5-dimensional indecomposable contact Lie algebras as double extensions

Journal of Geometry and Physics 100 (2016) 20–32

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5-dimensional indecomposable contact Lie algebras as double extensions M.C. Rodríguez-Vallarte, G. Salgado ∗ Fac. de Ciencias, UASLP, Av. Salvador Nava s/n, Zona Universitaria, CP 78290, San Luis Potosí, S.L.P., México

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Article history: Received 21 July 2015 Received in revised form 13 October 2015 Accepted 31 October 2015 Available online 12 November 2015 MSC: primary 17Bxx secondary 53D10 Keywords: Contact Lie algebras Double extension of Lie algebras Indecomposable Lie algebras

abstract In this work we shall show that a suitable double extension of a finite dimensional indecomposable contact Lie algebra is a contact Lie algebra again. In particular, with exception of the family of 5-dimensional indecomposable contact solvable Lie algebras A5,35 , any 5-dimensional indecomposable contact solvable Lie algebra can be obtained as a double extension of a 3-dimensional Lie algebra. The family A5,35 can be generalized to a family of (4n + 1)-dimensional indecomposable contact solvable Lie algebras that cannot be obtained neither as a suspension of a symplectic Lie algebra of codimension 1 or as a double extension of a contact Lie subalgebra of codimension 2. © 2015 Elsevier B.V. All rights reserved.

0. Introduction The notion of double extension plays a significant role providing an inductive construction of finite dimensional Lie algebras endowed with a geometric structure. This concept appears for first time in the book of V. Kac, Infinite dimensional Lie algebras (see [1], §2, exercise 2.10), in order to build finite dimensional quadratic Lie algebras, i.e., finite dimensional Lie algebras endowed with a non-degenerate invariant symmetric bilinear form. From [1] it follows that a solvable n-dimensional quadratic Lie algebra g is either an orthogonal direct sum of a quadratic solvable ideal of codimension 1 and a 1-dimensional quadratic ideal, or g can be constructed as a double extension by a hyperbolic pair of a quadratic solvable subalgebra of codimension 2. A few years later, A. Medina and P. Revoy generalize this idea to prove that every finite dimensional quadratic indecomposable Lie algebra can be obtained as a double extension of a quadratic subalgebra by an appropriate subspace (see [2]). As a consequence, this fact reduces the study of finite dimensional quadratic Lie algebras to the study of finite dimensional indecomposable quadratic Lie algebras. Similar ideas have been developed in order to give an inductive classification of finite dimensional Lie algebras endowed with other type of geometric structures. For example, A. Medina and P. Revoy proved in [3] that if g is a 2n-dimensional symplectic Lie algebra having a non-trivial center, g can be obtained as a symplectic double extension by an appropriate subspace of a symplectic Lie subalgebra of codimension 2. In [4], I. Bajo, S. Benayadi and A. Medina give an inductive description of symplectic quadratic Lie algebras in terms of quadratic double extensions. On the other hand, in [5–7], S. Benayadi and other collaborators have extended the notion of double extension to Z2 -homogeneous quadratic Lie superalgebras, presenting inductive classifications of them subject to specific conditions (see [5–7] for more details).

∗

Corresponding author. E-mail addresses: [email protected] (M.C. Rodríguez-Vallarte), [email protected], [email protected] (G. Salgado).

http://dx.doi.org/10.1016/j.geomphys.2015.10.014 0393-0440/© 2015 Elsevier B.V. All rights reserved.

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

21

Hence, given a finite dimensional Lie algebra g, it is natural to pose the problem of determining the geometric structures on g that can be recovered through an appropriate definition of double extension. In this work we focus on the case of contact Lie algebras, i.e., Lie algebras endowed with a 1-form α ∈ g∗ such that α ∧ (dα)n ̸= 0. The aim of this paper is to get a description as complete as possible of the 5-dimensional indecomposable contact Lie algebras as double extensions of 3-dimensional Lie algebras. For this we shall follow the notation introduced in [8], denoting by A5,ℓ a 5-dimensional Lie algebra with basis {e1 , e2 , e2 , e4 , e5 }, where the index ℓ indicates the number of the class to which the given Lie algebra belongs. Contact Lie algebras have been studied by several authors (see for example [9–11]), and the problem of constructing contact Lie algebras has been addressed in different ways. For example, in [12], B. Kruglikov described a method to construct an (n + 1)-dimensional non-degenerate Lie algebra from an n-dimensional non-degenerate one, as follows: given an n-dimensional Lie algebra g and a 1-form α ∈ g∗ , the genre of α is defined as the highest degree of a non-zero form in the following sequence

α, dα, α ∧ dα, (dα)2 , α ∧ (dα)2 , . . . Then, the pair (g, α) is said to be non-degenerate if the genre of α equals the dimension of g, that is n = dimR g. Observe that for even n, (g, α) is an exact symplectic Lie algebra with a symplectic structure given by dα , whereas for odd n, (g, α) is a contact Lie algebra with a contact structure α . Now consider an (n + 1)-dimensional Lie algebra g and suppose that it can be written as g = h ⊕ ⟨e⟩, where h is an n-dimensional ideal such that (h, α) is a non-degenerate pair. Letting π : g → h be the projection induced from the direct decomposition of g, set α+ = π ∗ α . Then the pair (g, α+ ) is called a suspension over (h, α) if it is a non-degenerate pair. For the even dimensional case, this process is called a contactization or classical contactization, whereas for the odd one, this process is called a symplectization. It has been pointed out by the same author that it is also possible to formulate this setting considering just that h is a Lie subalgebra of g, but a proof was not provided. The suspension method unifies contactization and symplectization, but it does not provide explicit calculations in order to get a contact Lie algebra from an exact symplectic one. Moreover, not every contact Lie algebra can be obtained in this way. For example, letting ρ : sl2 → R2 be the usual irreducible representation of sl2 into R2 , the Lie algebra A5,40 = sl2 nρ R2 cannot be obtained as a suspension of a symplectic Lie algebra of codimension 1. In [13], A. Diatta presented a contactization method to construct contact Lie groups (via their Lie algebras), from exact symplectic ones. For this, he first determined the Lie algebras g containing h as a codimension 1 subalgebra. Next, for an exact symplectic Lie algebra (h, dα), Diatta obtained the contact Lie algebras (g, η) containing h as a codimension 1 subalgebra such that i∗ η = α where i : (h, ω) → (g, η) is the natural inclusion, proving the claim made by Kruglikov (see [12]). The method works fine for 3-dimensional contact Lie algebras, since with exception of so3 , every 3-dimensional contact Lie algebra can be obtained in this way from the 2-dimensional affine Lie algebra af2 . But in dimension 5, the method does not provide a constructive way to get a complete classification of the 5-dimensional indecomposable solvable contact Lie algebras as contactizations of exact symplectic Lie algebras, at first sight, just the Lie algebras A5,36 and A5,37 can be obtained in this way. In order to get a complete classification, a characterization in terms of the derived ideal of a solvable contact Lie algebra g has to be done. More recently, in [14], M. Goze and E. Remm approached the problem of classification of finite dimensional contact Lie algebras in terms of deformations. They proved that any (2n + 1)-dimensional contact Lie algebra is a linear or quadratic deformation of the (2n + 1)-dimensional Heisenberg Lie algebra. They also determined the closed 2-forms of the Chevalley–Eilenberg cohomology of the Heisenberg algebras which give linear or quadratic deformations, and they found the classifications in dimension 3, 5, or 7 of contact Lie algebras given in [13]. In this work we prove that for each central extension gθ (e) of a 3-dimensional contact Lie algebra g by a closed 2-form θ ∈ (Λ2 g)∗ , there exists a derivation D ∈ Der(gθ (e)) acting non-trivially in ⟨e⟩, and such that the 5-dimensional Lie algebra g(D, θ) := ⟨D⟩ n (gθ (e)) is a contact Lie algebra. For this, in Section 3 we calculate the second scalar cohomology group H 2 (g, R) of a 3-dimensional Lie algebra g, and in Appendix A we compute the Lie algebra of derivations Der(gθ (e)). We say that the Lie algebra g(D, θ ) is a double extension of g by the pair (D, θ ). In the general case, given a 2n + 1-dimensional Lie algebra, if there exist a closed 2-form θ and a derivation D ∈ Der(gθ (e)) both satisfying the conditions stated in Theorem 2.2, we can prove that the double extension g(D, θ ) of g by the pair (D, θ ) is a contact Lie algebra. The original aim of this paper was to present an inductive construction of all real contact Lie algebras via a double extension, that is, letting n ≥ 2, our goal was to construct a (2n + 3)-dimensional real indecomposable contact Lie algebra from a (2n + 1)-dimensional real indecomposable contact one. It was not possible to achieve this goal since we found some relevant 5-dimensional counterexamples, namely, the real indecomposable contact Lie algebras A5,35 , A5,39 and A5,40 cannot be obtained as double extensions of real 3-dimensional contact Lie algebras. More specifically, the real contact solvable Lie algebra A5,39 is a double extension of the 3-dimensional real Lie algebra q1 (see Table 1 for more details), but q1 does not have a contact structure. On the other hand, the Lie algebra A5,40 = sl2 nρ R2 cannot be obtained as a double extension of a contact Lie algebra of codimension 2 (see Example 2.6 in Section 1), and the same holds for the family of indecomposable solvable contact Lie algebras A5,35 (see Example 2.4). Moreover, the family A5,35 can be generalized to an irreducible family of (4n + 1)-dimensional contact solvable Lie algebras that cannot also be obtained as a suspension of a symplectic Lie algebra of codimension 1 or as a double extension of a contact Lie algebra of codimension 2 (see Example 2.5). Hence, in Section 3, we can describe each of the 23 out of 24 non-isomorphic indecomposable 5-dimensional contact Lie algebras as a double extension of a 3-dimensional Lie algebra. The exception is A5,35 .

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1. Preliminaries All through the paper, g denotes a finite-dimensional real Lie algebra. For the usual definitions about Lie algebra cohomology, we follow [15]. Let g be a Lie algebra and ρ : g → gl(V ) be a representation. We denote by C q (g, V ) (q ≥ 2), the set of alternating q-linear applications ω : g × · · · × g → V . By definition C 0 (g, V ) = V and C 1 (g, V ) = HomR (g, V ). We say that ω ∈ C q (g, V ) is a q-form. Let g be a Lie algebra and ρ : g → gl(V ) be a representation. The exterior differential d : C q (g, V ) → C q+1 (g, V ) is defined as follows: If q = 0, then ω ∈ V and by definition, dω(x) = ρ(x)ω for all x ∈ g. If q ≥ 1, 1

dω(x1 , . . . , xq+1 ) =

q +1  (−1)i+1 ρ(xi )ω(x1 , . . . , xˆ i , . . . , xq+1 )

q + 1 i =1 1 

+

q + 1 i
(−1)i+j ω([xi , xj ], x1 , . . . , xˆ i , . . . , xˆ j , . . . , xq+1 ),

for all xi ∈ g. Given a Lie algebra g and a representation ρ : g → gl(V ), we define Z q (g, V ) = Ker(d : C q (g, V ) → C q+1 (g, V )), Bq (g, V ) = Im(d : C q−1 (g, V ) → C q (g, V )). We say that a q-form ω ∈ Z q (g, V ) is closed, and that a q-form η ∈ Bq (g, V ) is exact. The qth group of cohomology with coefficients in the representation ρ is defined by: H q (g, V ) =

Z q (g, V ) Bq (g, V )

,

q ≥ 1,

H 0 (g, V ) = Z 0 (g, V ). When ρ : g → gl(V ) is the trivial representation and dimR V = 1, the qth group of cohomology with coefficients in the representation ρ is called the qth group of scalar cohomology, and we denote C q (g, R) by (Λq g)∗ . A contact structure on a (2n + 1)-dimensional Lie algebra g is a 1-form α ∈ g∗ such that α ∧ (dα)n ̸= 0. We say that a contact Lie algebra is a Lie algebra endowed with a contact structure. Observe that if g is a contact Lie algebra, α ∧ (dα)n is a volume form for the corresponding Lie group. A symplectic structure on a 2n-dimensional Lie algebra g is a closed 2-form ω ∈ (Λ2 g)∗ such that ω has maximal rank, that is, dω = 0 and ωn ̸= 0 is a volume form on the corresponding Lie group. We say that a Lie algebra endowed with a symplectic structure is a symplectic or Frobenius Lie algebra. Moreover, the symplectic structure is called exact if ω = dϕ holds for some ϕ ∈ g∗ . We say that a finite dimensional Lie algebra g is decomposable if it can be written as a direct sum of non-trivial ideals. Otherwise, we say that g is indecomposable. It is easy to check that a decomposable Lie algebra g = m ⊕ n is a contact Lie algebra if and only if m is a contact Lie algebra and n is an exact symplectic Lie algebra or vice versa. Let g be a finite dimensional Lie algebra g and let θ ∈ (Λ2 g)∗ be a closed 2-form. Let SpanF {e} := ⟨e⟩ be a 1-dimensional real vector space generated by an element e that does not belong to g. Then, letting

[x, y]θ = [x, y]g + θ (x, y)e x, y ∈ g [x, e]θ = 0 x ∈ g,

(1)

the real vector space g ⊕ ⟨e⟩ is a Lie algebra. We denote this Lie algebra by gθ (e) and we say that it is a central extension of g by the closed 2-form θ . It is a well known fact that the elements of H 2 (g, R) are in a one to one correspondence with the isomorphism classes of central extensions of a given Lie algebra g (see [15], Theorem 26.2). Now given a central extension gθ (e) of g by the closed 2-form θ and a derivation D ∈ Der(gθ (e)), we can consider the semidirect product of the abelian Lie algebra ⟨D⟩ with gθ (e),

g(D, θ ) := ⟨D⟩ n gθ (e). We say that the Lie algebra g(D, θ ) is the double extension of g by the pair (D, θ ). 2. Double extension of contact Lie algebras In this section we want to determine the conditions for which the double extension of a contact Lie algebra is a contact Lie algebra again. In the next result, e∗ ∈ (gθ (e))∗ satisfies ⟨e∗ , e⟩ = 1 and ⟨e∗ , x⟩ = 0 for every x ∈ g.

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

23

Proposition 2.1. Let g be a finite dimensional contact Lie algebra with a contact structure α ∈ g∗ , and let θ ∈ (Λ2 g)∗ . Then there exists λ ∈ R such that:

α ∧ (dα + λθ )n ̸= 0. Proof. Let {e1 , . . . , e2n+1 } be a basis for g and {e1 , . . . , e2n+1 } be its dual basis. Since α ∈ g∗ is a contact structure on g, it follows that α ∧ (dα)n = a0 e1 ∧ · · · ∧ e2n+1 for some a0 ̸= 0. On the other hand, for each k ∈ {1, . . . , n}, α ∧ (dα)n−k ∧ θ k = ak e1 ∧ · · · ∧ e2n+1 with ak ∈ R. Hence, n    n k λ α ∧ (dα)n−k ∧ θ k

α ∧ (dα + λθ )n =

k

k=0



= a0 +

n    n k=1

k



ak λk (e1 ∧ · · · ∧ e2n+1 ).

k Let p(x) = a0 + k=1 k ak x ∈ R[x] be a non zero polynomial of degree n. Since p(x) has at most n real roots, there exists λ ∈ R such that p(λ) ̸= 0. Therefore, for such λ ∈ R, α ∧ (dα + λθ )n ̸= 0. 

n

n

Theorem 2.2. Let g be a finite dimensional contact Lie algebra with a contact structure α ∈ g∗ . Given a closed 2-form θ ∈ (Λ2 g)∗ , consider a central extension gθ (e) of g by θ and let β ∈ (gθ (e))∗ be defined by β = α + λe∗ , λ ∈ R. If there exists a derivation D ∈ Der(gθ (e)) such that β(D(e)) ̸= 0, then the double extension g(D, θ ) of g by the pair (D, θ ) is a contact Lie algebra with contact form β for some λ ∈ R. Proof. Let [θ ] ∈ H 2 (g, R). Since the elements of H 2 (g, R) are in a one to one correspondence with the isomorphism classes of central extensions of g, if [θ ] = 0 (for example if θ = −dα ), then the central extension of g by θ is isomorphic to a direct sum of Lie algebras:

gθ (e) ≃ g ⊕ ⟨e⟩. Hence, defining D ∈ Der(gθ (e)) by D|g ≡ 0 and D(e) = e, it follows that

g(D, θ ) ≃ g ⊕ ⟨e, D⟩, where [D, e] = D(e) = e. Since ⟨e, D⟩ is an exact symplectic Lie algebra, it follows that g(D, θ ) is a contact Lie algebra with contact form β = α + e∗ and clearly, λ = 1. Suppose now that [θ] ̸= 0. We shall use the Maurer–Cartan equations of g(D, θ ) in order to prove that β = α + λ e∗ ∈ (g(D, θ ))∗ is a contact structure for some λ ∈ R. For this we shall use the following notation: dθ stands for the Chevalley–Eilenberg differential operator associated with the trivial representation on g(D, θ ), and d stands for the Chevalley–Eilenberg differential operator associated with the trivial representation on g. 2n+1 i Let {e1 , . . . , e2n+1 , e, D} be a basis for g(D, θ ) and {e1 , . . . , e2n+1 , e∗ , D∗ } be its dual basis. Then α = i=1 αi e for αi ∈ R, and D ∈ Der(gθ (e)) has a matrix representation given by

 [D]gθ (e) =

v

[D|g ] ut



a

,

where v, u ∈ Rn , a ∈ R and [D|g ] is a matrix representation for D|g ∈ EndR (g), with ([D|g ])ij = Dij ∈ R. Take β = α + λ e∗ ∈ (g(D, θ ))∗ and suppose that β(D(e)) ̸= 0. Clearly, dθ β = dθ α + λdθ e∗ where dθ α = dα +

2n+1 1 

2 i =1





2n+1



1

αk Dki e ∧ D + ∗

i

2

k=1





2n+1



αi vi e∗ ∧ D∗ ,

i=1

and ∗

dθ e = −

λ



2n+1



2



2n+1

θ (ei , ej )e ∧ e − i

j

i


= dα −

+1  λ 2n

2 i
λ 2

θ,

θ (ei , ej )ei ∧ ej

∗

∗

ui e ∧ D − ae ∧ D

i=1

Now, let X be the 2-form in (Λ2 g)∗ given by X = dα −

i

∗

.

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M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

and let Y be the 2-form in (Λ2 g(D, θ ))∗ given by Y =



1



2



2n+1

 ui (λ) e ∧ D + a(λ) e ∧ D ∗

i

∗

,

∗

i=1

where 2n+1

 ui (λ) = λui +



αk Dki ,

k=1 2n+1

 a(λ) = λa +



αi vi = β(D(e)) ̸= 0.

i=1

Clearly, X k = 0 for k ≥ n + 1, whereas Y k = 0 for k ≥ 2. Then dθ β can be written as X + Y , and it follows that

(dθ β)n+1 = (X + Y )n+1 = (n + 1)X n ∧ Y . Hence,

β ∧ (dθ β)n+1 = (n + 1)(α + λe∗ ) ∧ X n ∧ Y = (n + 1)(α ∧ X n ∧ Y + λe∗ ∧ X n ∧ Y ). Recalling that Y ∈ (Λ2 g(D, θ ))∗ is a linear combination of ei ∧ D∗ (i = 1, . . . , 2n + 1) and e∗ ∧ D∗ , we have that

α∧X ∧Y =

 a(λ)



2

α ∧ dα −

λ 2

θ

n

∧ e∗ ∧ D ∗ ,     2n+1 1  λ n i ∗ ∗ n ∗  ui (λ)e ∧ D λe ∧ X ∧ Y = λe ∧ dα − θ ∧ n

2

λ

2 i=1

2n+1

= −

2 i=1



 ui (λ) ei ∧ dα −

λ 2

θ

n

∧ e∗ ∧ D ∗ .

Observe that

Γ (λ) =

+1  λ 2n

2 i=1

  λ n i  ui (λ) e ∧ dα − θ 2

is a (2n + 1)-form on g, depending on the coefficients of [D|g ] and λ ∈ R. Define now,

 K =

k ∈ {1, . . . , 2n + 1} | ek ∧



dα −

λ 2

θ

n

 = bk e1 ∧ · · · ∧ e2n+1 , with bk ̸= 0 .

Then, Γ (λ) can be written as follows:

 Γ (λ) =

 λ



bk uk (λ) e1 ∧ · · · ∧ e2n+1 .

k∈K

On the other hand, we have that

 

bk uk (λ) =

k∈K



bk

λuk +

k∈K



 

b k uk

 +

k∈K

Letting A =





k∈K

αs Dsk

s=1

 =λ



2n+1

bk uk and B =

+1  2n 

 αs Dsk bk .

k∈K s=1



k∈K

2n+1

bk uk (λ) = λA + B,

k∈K

and thus,

Γ (λ) = λ(λA + B) e1 ∧ · · · ∧ e2n+1 .

s=1

αs Dsk bk , it follows that

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

25

Therefore, β ∧ (dθ β)n+1 can be expressed as follows:

β ∧ (dθ β)

n+1

= (n + 1)

  a(λ) 2



α ∧ dα −

λ 2

θ

n

 − Γ (λ) ∧ e∗ ∧ D∗ .

 a(λ)

α ∧ (dα − λ2 θ )n − Γ (λ) = 0} is a measure zero subset of R. Then, with no loss of  a(λ) generality, this allows us to choose a scalar λ ∈ F, such that 2 α ∧ (dα − λ2 θ )n − Γ (λ) ̸= 0. Hence, β ∧ (dθ β)n+1 ̸= 0.  Finally, observe that {λ ∈ R |

2

Example 2.3. Let h3 be the 3-dimensional Heisenberg Lie algebra, that is, h3 = ⟨e1 , e2 , e3 ⟩ with bracket [e1 , e2 ] = e3 . Consider the central extension (h3 )θ (e) of h3 given by the closed 2-form θ = e1 ∧ e3 + 2e2 ∧ e3 ∈ (Λ2 h3 )∗ . Choose



1 0 D= 1 0

0 1 1 0

0 0 2 −1



0 0 ∈ Der((h3 )θ (e)), 0 3

then the double extension h3 (D, θ ) is a 5-dimensional contact Lie algebra with contact structure α = e2 + e∗ ∈ h∗3 . So given a (2n + 1)-dimensional contact Lie algebra g, if there exist a closed 2-form θ ∈ (Λ2 g)∗ and a derivation D ∈ Der(gθ (e)) both satisfying the conditions stated in Theorem 2.2, then the double extension g(D, θ ) of g by the pair (D, θ) is a (2n + 3)-dimensional contact Lie algebra. However, the converse does not necessarily hold. It is not true that every (2n + 3)-dimensional contact Lie algebra m can be obtained as a double extension of a (2n + 1)-dimensional contact Lie algebra g, as the following example shows: Example 2.4 (A contact Lie algebra that cannot be obtained as a double extension of a contact Lie algebra of codimension 2). Let a3 = ⟨e1 , e2 , e3 ⟩ be the 3-dimensional abelian Lie algebra R3 and let S , T ∈ Der(a3 ) = gl(R3 ) be derivations defined in such a way that:

 [S ] =

a 0 0

0 1 0

0 0 , a ∈ R, 1



 [T ] =

b 0 0

0 0 −1

0 1 , b ∈ R. 0



Now, let e4 := S, e5 := T and consider the semidirect product A5,35 := ⟨e4 , e5 ⟩ n a3 . Clearly, this is a family of indecomposable solvable Lie algebras having trivial center. Letting {e1 , . . . , e5 } be a basis of (A5,35 )∗ , it is easy to verify that both α = e1 + e2 ∈ (A5,35 )∗ and α = e1 + e3 ∈ (A5,35 )∗ define contact structures on A5,35 . Clearly, A5,35 cannot be constructed as a double extension of a 3-dimensional contact Lie algebra g by a pair (D, θ ) since it is not splittable with a 1-dimensional abelian subalgebra. In fact, Example 2.4 can be generalized to a family of (4n + 1)-dimensional contact solvable Lie algebras, as follows: Example 2.5 (A family of contact solvable Lie algebras that cannot be obtained as a double extension of a contact Lie algebra of codimension 2). Let a2n+1 be the (2n + 1)-dimensional abelian Lie algebra R2n+1 . We fix the following decomposition and notation:

a2n+1 = ⟨e1 ⟩ ⊕ R21 ⊕ · · · ⊕ R2n , where R2k = ⟨ek1 , ek2 ⟩ for k = 1, . . . , n. For i = 1, . . . , n, let Si , Ti ∈ Der(a2n+1 ) be the linear maps defined by: S i ( e 1 ) = ai e 1 , ai ∈ R , Ti (e1 ) = bi e1 , bi ∈ R,

Si |R2 = δik IdR2 for k = 1, . . . , n, k

k

Ti |R2 = δik J for k = 1, . . . , n, k

where J : R2k → R2k is the linear map defined by J (ek1 ) = −ek2 and J (ek2 ) = ek1 . It is easy to verify that:

[Si , Sj ] = [Si , Tr ] = [Tr , Ts ] = 0 for all i, j, r , s ∈ {1, 2, . . . , n}, i.e., ⟨S1 , . . . , Sn , T1 , . . . , Tn ⟩ is an abelian Lie subalgebra of Der(a2n+1 ). So, we can define a (4n + 1)-dimensional Lie algebra as follows: a¯ ,b¯

A4n+1 := ⟨S1 , . . . , Sn , T1 , . . . , Tn ⟩ n a2n+1 , where a¯ = (a1 , . . . , an ), b¯ = (b1 , . . . , bn ) ∈ Rn . a¯ ,b¯

Clearly, Z (A4n+1 ) = {0}. For all k = 1, . . . , n, let f1k = Sk , f2k = Tk , and {e∗1 , (ek1 )∗ , (ek2 )∗ , (f1k )∗ , (f2k )∗ | k = 1, . . . , n} be the

n n a¯ ,b¯ a¯ ,b¯ a¯ ,b¯ dual basis of A4n+1 . Observe that both α = e∗1 + k=1 (ek2 )∗ ∈ (A4n+1 )∗ and β = e∗1 + k=1 (ek1 )∗ ∈ (A4n+1 )∗ define contact a¯ ,b¯

structures on this Lie algebra. However, A4n+1 cannot be constructed as a double extension of a contact Lie subalgebra of codimension 2.

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Although Example 2.5 exhibits the existence of a (4n + 1)-dimensional contact Lie algebra that cannot be obtained as a double extension of a (4n − 1)-dimensional contact one, in general, the double extension has the advantage of providing a constructive way to obtain a (2n + 1)-dimensional contact Lie algebra m starting from a (2n − 1)-dimensional contact one g, as far as we know the space of closed 2-forms H 2 (g, R) and the Lie algebra of derivations Der(gθ (e)) of the central extension gθ (e) of g by θ . In Section 3 we will see that, with exception of the Lie algebras A5,35 and A5,40 , the double extension allows us to obtain a list as complete as possible of the 5-dimensional non-isomorphic indecomposable contact Lie algebras. In contrast, the known methods to obtain contact Lie algebras provide a small number of them. For example, suppose that m is a (n + 1)-dimensional Lie algebra that can be written as m = g ⊕ ⟨e⟩, where g is an n-dimensional ideal such that (g, α) is a non-degenerate pair (see [12] for more details). Let π : m → g be the projection induced from the direct decomposition and set α+ = π ∗ α , then the pair (m, α+ ) is called a suspension over (g, α) if it is a non-degenerate pair. Observe that for the even dimensional case, this process is called contactization or classical contactization, whereas for the odd one, this process is called symplectization. The suspension method unifies contactization and symplectization, since it allows to get a contact but it does not provide explicit calculations in order to get a contact Lie algebra from an exact symplectic one. Moreover, not every contact Lie algebra can be obtained as a suspension, as the following example shows: Example 2.6 (A contact Lie algebra that cannot be obtained as a suspension of a symplectic Lie algebra or as a double extension of codimension 2). Let sl2 = ⟨H , E , F ⟩ be the 3-dimensional real special linear Lie algebra with brackets

[H , E ] = 2E ,

[H , F ] = −2F ,

[E , F ] = H ,

and let ρ : sl2 → gl(R ) be the irreducible representation defined by: 2

ρ(H ) :=



1 0



0 , −1

ρ(E ) :=



0 0



1 , 0

ρ(F ) :=



0 1



0 . 0

Let A5,40 := sl2 nρ R2 . A straightforward calculation shows that A5,40 is a contact Lie algebra. However, it cannot be obtained as a suspension since sl2 nρ R2 does not have a symplectic Lie subalgebra of codimension 1. Suppose now that A5,40 can be written as a double extension of a 3-dimensional Lie algebra, say, A5,40 = ⟨D⟩ n sl2 (e) for some D ∈ Der(sl2 (e)). Since ⟨e⟩ is a 1-dimensional trivial sl2 -module, it follows that the irreducible sl2 -module R2 can be split as ⟨D⟩ ⊕ ⟨e⟩ and clearly, this is not possible. In [13] it is presented a contactization method to construct a contact Lie algebra m from an exact symplectic Lie algebra g, generalizing the suspension method presented in [12]. For this, the first task is to determine the Lie algebras m containing g as a subalgebra of codimension 1. Then, given an exact symplectic Lie algebra (g, dα), the next step is to obtain the contact Lie algebras (m, η) containing g as a subalgebra of codimension 1 and such that i∗ η = α where i : (g, dα) → (m, η) is the natural inclusion. This method works fine for dimension 3, since with the exception of so3 , every 3-dimensional contact Lie algebra can be obtained in this way from the 2-dimensional affine Lie algebra af2 . But in dimension 5, the method does not provide a constructive way to get a list as complete as possible of the 5-dimensional indecomposable non-isomorphic solvable contact Lie algebras. For example, in the classification of the 5-dimensional contact solvable Lie algebras presented in [13], at first sight, just the Lie algebras A5,36 and A5,37 can be obtained through the contactization method described above. In order to obtain a complete classification, a characterization of m in terms of the derived ideal has to be done. 3. 5-dimensional contact Lie algebras In this section we provide a list as complete as possible of the 5-dimensional non-isomorphic indecomposable Lie algebras as double extensions of 3-dimensional Lie algebras. In dimension 5, the classification of real contact Lie algebras is known (see [13] for more details): there are 3 non-solvable contact Lie algebras of dimension 5, namely, af2 ⊕ sl2 , af2 ⊕ so3 (both of them decomposable), and sl2 nρ R2 (indecomposable); whereas there are 24 indecomposable solvable contact Lie algebras of dimension 5. Example 2.6 shows that sl2 nρ R2 is a non-solvable 5-dimensional indecomposable contact Lie algebra that cannot be obtained neither as a double extension of a contact Lie algebra of codimension 2 nor a suspension of a symplectic Lie algebra of codimension 1. In the same spirit, Example 2.4 shows that the Lie algebra A5,35 cannot also be obtained in that way. Hence, with exception of A5,35 , from the list of 24 indecomposable solvable contact Lie algebras classified in [13], we can describe 23 of them as a double extension of a 3-dimensional Lie algebra. For the sake of completeness, we shall include the following well known result: Theorem 3.1. Let g be a 3-dimensional real Lie algebra over the ground field R, with basis {e1 , e2 , e3 }. Then, g is isomorphic to: 1. 2. 3. 4. 5.

a3 := R3 , h3 , [e1 , e2 ] = e3 , q0 , [e1 , e2 ] = e2 , q1λ , [e1 , e2 ] = λe2 + e3 , [e1 , e3 ] = −e2 + λe3 , λ ∈ R, qλ , [e1 , e2 ] = e2 , [e1 , e3 ] = λe3 , λ ̸= 0,

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

27

6. p, [e1 , e2 ] = e2 , [e1 , e3 ] = e2 + e3 , 7. sl2 , [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 ,

su2 or so3 , [e1 , e2 ] = −e3 , [e1 , e3 ] = e2 , [e2 , e3 ] = −e1 . Observe that, with exception of q1 , every 3-dimensional non-abelian real Lie algebra is a contact Lie algebra. Now, in order to describe a 5-dimensional indecomposable Lie algebra m as a double extension of a 3-dimensional Lie algebra g as those given in Theorem 3.1, we shall first compute the second scalar cohomology group H 2 (g, R). Straightforward calculations prove the following result. Theorem 3.2. Let g be a 3-dimensional real Lie algebra with basis {e1 , e2 , e3 }, as in Theorem 3.1, and let {e1 , e2 , e3 } be its dual basis. Let H 2 (g, R) be its second scalar cohomology group. Then (1) dim H 2 (g, R) = 3 if and only if g = a3 . In this case, H 2 (a3 , R) = ⟨[e1 ∧ e2 ], [e1 ∧ e3 ], [e2 ∧ e3 ]⟩. (2) dim H 2 (g, R) = 2 if and only if g = h3 . In this case, H 2 (h3 , R) = ⟨[e1 ∧ e3 ], [e2 ∧ e3 ]⟩. (3) dim H 2 (g, R) = 1 if and only if g is one of the following Lie algebras: q−1 , q0 , q10 . In this case, H 2 (q−1 , R) = ⟨[e2 ∧ e3 ]⟩. H 2 (q0 , R) = ⟨[e1 ∧ e3 ]⟩. H 2 (q10 , R) = ⟨[e2 ∧ e3 ]⟩. (4) dim H 2 (g, R) = 0 if g is different from a3 , h3 , q−1 , q0 , q10 . Convention 3.3. When H 2 (g, R) ̸= {0}, for some i, j ∈ {1, 2, 3} with i < j, we can choose a 2-closed form ei ∧ ej as a representative of each cohomology class. In order to simplify the following calculations, such ei ∧ ej shall be denoted by θij . Suppose now that g is a 3-dimensional contact Lie algebra with a basis {e1 , e2 , e3 } as in Theorem 3.1. From Theorem 3.2 we can compute a central extension gθ (e) of g by a closed 2-form θ . Now, for such central extension, if there exists a suitable derivation D ∈ Der(gθ (e)) satisfying the conditions stated in Theorem 2.2, then the double extension g(D, θ ) is also endowed with a contact form. Thus, in order to prove that a 5-dimensional indecomposable contact Lie algebra can be obtained as a double extension of a 3-dimensional Lie algebra g as those given in Theorem 3.1, we shall determine the structure of the Lie algebra of derivations Der(gθ (e)) for all possible central extensions gθ (e). For sake of completeness, in Appendix A we provide the structure of the Lie algebra of derivations Der(gθ (e)). With exception of A5,40 = sl2 nρ R2 and A5,35 , Theorem 3.4 provides a list as complete as possible of 5-dimensional nonisomorphic indecomposable contact Lie algebras described as a double extension of a 3-dimensional Lie algebra. The proof follows by straightforward calculations in each Lie algebra. We follow the notation and classification of the 5-dimensional indecomposable solvable Lie algebras presented in [8]. Hence A5,ℓ is a 5-dimensional indecomposable solvable Lie algebra, where the index ℓ indicates the number of the class to which the given Lie algebra belongs. Letting {f1 , . . . , f5 } be a basis for 5 i A5,ℓ and {f 1 , . . . , f 5 } be its dual basis, any α ∈ (A5,ℓ )∗ can be written as α = i=1 ai f with ai ∈ R. Then we first compute ∗ the condition for which any α ∈ (A5,ℓ ) is a contact structure on (A5,ℓ ). Now, for each A5,ℓ , we provide a suitable change of basis in order to recognize inside an underlying 3-dimensional Lie algebra g, as those of Theorem 3.1. On the other hand, from Theorem 3.2 we can obtain all possible central extensions gθ (e) of such 3-dimensional Lie algebras g by a closed 2-form θ , whereas from the results presented in Appendix A we can determine the structure of the Lie algebra of derivations for the central extensions, Der(gθ (e)). Finally, the next step is to identify A5,ℓ as a double extension g(D, θ ) of a 3-dimensional Lie algebra g by a pair (D, θ ), where θ is a closed 2-form and D ∈ Der(gθ (e)) is a derivation satisfying the conditions stated in Theorem 2.2. We present this information in Table 1, where the double extension data g(D, θ ) is given by the triple (g, θ , D). Observe that, with exception of A5,40 = sl2 nρ R2 , every Lie algebra A5,ℓ is solvable and it has trivial center. Theorem 3.4. Let A5,ℓ be a 5-dimensional indecomposable contact Lie algebra with basis {f1 , . . . , f5 }. Then, with exception of A5,35 , A5,39 and A5,40 , each A5,ℓ can be obtained as a double extension of a 3-dimensional contact Lie algebra g, as follows from Table 1. Acknowledgments The authors would like to acknowledge the support received by PROMEP grant UASLP-CA-228 and CONACyT Grant 222870. MCRV was also supported by CONACyT Grant 154340. The authors would also like to thank the referee for his/her comments and questions, as they gave them the opportunity to produce a better exposition of their results, improving the original presentation.

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Table 1 5-dimensional indecomposable contact Lie algebras as double extensions of 3-dimensional Lie algebras. A5,4 Condition for α ∈ (A5,4 )∗ : Change of basis: Double extension data:

[f2 , f4 ] = f1 , [f3 , f5 ] = f1 . a1 ̸= 0. f1 → e3 , f2 → e1 , f3 → e4 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,5 Condition for α ∈ (A5,5 )∗ : Change of basis: Double extension data:

[f3 , f4 ] = f1 , [f2 , f5 ] = f1 , [f3 , f5 ] = f2 . a1 ̸= 0. f1 → e3 , f2 → e4 , f3 → e1 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,6 Condition for α ∈ (A5,6 )∗ : Change of basis: Double extension data:

[f3 , f4 ] = f1 , [f2 , f5 ] = f1 , [f3 , f5 ] = f2 , [f4 , f5 ] = f3 . a1 ̸= 0. f1 → e3 , f2 → e4 , f3 → e1 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,19 , a ∈ R, b ̸= 0 Condition for α ∈ (A5,19 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = (a + 1)f1 , [f2 , f5 ] = f2 , [f3 , f5 ] = af3 , [f4 , f5 ] = bf4 . a1 a4 (a + 1 − b) ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,20 , a ∈ R Condition for α ∈ (A5,20 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = (a + 1)f1 , [f2 , f5 ] = f2 , [f3 , f5 ] = af3 , [f4 , f5 ] = f1 + (a + 1)f4 . a1 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,21 Condition for α ∈ (A5,21 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = 2f1 , [f2 , f5 ] = f2 + f3 , [f3 , f5 ] = f3 + f4 , [f4 , f5 ] = f4 . a1 a4 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,22 Condition for α ∈ (A5,22 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f2 , f5 ] = f3 , [f4 , f5 ] = f4 . a1 a4 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,23 , b ̸= 0 Condition for α ∈ (A5,23 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = 2f1 , [f2 , f5 ] = f2 + f3 , [f3 , f5 ] = f3 , [f4 , f5 ] = bf4 . a21 a4 (2 − b) ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,24 , ε = ±1 Condition for α ∈ (A5,24 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = 2f1 , [f2 , f5 ] = f2 + f3 , [f3 , f5 ] = f3 , [f4 , f5 ] = ε f1 + 2f4 . a1 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,25 , p ∈ R, b ̸= 0 Condition for α ∈ (A5,25 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = 2pf1 , [f2 , f5 ] = pf2 + f3 , [f3 , f5 ] = pf3 − f2 , [f4 , f5 ] = bf4 . a1 a4 (b − 2p) ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,26 , p ∈ R, ε = ±1 Condition for α ∈ (A5,26 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = 2pf1 , [f2 , f5 ] = pf2 + f3 , [f3 , f5 ] = pf3 − f2 , [f4 , f5 ] = ε f1 + 2pf4 . a1 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,27 Condition for α ∈ (A5,27 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = f1 , [f3 , f5 ] = f3 + f4 , [f4 , f5 ] = f1 + f4 . a1 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,28 , a ∈ R Condition for α ∈ (A5,28 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f5 ] = (a + 1)f1 , [f2 , f5 ] = af2 , [f3 , f5 ] = f3 + f4 , [f4 , f5 ] = f4 . aa1 a4 ̸= 0. f1 → e3 , f2 → e1 , f3 → e2 , f4 → e4 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,29 Condition for α ∈ (A5,29 )∗ : Change of basis: Double extension data:

[f2 , f4 ] = f1 , [f1 , f5 ] = f1 , [f2 , f5 ] = f2 , [f4 , f5 ] = f3 . a1 a3 ̸= 0. f1 → e3 , f2 → e1 , f3 → e4 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

29

Table 1 (continued) A5,30 , a ∈ R Condition for α ∈ (A5,30 )∗ : Change of basis: Double extension data:

[f2 , f4 ] = f1 , [f1 , f5 ] = (a + 2)f1 , [f3 , f4 ] = f2 , [f2 , f5 ] = (a + 1)f2 , [f3 , f5 ] = af3 , [f4 , f5 ] = f4 . 2a21 a3 − a1 a22 ̸= 0. f1 → e3 , f2 → e1 , f3 → e4 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,31 Condition for α ∈ (A5,31 )∗ : Change of basis: Double extension data:

[f2 , f4 ] = f1 , [f1 , f5 ] = 3f1 , [f3 , f4 ] = f2 , [f2 , f5 ] = 2f2 , [f3 , f5 ] = f3 , [f4 , f5 ] = f3 + f4 . 2a21 a3 − a1 a22 ̸= 0. f1 → e3 , f2 → e1 , f3 → e4 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,32 , a ∈ R Condition for α ∈ (A5,32 )∗ : Change of basis: Double extension data:

[f2 , f4 ] = f1 , [f1 , f5 ] = f1 , [f3 , f4 ] = f2 , [f2 , f5 ] = f2 , [f3 , f5 ] = af1 + f3 . aa1 ̸= 0. f1 → e4 , f2 → e3 , f3 → e1 , f4 → e2 , f5 → e5 . (h3 = ⟨e1 , e2 , e3 ⟩, θ23 , ad(e5 )).

A5,33 , a2 + b2 ̸= 0 Condition for α ∈ (A5,33 )∗ : Change of basis: Double extension data:

[f1 , f4 ] = f1 , [f2 , f5 ] = f2 , [f3 , f4 ] = bf3 , [f3 , f5 ] = af3 . a1 a2 a3 (1 − (a + b)) ̸= 0. f1 → e2 , f2 → e4 , f3 → e3 , f4 → −e1 , f5 → e5 . (qb = ⟨e1 , e2 , e3 ⟩ b ̸= 0, 0, ad(e5 )).

A5,34 , a ∈ R Condition for α ∈ (A5,34 )∗ : Change of basis: Double extension data:

[f1 , f4 ] = af1 , [f1 , f5 ] = f1 , [f2 , f4 ] = f2 , [f3 , f5 ] = f2 , [f3 , f4 ] = f3 . a1 a22 (a − 1) ̸= 0. f1 → e2 , f2 → e4 , f3 → e3 , f4 → e5 , f5 → −e1 . (q0 = ⟨e1 , e2 , e3 ⟩, −θ13 , ad(e5 )).

A5,35 , a2 + b2 ̸= 0 Condition for α ∈ (A5,35 )∗ : Change of basis: Double extension data:

[f1 , f4 ] = bf1 , [f1 , f5 ] = af1 , [f2 , f4 ] = f2 , [f3 , f5 ] = f2 , [f3 , f4 ] = f3 , [f2 , f5 ] = −f3 . a1 (a22 + a23 )(b − 1) ̸= 0.

A5,36 Condition for α ∈ (A5,36 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f4 ] = f1 , [f2 , f4 ] = f2 , [f2 , f5 ] = −f2 , [f3 , f5 ] = f3 . a21 a5 + a1 a2 a3 ̸= 0. f1 → e4 , f2 → e2 , f3 → e3 , f4 → e5 , f5 → e1 . (q−1 = ⟨e1 , e2 , e3 ⟩, θ23 , ad(e5 )).

A5,37 Condition for α ∈ (A5,37 )∗ : Change of basis: Double extension data:

[f2 , f3 ] = f1 , [f1 , f4 ] = 2f1 , [f2 , f4 ] = f2 , [f3 , f5 ] = f2 , [f3 , f4 ] = f3 , [f2 , f5 ] = −f3 . 2a21 a5 + a1 a23 + a1 a22 ̸= 0. f1 → e4 , f2 → e3 , f3 → −e2 , f4 → e5 , f5 → e1 . (q10 = ⟨e1 , e2 , e3 ⟩, −θ23 , ad(e5 )).

A5,38 Condition for α ∈ (A5,38 )∗ : Change of basis: Double extension data:

[f1 , f4 ] = f1 , [f2 , f5 ] = f2 , [f4 , f5 ] = f3 . a1 a2 a3 ̸= 0. f1 → e2 , f2 → e3 , f3 → e4 , f4 → −e1 , f5 → e5 . (q0 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,39 Condition for α ∈ (A5,39 )∗ : Change of basis: Double extension data:

[f1 , f4 ] = f1 , [f2 , f5 ] = f1 , [f2 , f4 ] = f2 , [f1 , f5 ] = −f2 , [f4 , f5 ] = f3 . (a21 + a22 )a3 ̸= 0. f1 → e2 , f2 → e3 , f3 → e4 , f4 → −e1 , f5 → e5 . (q1 = ⟨e1 , e2 , e3 ⟩, 0, ad(e5 )).

A5,40 Condition for α ∈ m∗ : Change of basis: Double extension data:

[f1 , f2 ] = 2f1 , [f1 , f3 ] = −f2 , [f2 , f3 ] = 2f3 , [f2 , f4 ] = f4 , [f3 , f5 ] = f4 , [f1 , f4 ] = f5 , [f2 , f5 ] = −f5 . a3 a25 + a2 a4 a5 − 3a1 a24 ̸= 0.

Appendix A. Structure of the Lie algebra of derivations Der(gθ (h)) for a central extension gθ (e4 ) of 3-dimensional real Lie algebra g. Let g be a 3-dimensional real Lie algebra. Then, from Theorem 3.1 it follows that there exists a basis {e1 , e2 , e3 } for which g is isomorphic to one of the following, (1) a3 , (2) h, [e1 , e2 ] = e3 , (3) q0 , [e1 , e2 ] = e2 , (4) q1λ , [e1 , e2 ] = λe2 + e3 , [e1 , e3 ] = −e2 + λe3 , (5) qλ , [e1 , e2 ] = e2 , [e1 , e3 ] = λe3 ,

λ ̸= 0,

λ ∈ R,

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M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

(6) p, [e1 , e2 ] = e2 , [e1 , e3 ] = e2 + e3 , (7) sl2 , [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 ,

su2 (R) or so3 (R); [e1 , e2 ] = −e3 , [e1 , e3 ] = e2 , [e2 , e3 ] = −e1 . In this section we compute the Lie algebra of derivations Der(gθ (e)) of a central extension gθ (e) of g by a closed 2-form θ . For this, we use the notation introduced in Convention 3.3: for some i, j ∈ {1, 2, 3} with i < j, θij stands for a non trivial representative ei ∧ ej of a cohomology class of H 2 (g, R). Then, gθ (e) denotes the central extension of g by a closed 2-form θ , with Lie bracket given by

[x, y]gθ (e4 ) = [x, y]g + θ (x, y)e4 ,

[θ ] ∈ H 2 (g, R),

and for a derivation D ∈ Der(gθ (e)), taking Dij ∈ R for i, j = 1, . . . , 4, [D] = (Dij ) denotes its matrix representation in the chosen basis. The following results are obtained from straightforward calculations in each Lie algebra. Proposition A.1. For the 3-dimensional abelian Lie algebra a3 , let θ = aθ12 + bθ13 + c θ23 , [θ] ∈ H 2 (a3 , R) for some a, b, c ∈ R, and consider a central extension (a3 )θ (e) = ⟨e1 , e2 , e3 , e⟩ with brackets [e1 , e2 ] = ae, [e1 , e3 ] = be, [e2 , e3 ] = ce. Then D ∈ Der((a3 )θ (e)) if and only if [D] = (Dij ), where Dij ∈ R (i, j = 1, . . . , 4) satisfy the following relations: (1) (2) (3) (4)

aDi4 = bDi4 = cDi4 = 0 for i = 1, 2, 3, aD44 = a(D11 + D22 ) + bD32 − cD31 , bD44 = b(D11 + D33 ) + aD23 + cD21 , cD44 = c (D22 + D33 ) − aD13 + bD12 .

Proposition A.2. For the 3-dimensional Heisenberg Lie algebra h3 , let θ = aθ13 + bθ23 , [θ ] ∈ H 2 (h3 , F) for some a, b ∈ R, and consider a central extension (h3 )θ (e) = ⟨e1 , e2 , e3 , e⟩ with brackets [e1 , e2 ] = e3 , [e1 , e3 ] = ae, [e2 , e3 ] = be. Then D ∈ Der((h3 )θ (e)) if and only if



D11 D21 [D] =  D31 D41

D12 D22 D32 D42

0 0 D33 D43



0 0  , D34  D44

where Dij ∈ R (i, j = 1, . . . , 4) satisfy the following relations: (1) (2) (3) (4) (5)

D33 = D11 + D22 , aD34 = bD34 = 0, D43 = aD32 − bD31 , aD44 = a(D11 + D33 ) + bD21 , bD44 = aD12 + b(D22 + D33 ).

Proposition A.3. For the Lie algebra q0 , let θ = aθ13 , [θ] ∈ H 2 (q0 , R) for some a ∈ R, and consider a central extension (q0 )θ (e) with brackets [e1 , e2 ] = e2 , [e1 , e3 ] = ae. Then D ∈ Der((q0 )θ (e)) if and only if



0 D21 [D ] =  D31 D41

0 D22 0 0

0 0 D33 D43



0 0  , D34  D44

where Dij ∈ R (i, j = 1, . . . , 4) satisfy the following relations: 1. aD34 = 0, 2. aD44 = aD33 . Proposition A.4. For the Lie algebra q10 , let θ = bθ23 , [θ ] ∈ H 2 (q10 , R) for some b ∈ R, and consider a central extension (q10 )θ (e), with brackets [e1 , e2 ] = e3 , [e1 , e3 ] = −e2 , [e2 , e3 ] = be. Then D ∈ Der((q10 )θ (e)) if and only if



0 D21 [D ] =  D31 D41

0 D22 −D23 −bD21

0 D23 D22 −bD31



0 0  , 0  D44

where Dij ∈ R (i, j = 1, . . . , 4) and bD33 = 2bD22 .

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

31

Proposition A.5. For the Lie algebra q1λ , consider the trivial central extension q1λ (e), λ ̸= 0, with brackets [e1 , e2 ] = λe2 + e3 , [e1 , e3 ] = −e2 + λe3 . Then D ∈ Der(q1λ (e)) if and only if



0 D21 [D ] =  D31 D41

0 D22 −D23 0

0 D23 D22 0



0 0  , 0  D44

where Dij ∈ R (i, j = 1, 2, 3). Proposition A.6. For the Lie algebra q−1 , let θ = bθ23 , [θ ] ∈ H 2 (q−1 , R) for some b ∈ R, and consider a central extension (q−1 )θ (e) with brackets [e1 , e2 ] = e2 , [e1 , e3 ] = −e3 , [e2 , e3 ] = be. Then D ∈ Der((q−1 )θ (e)) if and only if



0 D21 [D ] =  D31 D41

0 D22 0 −bD31



0 0 D33 −bD21

0 0  , 0  D44

where Dij ∈ R (i, j = 1, . . . , 4) and bD44 = b(D22 + D33 ). Proposition A.7. For the Lie algebra qλ with λ ̸= {0, −1}, consider a trivial central extension qλ (e) with brackets [e1 , e2 ] = e2 , [e1 , e3 ] = λe3 . Then D ∈ Der(qλ (e)) if and only if



D11 D21 [D ] =  D31 D41

D12 D22 D32 0



D13 D23 D33 0

0 0  0  D44

where Dij ∈ R (i, j = 1, . . . , 4). Moreover, letting A = D|qλ ∈ Der(qλ ), it follows that for λ ̸= 1,

 [A] =

0 D21 D31

0 D22 0

0 0 D33



0 D22 D32

0 D23 D33



,

whereas for λ = 1,

 [A] =

0 D21 D31

.

Proposition A.8. For the Lie algebra p, consider a trivial central extension p(e) with brackets [e1 , e2 ] = e2 , [e1 , e3 ] = e2 + e3 . Then D ∈ Der(p(e)) if and only if



0 D21 [D ] =  D31 D41

0 D22 0 0

0 D23 D22 0



0 0  , 0  D44

where Dij ∈ R (i, j = 1, . . . , 4). Proposition A.9. For the Lie algebra sl2 consider a trivial central extension sl2 (e) with brackets [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 . Then D ∈ Der(sl2 (e)) if and only if

[D ] =

  D 0

0 D44



where D44 ∈ R and  D ∈ Der (sl2 ) if and only if

 [ D] =

0 −2D13 2D12

−D12 D22 0

where Dij ∈ R (i, j = 1, 2, 3).

D13 0 −D22 ,



32

M.C. Rodríguez-Vallarte, G. Salgado / Journal of Geometry and Physics 100 (2016) 20–32

Proposition A.10. For the Lie algebra so3 consider a trivial central extension so3 (e) with brackets [e1 , e2 ] = −e3 , [e1 , e3 ] = e2 , [e2 , e3 ] = −e1 . Then D ∈ Der(so3 (e)) if and only if

[D] =

  D 0

0 D44



with D44 ∈ F and  D ∈ Der (so3 ) if and only if

 0  [D] = −D12 −D13

D12 0 −D23

D13 D23 0

 ,

where Dij ∈ R (i, j = 1, 2, 3). References [1] V. Kac, Infinite-dimensional Lie algebras, Third edition, Cambridge University Press, Cambridge, 1990. [2] A. Medina, P. Revoy, Algèbres de Lie et produit scalaire invariant. (French), Ann. Scient. École. Norm. Sup. (4) 18 (3) (1985) 553–561. [3] A. Medina, P. Revoy, Groupes de Lie à structure symplectique invariante, (Berkeley, CA, 1989), in: Math. Sci. Res. Inst. Publ., 20, Springer, New York, 1991, pp. 247–266. [4] I. Bajo, S. Benayadi, A. Medina, Symplectic structures on quadratic Lie algebras, J. Algebra. 316 (2007) 174–188. [5] H. Albuquerque, E. Barreiro, S. Benayadi, Odd-quadratic Lie superalgebras, J. Geom. Phys. 60 (2) (2010) 230–250. [6] H. Benamor, S. Benayadi, Double extension of quadratic Lie superalgebras, Comm. Algebra. 27 (1999) 67–68. [7] S. Benayadi, Quadratic Lie superalgebras with the completely reducible action of the even part on the odd part, J. Algebra. 223 (2000) 344–366. [8] P. Basarab-Horwath, V. Lahno, R. Zhdanov, The structure of Lie algebras and the classification problem for partial differential equations, Acta Appl. Math. 69 (2001) 43–94. [9] D.E. Blair, Contact Manifolds in Riemannian Geometry, in: Lect. Notes Math., 509, Springer-Verlag, Berlin, New York, 1976. [10] W.M. Boothby, H.C. Wang, On contact manifolds, Ann. of Math. 68 (1958) 721–734. [11] Y. Khakimdjanov, M. Goze, A. Medina, Symplectic or contact structures on Lie groups, Differential Geom. Appl. 21 (1) (2004) 41–54. [12] B. Kruglikov, Symplectic and contact Lie algebras with an application to Monge-Ampère equations, Proc. Steklov Inst. Math. 221 (2) (1998) 221–235. [13] A. Diatta, Left invariant contact structures on Lie groups, Differential Geom. Appl. 26 (2008) 544–552. [14] M. Goze, E. Remm, Contact and Frobeniusian forms on Lie gropus, Differential Geom. Appl. 35 (2014) 74–94. [15] C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948) 85–124.