study the particular cases of dimension 4, 5 and 6. The previous remarks show that: • If dim g = 4 then g is isomorphic to one of the following Lie algebra given by its Maurer- where I 3 is an abelian ideal of dimension 3.
• If dim g = 5 then g is isomorphic to one of the following Lie algebra where I 4 is an abelian ideal of dimension 4.
• If dim g = 6 then g is isomorphic to one of the following Lie algebra where I 5 is an abelian ideal of dimension 5.

Remarks
1. Among the Lie algebras g = t ⊕ I n−1 where I n−1 is an abelian ideal of dimension n − 1 there exist a family of nilpotent Lie algebras which are the "model" for a given characteristic sequence (see [23]). They are the nilpotent Lie algebras L n,c , c ∈ {(n − 1, 1), (n − 3, 2, 1), · · · , (2, 1, · · · , 1)} defined by The characteristic sequence c corresponds de c(U) and {X 1 , · · · , X n k−1 } is a Jordan basis of adU. We shall return to this notion in the next section.
2. Let U(g) be the universal enveloping algebra of g and consider the category U(g)−Mod of right U(g)-module. Then if g is a Lie algebra described in this section (that is with coadjoint orbits of dimension 0 or 2) thus any U(g)-mod satisfy the property that "any injective hulls of simple right U(g)-module are locally Artinian" (see [18]). 4. Lie algebras whose non singular coadjoint orbit are of dimension 4 We generalize some results of the previous section, considering here real Lie algebras such that for a fixed p ∈ N, dim O(ω) = 2p or 0 for any ω ∈ g * . We are interested, in this section, in the case p = 2. The Cartan class of any non closed linear form is equal to 5 or 4.
Lemma 13. Let g be a Lie algebra whose Cartan class of any non trivial and non closed linear form is 4 or 5. Then g is solvable.
Proof. If g is a simple Lie algebra of rank r and dimension n, then the Cartan class of any linear form ω ∈ g * satisfies c ≤ n − r + 1 and this upper bound is reached. Then n − r + 1 = 4 or 5 and the only possible case is for r = 2 and g = so (4). Since this algebra is compact, the Cartan class is odd. We can find a basis of so(4) whose corresponding Maurer-Cartan equations are , If each of the linear forms of this basis has a Cartan class equal to 5, it is easy to find a linear form, for example ω 1 + ω 6 , of Cartan class equal to 3. Then g is neither simple nor semi-simple. This implies also that the Levi part of a non solvable Lie algebra is also trivial, then g is solvable.
As consequence, g contains a non trivial abelian ideal. From the result of the previous section, the codimension of this ideal I is greater or equal to 2 and g = t ⊕ I. Assume in a first step that dim t = 3. In this case g/I is abelian, that is [t, t] ⊂ I. This implies that for any ω ∈ t * we have dω = 0 and we obtain, considering the dimension of [t, t], the following Lie algebras which are nilpotent because the Cartan class is always odd: which is of dimension 7 sometimes called the Kaplan Lie algebra or the generalized Heisenberg algebra. (2) which is of dimension 8.
Assume now that dim t = 4. In this case g/I is abelian or isomorphic to the solvable Lie algebra whose Maurer-Cartan equations are    dω 2 = dω 4 = 0, with a = 0. Let us assume that g/I is not abelian. Let {X 1 , · · · , X n } be a basis of g such that {X 1 , · · · , X 4 } is the basis of t dual of ω 1 , · · · , ω 4 } and {X 5 , · · · , X n } a basis of I. Since I is maximal, then [X 1 , I] and [X 3 , I] are not trivial. There exists a vector of I, for example, X 5 such that [X 1 , X 5 ] = 0. Let us put [X 1 , X 5 ] = Y with Y ∈ I and let be ω its dual form. Then with ω 1 ∧ ω 5 ∧ θ = 0 and ω 3 ∧ ω 4 ∧ θ = ω 3 ∧ ω 2 ∧ θ = 0 if not there exists a linear form of class greater that 5. This implies they there exists ω 6 independent with ω 5 dω = ω 1 ∧ ω 5 + bω 3 ∧ ω 6 with b = 0. Now the Jacobi condition's which are equivalent to d(dω) = 0 implies that we cannot have mega = ω 5 and omega = ω 7 . Then we put ω = ω 7 . This implies We deduce Proposition 14. If g = t ⊕ I where I is a maximal abelian ideal of codimension 4, then g is isomorphic to the Lie algebra whose Maurer-Cartan equations are with a 1 a 2 a 3 a 4 = 0.
If dim t ≥ 5, then dim A(ω) = 4 or 0 and the codimension of I is greater than n − 4. Then dim t ≤ 4.
It remains to describe the action of t on I when we consider the second case. Assume that g = t ⊕ I and dim t = 2. Let {T 1 , T 2 } be a basis of t. Then g = g/K{T 2 } is a Lie algebra whose any non closed linear form is of class 2 or 3. Such Lie algebra is described in Proposition 12.
Proposition 16. Let g be a Lie algebra whose the dimension of the coadjoint orbit of any non closed linear form is 4 such that g = t⊕I where I is a codimension 2 abelian ideal. Then g is a one dimensional extension by a derivation f of g such that f (T 1 ) = 0, Im(f ) = Im(adT 1 ) and for any Y ∈ Im(adT 1 ), there exits X 1 , X 2 ∈ I linearly independent such that

Examples.
• dim g = 4. Then dim g = 3 and it is isomorphic to to one of the two algebras whose Lie brackets are given by In the first case, it is easy to see that we cannot find derivation of g satisfying Proposition 16. In the second case the matrix of f in the basis We have no solution if a = 1. If a = 1 the f satisfies (e − b) 2 + 4cd < 0.
In particular for b = e = 0 and c = 1 we obtain Proposition 17. Any 4-dimensional Lie algebra whose the coadjoint orbit of non closed linear form are of dimension 4 is isomorphic to tthe following Lie algebra g 4 (λ) whose Maurer-Cartan equations are    dα 1 = dα 2 = 0, • dim g = 5. Let us put g = K{T 1 } ⊕ I. Let h 1 be the restriction of adT 1 to I. It is an endomorphism of I and since dim I = 3, it admits an eigenvalues λ. Assume that λ = 0 and let X 1 be an associated eigenvector. Then, since f is a derivation commuting with adT 1 , Then f (X 1 ) is also an eigenvector associated with λ . By hypothesis X 1 and f (X 1 ) are independent and λ is a root of order 2. Thus h 1 is semi-simple. Let λ 2 be the third eigenvalue. If X 3 is an associated eigenvector, as above f (X 3 ) is also an eigenvector and λ 2 is a root of order 2 except if λ 2 = λ 1 . Then Lemma 18. If dim I is odd, then if the restriction h 1 of adT 1 to I admits a nonzero eigenvalue, we have h 1 = λId.
Proof. We have proved this lemma for dim I = 3. By induction we find the general case.
We assume that h 1 = λId with λ = 0. The derivation f of I is of rank 3 because f and h 1 have the same rank by hypothesis. Since f is an endomorphism in a 3-dimensional space, it admits a non zero eigenvalue µ. Let Y be an associated eigenvector, then This implies that there exists Y such that f (Y ) and h 1 (Y ) are not linearly independent. This is a contradiction. We deduce that λ = 0.
As consequence, any eigenvalues of h 1 are null and h 1 is a nilpotent operator. In particular dim Im(h 1 ) ≤ 2. If this rank is equal to 2, the kernel is of dimension 1. Let X 1 be a generator of this kernel. Then [T 1 , X 1 ] = 0 this implies that f (X 1 ) = 0 because f and h 1 have the same image. We deduce that the subspace of g generated by X 1 is an belian ideal and g is not indecomposable. Then dim Im(h 1 ) = 1 and g is the 5-dimensional Heisenberg algebra. Proposition 19. Any 5-dimensional Lie algebra whose the coadjoint orbit of non closed linear form are of dimension 4 is isomorphic to the 5-dimensional Heisenberg algebra whose Maurer-Cartan equations are Solvable non nilpotent case. Since the Cartan class of any linear form on a solvable Lie algebra is odd if and only if this Lie algebra is nilpotent, then if we assume that g is solvable non nilpotent, there exists a linear form of Cartan class equal to 4. We assume also that g = t ⊕ I with dim t = 2 and satisfying Proposition 16. The determination of these Lie algebras is similar to (8) without the hypothesis [X 1 , I] = 0 and [X 3 , I] = 0. In this case X 1 and X 3 are also in I. We deduce immediately: Proposition 20. Let g = I with dim t = 2 and I an abelian ideal be a solvable non nilpotent Lie algebra whose the dimensions of non singular coadjoint orbits are equal to 4. Then g is isomorphic to the followingg Lie algebra whose Maurer-cartan equations are with a 1 · · · a 2l−3 = 0.
Nilpotent case. Let us describe nilpotent algebras of type t ⊕ I where I is a maximal abelian ideal with dim A(dω) = n − 4 or n for any ω ∈ g * . Let us recall also that the Cartan class of any non trivial linear form is odd then here equal to 5. In the previous examples, we have seen that for the 5-dimensional case, we have obtained only the Heisenberg algebra. Before to study the general case, we begin by a description of an interesting example. Let us consider the following nilpotent Lie algebra, denoted by h(p, 2) given by This Lie algebra is nilpotent of dimension 3p + 2 and it has been introduced in [17] in the study of Pfaffian system of rank greater than 1 and of maximal class.
Proposition 21. For any non closed linear form on h(p, 2), the dimension of the coadjoint orbit is equal to 4.
To study the general case, we shall use the notion of characteristic sequence which is an invariant up to isomorphism of nilpotent Lie algebras (see for example [23] for a presentation of this notion. For any X ∈ g, let c(X) be the ordered sequence, for the lexicographic order, of the dimensions of the Jordan blocks of the nilpotent operator adX. The characteristic sequence of g is the invariant, up to isomorphism, c(g) = max{c(X), X ∈ g − C 1 (g)}.
Theorem 22. Let g be a nilpotent Lie algebra such that the dimension of the coadjoint orbit of non closed form is 4 admitting the decomposition g = t ⊕ I where I is an abelian ideal of codimension 2. Then t admits a basis {T 1 , T 2 } of characteristic vector of g with the same characteristic sequence and Im(adT 1 ) = Im(adT 2 ).
Proof. Let T be a non null vector of t such that g = g/K{T } is a nilpotent Lie algebra given in (??). Then g = t 1 ⊕ I and if T 1 ∈ t 1 , T 1 = 0, then T 1 is a characteristic vector of g. Then T 1 can be considered as a characteristic vector of g. Let be T 2 ∈ t such that adT 1 and adT 2 have the same image in I. Then T 2 is also a characteristic vector with same characteristic sequence, if not c(T 1 ) will be not maximal.
Definition 25. A formal quadratic deformation g t of h 2p+1 is a (2p + 1)-dimensional Lie algebra whose Lie bracket µ t is given by where the maps ϕ i are bilinear on h 2p+1 with values in h 2p+1 and satisfying In this definition δ µ denotes the coboundary operator of the Chevalley-Eilenberg cohomology of a Lie algebra g whose Lie bracket is µ with values in g, and if ϕ and ψ are bilinear maps, then ϕ • ψ is the trilinear map given by In particular ϕ • ϕ = 0 is equivalent to Jacobi Identity and ϕ, in this case, is a Lie bracket and the coboundary operator writes Theorem 26. [19] Any (2p + 1)-dimensional contact Lie algebra g is isomorphic to a quadratic formal deformation of h 2p+1 .

As consequence, we have
Corollary 29. Any (2p + 1)-dimensional contact nilpotent Lie algebra is isomorphic to a central extension of a 2p-dimensional symplectic Lie algebra by its symplectic form.
Proof. Let t be the 2p-dimensional vector space generated by {X 1 , · · · , X 2p }. The restriction to t of the 2-cocycle ϕ 1 is with values in t. Since ϕ 1 • ϕ 1 = 0, it defines on t a structure of 2p-dimensional Lie algebra. If {ω 1 , · · · , ω 2p+1 } is the dual basis of the given classical basis of h 2p+1 , then θ = ω 1 ∧ ω 2 + · · · + ω 2p−1 ∧ ω 2p is a 2-form on t. We denote by d ϕ 1 the differential operator on the Lie algebra (t, ϕ 1 ), that is d ϕ 1 ω(X, Y ) = −ω(ϕ 1 (X, Y )) for all X, Y ∈ t and ω ∈ t * . Since µ 0 is the Heisenberg Lie algebra multiplication, the condition δ µ 0 ϕ 1 = 0 is equivalent to d ϕ 1 (θ) = 0. It implies that θ is a closed 2-form on t and g is a central extension of t by θ. ♣ We deduce: Theorem 30. [19] Let g be a (2p + 1)-dimensional k-step nilpotent Lie algebra. Then there exists on g a coadjoint orbit of dimension 2p if and only if g is a central extension of a (2p)-dimensional (k − 1)-step nilpotent symplectic Lie algebra t, the extension being defined by the 2-cocycle given by the symplectic form.
Since the classification of nilpotent Lie algebras is known up the dimension 7, the previous result permits to establish the classification of contact nilpotent Lie algebras of dimension 3, 5 and 7 using the classification in dimension 2,4 and 6. For example, the classification of 5-dimensional nilpotent Lie algebras with an orbit of dimension 4 is the following: • g is 4-step nilpotent: We shall now study a contact structure in respect of the characteristic sequence of a nilpotent Lie algebra. For any X ∈ g, let c(X) be the ordered sequence, for the lexicographic order, of the dimensions of the Jordan blocks of the nilpotent operator adX. The characteristic sequence of g is the invariant, up to isomorphism, c(g) = max{c(X), X ∈ g − C 1 (g)}.
This invariant was introduced in [2] in order to classify 7-dimensional nilpotent Lie algebras. A link between the notions of breath of nilpotent Lie algebra introduced in [21] and characteristic sequence is developed in [23]. If c(g) = (c 1 , c 2 , · · · , c k , 1) is the characteristic sequence of g then g is c 1 -step nilpotent. Assume now that c 1 = c 2 = · · · = c l and c l+1 < c l . Then the dimension of the center of g is greater that l because in each Jordan blocks corresponding to c 1 , · · · , c l , the last vector is in C c 1 (g) which is contained in the center of g. We deduce Proposition 31. Let g be a contact nilpotent Lie algebra. Then its characteristic sequence is of type c(g) = (c 1 , c 2 , · · · , c k , 1) with c 2 = c 1 .
Let us note also, that in [24], we construct the contact nilpotent filiform Lie algebras, that is with characteristic sequence equal to (2p, 1).

5.1.2.
The non nilpotent case. It is equivalent to consider Lie algebras with a contact form defined by a quadratic non linear deformations of the Heisenberg algebra. We refer to [19] to the description of this class of Lie algebras. An interesting particular case consists to determinate all the (2p + 1)-dimensional Lie algebras (p = 0), such that all the coadjoint orbits of non trivial elements are of dimension 2p.
Lemma 32. [14] Let g a simple Lie algebra of rank r and dimension n. Then any non trivial linear form α on g satisfies cl(α) ≤ n − r + 1. Moreover, if g is of classical type, we have cl(α) ≥ 2r.
In particular, a simple Lie algebra admits a contact form if its rank is equal to 1 and this Lie algebra is isomorphic to sl(2, R) or so(3). Now, if g is a (2p + 1)-dimensional Lie algebras whose the dimension of the coadjoint orbit of any non trivial linear form is equal to 2p, then the Cartan class of non trivial linear form is 2p or 2p + 1. Such Lie algebra cannot be solvable. From the previous lemma, the Levi semisimple subalgebra is of rank 1 and the radical cannot be of dimension greater than 1. Then g is simple of rank 1 and we have Proposition 33. Any (2p+1)-dimensional Lie algebra whose the dimension of the coadjoint orbits of non trivial forms are equal to 2p is simple of rank 1 and isomorphic to sl(2, R) or so(3).

(2p)-dimensional
Lie algebras with a 2p-dimensional coadjoint orbit. Such Lie algebra is frobeniusian. Since the Cartan class of a linear form on a nilpotent Lie algebra is always odd, this Lie algebra is not nilpotent. In the contact case, we have seen that any contact Lie algebra is a deformation of the Heisenberg algebra. On other words, any contact Lie algebra can be contracted on the Heisenberg algebra. In the frobeniusian case, we have a similar but more complicated situation. We have to determinate an irreducible family of frobeniusian Lie algebras with the property that any frobeniusian Lie algebra can be contracted on a Lie algebra of this family.
In a first step, we recall the notion of contraction of Lie algebras. Let g 0 be a n-dimensional Lie algebra whose Lie bracket is denoted by µ 0 . We consider {f t } t∈]0,1] a sequence of isomorphisms in K n with K = R or C. Any Lie bracket corresponds to a Lie algebra g t which is isomorphic to g 0 . If the limit lim t→0 µ t exists (this limit is computed in the finite dimensional vector space of bilinear maps in K n ), it defines a Lie bracket µ of a n-dimensional Lie algebra g called a contraction of g 0 .
Remark. Let L n be the variety of Lie algebra laws over C n provided with its Zariski topology. The algebraic structure of this variety is defined by the Jacobi polynomial equations on the structure constants. The linear group GL (n, C) acts on C n by changes of basis. A Lie algebra g is contracted to the g 0 if the Lie bracket of g is in the closure of the orbit of the Lie bracket of g 0 by the group action (for more details see [8,16]).
In the following, we define Lie algebras, not with its brackets, but with its Maurer-Cartan equations. We assume here that K = C.
Then any 2p-dimensional Frobenius Lie algebra can be contracted in an element of the family F = {g a 1 ,··· ,a p−1 } a i ∈C . Moreover, any element of F cannot be contracted in another element of this family.
Proposition 35. The parameter {a 1 , · · · , a p−1 } which are the invariants of Frobenius Lie algebras up to contraction are the eigenvalues of the principal element of g a 1 ,··· ,a p−1 .

5.2.2.
Classification of real Frobenius Lie algebras up to contraction. We have seen that, in the complex case, the classification up to contraction of 2p-dimensional Lie algebras is in correspondence with the reduced matrix of the principal element. We deduce directly the classification in the real case: Theorem 36. Let g a 1 ,··· ,as,b 1 ,··· ,b 2p−s−1 , a i , b j ∈ R be the 2p-dimensional Lie algebras given by            [X 1 , X 2 ] = [X 2k−1 , X 2k ] = X 1 , k = 2, · · · , p,