Homology, Homotopy and Applications, vol.7(1), 2005, pp.9–37

PRODUCT STRUCTURES ON FOUR DIMENSIONAL SOLVABLE LIE ALGEBRAS

A. ANDRADA, M. L. BARBERIS, I. G. DOTTI and G. P. OVANDO

(communicated by Charles Weibel)

Abstract It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the three dimensional subalgebras.

Introduction

A product structure on a smooth manifold M is an endomorphism E of its tangent bundle satisfying E^{2} = Id together with

E[X, Y] = [EX, Y] + [X, EY] − E[EX, EY] for all vector fields X, Y on M. (1)

A product structure on M gives rise to a splitting of the tangent bundle T M into the Whitney sum of two subbundles T^{±}M corresponding to the ±1 eigenspaces of E. The distributions on M defined by T^{+}M and T^{−}M are completely integrable. When T^{+}M and T^{−}M have the same rank the product structure is called a paracomplex structure.

Product structures on manifolds were considered by many authors from different points of view. Examples of Riemannian almost product structures were given in [Miq] and a survey on paracomplex geometry can be found in [CFG]. The classification of Riemannian almost product manifolds according to a certain decomposition of the space of tensors was done in [N]. In [LM] the authors give a new look at singular and non holonomic Lagrangian systems in the framework of almost product structures. Complex product structures on Lie groups were considered in [AS] and [BV].

In this paper we consider product structures on four dimensional solvable Lie groups. Such groups provide an important source of applications in geometry. Invariant structures on the group, for instance, special metrics [Al], [B2], [DS], [F1], [F2], [J], complex and Kähler structures [ACFM], [AFGM], [O1], [SJ], [FG], hypercomplex and hypersymplectic structures [An], [B1], can be read off in ℝ^{4},

The authors were partially supported by CONICET and SECYT-UNC (Argentina). Received December 30, 2003, revised December 28, 2004; published on April 7, 2005. 2000 Mathematics Subject Classification: Primary 53C15; Secondary 22E25. Key words and phrases: solvable Lie algebra, product structure, paracomplex structure. \begin{matrix}c \\ ◯ \end{matrix} 2005, A. Andrada, M. L. Barberis, I. G. Dotti and G. P. Ovando. Permission to copy for private use granted.

the universal covering group, giving often explicit descriptions of the corresponding structure.

A left-invariant product structure on a Lie group is determined by its restriction to the corresponding Lie algebra, considered as the tangent space at the identity. A product structure on a Lie algebra 𝔤 is a linear endomorphism E : 𝔤 ⟶ 𝔤 satisfying E^{2} = Id (and not equal to ±Id) and

E[x, y] = [Ex, y] + [x, Ey] − E[Ex, Ey] for all x, y ∈ 𝔤.  (2)

A product structure on 𝔤 gives rise to a decomposition of 𝔤 into

𝔤 = 𝔤_{+} ⊕ 𝔤_{−}, E|𝔤_{+} = Id, E|𝔤_{−} = −Id,  (3)

where both, 𝔤_{+} and 𝔤_{−}, are Lie subalgebras of 𝔤. This will be denoted 𝔤 = 𝔤_{+} ▷◁ 𝔤_{−}, since the structure of 𝔤 is that of a double Lie algebra ([LW]). In case both 𝔤_{+} and 𝔤_{−} have the same dimension we say that 𝔤 carries a paracomplex structure.

The outline of this paper is as follows. In Section 1 we describe all non-isomorphic four dimensional solvable Lie algebras over ℝ. This was studied by Mubarakzyanov [Mu] and Dozias [D]. We found citations of the theorems obtained by Mubarakzyanov in [PSWZ], pp. 988 and Dozias in [Ve], pp. 180. We include a proof of the classification theorem since it will be frequently used to obtain the results throughout the article. Appendix II contains comparisons with the tables given by the various authors [Mu], [D], [SJ], [O1], [PSWZ].

In Section 2 we consider product structures on four dimensional Lie algebras. We determine all four dimensional solvable Lie algebras admitting a paracomplex structure (see Table 2). Among these, we study the case when one of the subalgebras is an ideal of 𝔤. We also exhibit decompositions where one of the subalgebras is three dimensional (see Table 3).

An important subclass of paracomplex structures is given by Manin triples and complex product structures (see Section 3). A paracomplex structure 𝔤 = 𝔤_{+} ▷◁ 𝔤_{−} is a Manin triple if there exists a non degenerate invariant symmetric bilinear form on 𝔤 such that 𝔤_{±} are isotropic subalgebras. It is shown that there is only one non abelian four dimensional solvable Lie algebra giving rise to a Manin triple. On the other hand, given a product structure E and a complex structure J on 𝔤 such that JE = −EJ, {J, E} is called a complex product structure on 𝔤. We determine all four dimensional solvable Lie algebras admitting complex product structures (see Table 4), giving an alternative proof of a result by Blazić and Vukmirović ([BV]).

1. Classification of four dimensional solvable Lie algebras

In this section we obtain the classification of four dimensional solvable Lie algebras. The proof follows the lines of [Mi] for the classification of three dimensional solvable Lie algebras, that is, we obtain the four dimensional solvable Lie algebras as extensions of the three dimensional unimodular Lie algebras ℝ^{3}, the Heisenberg algebra 𝔥_{3}, the Poincaré algebra 𝔢(1, 1) or the Euclidean algebra 𝔢(2). Both, [O1] and [SJ], obtain the four dimensional solvable Lie algebras as extensions of nilpotent Lie algebras of dimension at most three. In Appendix I we exhibit matrix realizations and Appendix II contains comparisons with the tables given by the various authors [Mu], [D], [SJ], [O1], [PSWZ].

1.1. Algebraic preliminaries

A Lie algebra 𝔤 which satisfies the condition tr(ad(x)) = 0 for all x ∈ 𝔤 will be called a unimodular Lie algebra. If 𝔤 is a Lie algebra, then using the Jacobi identity we see that tr(ad[x, y]) = 0 for all x, y ∈ 𝔤. Hence, the map χ : 𝔤 → ℝ defined by

χ(x) = tr(ad(x)),  x ∈ 𝔤,  (4)

is a Lie algebra homomorphism. In particular, its kernel 𝔲 = ker(χ) is an ideal containing the commutator ideal [𝔤, 𝔤]. The ideal 𝔲 will be called the unimodular kernel of 𝔤. It is easy to check that 𝔲 itself is unimodular.

We now introduce some notation that will be used throughout the paper (compare with [GOV]). 𝔞𝔣𝔣(ℝ): [e_{1}, e_{2}] = e_{2}, the two dimensional non-abelian Lie algebra of the group of

affine motions of the real line; 𝔥_{3}: [e_{1}, e_{2}] = e_{3}, the three-dimensional Heisenberg algebra; 𝔯_{3}: [e_{1}, e_{2}] = e_{2}, [e_{1}, e_{3}] = e_{2} + e_{3}; 𝔯_{3,λ}: [e_{1}, e_{2}] = e_{2}, [e_{1}, e_{3}] = λe_{3}; 𝔯′_{3,λ}: [e_{1}, e_{2}] = λe_{2} − e_{3}, [e_{1}, e_{3}] = e_{2} + λe_{3}; Remark. Observe that 𝔯_{3,−1} is the Lie algebra 𝔢(1, 1) of the group of rigid motions of Minkowski 2-space, 𝔯_{3,0} = ℝ × 𝔞𝔣𝔣(ℝ) and 𝔯_{3,1} is the Lie algebra of the solvable group which acts simply and transitively on the real hyperbolic space ℝH^{3}. Also 𝔯′_{3,0} is the Lie algebra 𝔢(2) of the group of rigid motions of Euclidean 2-space. Other authors denote 𝔞𝔣𝔣(ℝ) by 𝔰𝔬𝔩_{2} and 𝔢(1, 1) by 𝔰𝔬𝔩_{3}.

We recall the classification of solvable Lie algebras of dimension ⩽ 3. A proof can be found, for example, in [Mi] or [GOV].

Theorem 1.1. Let 𝔤 be a real solvable Lie algebra, dim 𝔤 ⩽ 3. Then 𝔤 is isomorphic to one and only one of the following Lie algebras: ℝ, ℝ^{2}, 𝔞𝔣𝔣(ℝ), ℝ^{3}, 𝔥_{3}, 𝔯_{3}, 𝔯_{3,λ}, |λ| ⩽ 1 and 𝔯′_{3,λ}, λ ⩾ 0. Among these, the unimodular ones are ℝ, ℝ^{2}, ℝ^{3}, 𝔥_{3}, 𝔯_{3,−1}, and 𝔯′_{3,0}.

The proof of Theorem 1.5 in next section is based on the knowledge of the algebra of derivations of solvable unimodular three dimensional Lie algebras. This is the content of the next lemma, whose proof is straightforward.

Lemma 1.2. The algebra of derivations of 𝔢(2), 𝔢(1, 1) and 𝔥_{3} are

Der 𝔢(2) = \{(\begin{matrix}0 & 0 & 0 \\ c & a & −b \\ d & b & a \end{matrix}) : a, b, c, d ∈ ℝ\} ,  (5)

with respect to the basis e_{i}, i = 1, 2, 3, such that [e_{1}, e_{2}] = e_{3}, [e_{1}, e_{3}] = −e_{2};

Der 𝔢(1, 1) = \{(\begin{matrix}0 & 0 & 0 \\ c & a & 0 \\ d & 0 & b \end{matrix}) : a, b, c, d ∈ ℝ\} ≅ 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ),  (6)

with respect to the basis e_{i}, i = 1, 2, 3 such that [e_{1}, e_{2}] = e_{2}, [e_{1}, e_{3}] = −e_{3};

Der 𝔥_{3} = \{( \begin{matrix}A & {0 \atop 0} \\ b c & tr A \end{matrix}) : A ∈ 𝔤𝔩(2, ℝ), b, c ∈ ℝ\} ,  (7)

with respect to the basis e_{i}, i = 1, 2, 3 such that [e_{1}, e_{2}] = e_{3}.

1.2. Classification theorem

In this section we obtain all four dimensional solvable Lie algebras as semidirect extensions of three dimensional unimodular Lie algebras. The classification theorem is then reduced to the study of the derivations of these three dimensional algebras. The proof will follow the lines of [Mi] for the three dimensional case, but instead of the rational form, we make use of the Jordan normal form over ℝ.

Given a Lie algebra 𝔤 and an ideal 𝔳 of codimension one in 𝔤, let e_{0} ∈ 𝔤 \ 𝔳. Then we denote

𝔤 = ℝe_{0} ⋉_{φ} 𝔳,  (8)

where φ : ℝe_{0} → Der 𝔳 is a linear map such that φ(e_{0}) = ad(e_{0}). Observe that the splitting of the short exact sequence

0 → 𝔳 → 𝔤 → ℝ → 0,

is an immediate consequence of the fact that ℝ is one dimensional.

The following result proves the desired decomposition, that is, any four dimensional solvable real Lie algebra is a semidirect product of ℝ and a three-dimensional unimodular ideal. Thus this proposition is a first step in the classification (compare with Proposition 2.1 in [DS]):

Proposition 1.3. Let 𝔤 be a four-dimensional solvable real Lie algebra. Then there is a short exact sequence

0 → 𝔳 → 𝔤 → ℝ → 0,

where 𝔳 is an ideal of 𝔤 isomorphic to either ℝ^{3}, 𝔥_{3}, 𝔢(1, 1) or 𝔢(2), that is, 𝔤 ≅ ℝe_{0} ⋉_{φ} 𝔳.

Proof. Consider the Lie algebra homomorphism χ : 𝔤 → ℝ defined in (4). If 𝔤 is not unimodular then its unimodular kernel 𝔲 has dimension three, therefore it is isomorphic to ℝ^{3}, 𝔥_{3}, 𝔢(1, 1) or 𝔢(2) and the proposition follows with 𝔳 = 𝔲.

We assume now that 𝔤 is unimodular. The commutator ideal 𝔤′ is nilpotent and dim 𝔤′ ⩽ 3, hence it follows that 𝔤′ is isomorphic to {0}, ℝ, ℝ^{2}, ℝ^{3} or 𝔥_{3}. In the last two cases the proposition follows by taking 𝔳 = 𝔤′. If 𝔤′ = {0} then 𝔤 is abelian so that 𝔳 = ℝ^{3} is an ideal of 𝔤.

If 𝔤′ is isomorphic to ℝ, 𝔤′ = ℝe_{3}, then there exist elements e_{1}, e_{2} in 𝔤 such that [e_{1}, e_{2}] = e_{3}. The set e_{1}, e_{2}, e_{3} is linearly independent since 𝔤 is unimodular. Therefore, the Lie subalgebra generated by e_{1}, e_{2}, e_{3} is an ideal isomorphic to 𝔥_{3}.

If 𝔤′ is isomorphic to ℝ^{2} then either i) there exists x not in 𝔤′ such that ad(x)_{|𝔤′} is non singular, or ii) for all x ∈ 𝔤 the transformation ad(x) is singular. Making use of the Jordan form of the corresponding complex transformation we get in both cases i) and ii), that χ(x) = λ_{1} + λ_{2} = 0, for λ_{i} ∈ ℂ, i = 1 or 2. Thus in case i) there is a basis of 𝔤′ such that the action of x is given as follows (up to a nonzero multiple):

a) ad(x)|_{𝔤′} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix})  or  b) ad(x)|_{𝔤′} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix})

where case b) corresponds to the eigenvalues i, −i. Thus ℝx ⊕ 𝔤′ is an ideal of 𝔤 isomorphic to 𝔢(1, 1) or to 𝔢(2), respectively.

In case ii), since λ_{1} or λ_{2} is zero, then the unimodular condition imposes that both eigenvalues vanish and so, for a fixed x not in 𝔤′, there is a basis of 𝔤′ such that the action of ad(x)|_{𝔤′} takes one of the following forms:

a) ad(x)|_{𝔤′} = 0  or  b) ad(x)|_{𝔤′} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix})

Therefore, ℝx ⊕ 𝔤′ is an ideal of 𝔤 isomorphic to ℝ^{3} in case a) or 𝔥_{3} in case b). This completes the proof.

The following lemma will be used in the proof of the classification theorem.

Lemma 1.4. Let 𝔤_{1} = ℝe_{0} ⋉_{φ_{1}} ℝ^{3} and 𝔤_{2} = ℝe_{0} ⋉_{φ_{2}} ℝ^{3} such that [𝔤_{i}, 𝔤_{i}] = ℝ^{3}, i = 1, 2. Then 𝔤_{1} ≅ 𝔤_{2} if and only if there exists γ ≠ 0 such that φ_{1}(e_{0}) and γφ_{2}(e_{0}) are conjugate in GL(3, ℝ).

Proof. Assume first that there exists a Lie algebra isomorphism ψ : 𝔤_{1} → 𝔤_{2}; then ψ : ℝ^{3} → ℝ^{3} and ψ(e_{0}) = γe_{0} + w, where γ ∈ ℝ ∖ {0} and w ∈ ℝ^{3}. If v ∈ ℝ^{3}, we calculate

[ψ(e_{0}), ψ(v)] = γφ_{2}(e_{0})ψ(v),
ψ([e_{0}, v]) = ψ(φ_{1}(e_{0})v),

and therefore γφ_{2}(e_{0})ψ(v) = ψ(φ_{1}(e_{0})v) for all v ∈ ℝ^{3}, that is, γφ_{2}(e_{0}) = ψφ_{1}(e_{0})ψ^{−1}.

The converse is straightforward.

Dozias and Mubarakzyanov gave in [D] and [Mu] a classification of four dimensional solvable Lie algebras. We prove below this result to make this article self contained. The proof uses Proposition 1.3 together with Lemma 1.2.

Theorem 1.5. Let 𝔤 be a four-dimensional solvable real Lie algebra. Then 𝔤 is isomorphic to one and only one of the following Lie algebras: ℝ^{4}, 𝔞𝔣𝔣(ℝ)×𝔞𝔣𝔣(ℝ), ℝ× 𝔥_{3}, ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ}, |λ| ⩽ 1, ℝ × 𝔯′_{3,λ}, λ ⩾ 0, or one of the Lie algebras with brackets given below in the basis e_{i}, i = 0, 1, 2, 3:

𝔫_{4}: [e_{0}, e_{1}] = e_{2}, [e_{0}, e_{2}] = e_{3};

𝔞𝔣𝔣(ℂ): [e_{0}, e_{2}] = e_{2}, [e_{0}, e_{3}] = e_{3}, [e_{1}, e_{2}] = e_{3}, [e_{1}, e_{3}] = −e_{2}; 𝔯_{4}: [e_{0}, e_{1}] = e_{1}, [e_{0}, e_{2}] = e_{1} + e_{2}, [e_{0}, e_{3}] = e_{2} + e_{3}; 𝔯_{4,λ}: [e_{0}, e_{1}] = e_{1}, [e_{0}, e_{2}] = λe_{2}, [e_{0}, e_{3}] = e_{2} + λe_{3}; 𝔯_{4,μ,λ}: [e_{0}, e_{1}] = e_{1}, [e_{0}, e_{2}] = μe_{2}, [e_{0}, e_{3}] = λe_{3}, μλ ≠ 0, −1 < μ ⩽ λ ⩽ 1

or −1 = μ ⩽ λ < 0 ;

𝔯′_{4,μ,λ}: [e_{0}, e_{1}] = μe_{1}, [e_{0}, e_{2}] = λe_{2} − e_{3}, [e_{0}, e_{3}] = e_{2} + λe_{3}, μ > 0;

𝔡_{4}: [e_{0}, e_{1}] = e_{1}, [e_{0}, e_{2}] = −e_{2}, [e_{1}, e_{2}] = e_{3};

𝔡_{4,λ}: [e_{0}, e_{1}] = λe_{1}, [e_{0}, e_{2}] = (1 − λ)e_{2}, [e_{0}, e_{3}] = e_{3}, [e_{1}, e_{2}] = e_{3}, λ ⩾ \frac{1}{2};

𝔡′_{4,λ}: [e_{0}, e_{1}] = λe_{1} − e_{2}, [e_{0}, e_{2}] = e_{1} + λe_{2}, [e_{0}, e_{3}] = 2λe_{3}, [e_{1}, e_{2}] = e_{3}, λ ⩾ 0;

𝔥_{4}: [e_{0}, e_{1}] = e_{1}, [e_{0}, e_{2}] = e_{1} + e_{2}, [e_{0}, e_{3}] = 2e_{3}, [e_{1}, e_{2}] = e_{3}.

Among these, the unimodular algebras are: ℝ^{4}, ℝ × 𝔥_{3}, ℝ × 𝔯_{3,−1}, ℝ × 𝔯′_{3,0}, 𝔫_{4}, 𝔯_{4,−1/2}, 𝔯_{4,μ,−1−μ} (−1 < μ ⩽ −1/2), 𝔯′_{4,μ,−μ/2}, 𝔡_{4}, 𝔡′_{4,0}.

Proof. In view of Proposition 1.3 there exists a three dimensional ideal 𝔳 of 𝔤 isomorphic to ℝ^{3}, 𝔢(2), 𝔢(1, 1) or 𝔥_{3}. We will analyze below the different cases.

1.3. Case 𝔳 = ℝ^{3}.

We introduce first the following 3 × 3 real matrices which will be needed in the next paragraphs:

A_{1}^{μ,λ} = (\begin{matrix}1 & 0 & 0 \\ 0 & μ & 0 \\ 0 & 0 & λ \end{matrix}) ,  A_{2}^{λ} = (\begin{matrix}1 & 0 & 0 \\ 0 & λ & 1 \\ 0 & 0 & λ \end{matrix}) ,  (9)

A_{3} = (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) ,  A_{4}^{μ,λ} = (\begin{matrix}μ & 0 & 0 \\ 0 & λ & 1 \\ 0 & −1 & λ \end{matrix}) .  (10)

By assumption, 𝔤 = ℝe_{0}⋉_{φ}ℝ^{3} where φ(e_{0}) = ad(e_{0}). Suppose first that φ(e_{0}) has real eigenvalues. We have the following possibilities for φ(e_{0}), where the eigenvalues are ordered such that |λ_{1}| ⩽ |λ_{2}| ⩽ |λ_{3}|:

i) φ(e_{0}) = (\begin{matrix}λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix}) , ii) φ(e_{0}) = (\begin{matrix}λ_{1} & 0 & 0 \\ 0 & λ_{2} & 1 \\ 0 & 0 & λ_{2} \end{matrix}) ,

iii) φ(e_{0}) = (\begin{matrix}λ & 1 & 0 \\ 0 & λ & 1 \\ 0 & 0 & λ \end{matrix}) .

Case i) \begin{cases}λ_{i} = 0, i = 1, 2, 3, & then 𝔤 ≅ ℝ^{4}; \\ λ_{1} = 0, λ_{3} ≠ 0, & then 𝔤 ≅ 𝔯_{3,λ} × ℝ; where λ = \frac{λ_{2}}{λ_{3}} ; \\ λ_{1}λ_{2}λ_{3} ≠ 0 & then 𝔤 ≅ ℝ ⋉_{φ_{1}} ℝ^{3}, φ_{1}(e_{0}) = A_{1}^{μ,λ}as shown in (9), \\ & that is, 𝔤 ≅ 𝔯_{4,μ,λ}. \end{cases}

The last isomorphism in Case i) follows by dividing e_{0} by λ_{3} and by reordering suitably the basis {e_{1}, e_{2}, e_{3}} of ℝ^{3}, we may assume that −1 ⩽ μ ⩽ λ ⩽ 1.

Case ii) \begin{cases}λ_{1} = λ_{2} = 0 & then 𝔤 ≅ ℝ × 𝔥_{3}; \\ λ_{1} = 0, λ_{2} ≠ 0 & then 𝔤 ≅ ℝ × 𝔯_{3}; \\ λ_{1} ≠ 0, & then 𝔤 ≅ ℝ ⋉_{φ_{2}^{λ}} ℝ^{3}, φ_{2}^{λ}(e_{0}) = A_{2}^{λ} as shown in (9), \\ & that is, 𝔤 ≅ 𝔯_{4,λ}. \end{cases}

Case iii) \begin{cases}λ = 0, & then 𝔤 ≅ 𝔫_{4}; \\ λ ≠ 0, & then 𝔤 ≅ ℝ ⋉_{φ_{3}} ℝ^{3}, φ_{3}(e_{0}) = A_{3} as shown in (10), that is, \\ & 𝔤 ≅ 𝔯_{4}. \end{cases}

The last isomorphism in case iii) follows by taking e_{0}/λ.

In case φ(e_{0}) has only one real eigenvalue, μ, then we may assume that φ(e_{0}) = A_{4}^{μ,λ} as in (10) and we have:

μ = 0, then 𝔤 ≅ ℝ × 𝔯′_{3,λ};
μ ≠ 0, then 𝔤 ≅ 𝔯′_{4,μ,λ}, μ > 0.

Observe that the last isomorphism follows by changing e_{0} by −e_{0}.

1.4. Case 𝔳 = 𝔢(2).

Assume that 𝔤 = ℝe_{0} ⋉_{φ} 𝔢(2) where φ(e_{0}) = ad(e_{0}) ∈ Der 𝔢(2) is as in (5). Then setting e′_{0} = e_{0} − be_{1} + de_{2} − ce_{3} , it follows that

[e′_{0}, e_{1}] = 0, [e′_{0}, e_{2}] = ae_{2}, [e′_{0}, e_{3}] = ae_{3};

therefore, 𝔤 ≅ ℝ × 𝔢(2) = ℝ × 𝔯′_{3,0} or 𝔤 ≅ 𝔞𝔣𝔣(ℂ) depending on a = 0 or a ≠ 0, respectively.

1.5. Case 𝔳 = 𝔢(1, 1).

Assume that 𝔤 = ℝe_{0} ⋉_{φ} 𝔢(1, 1) where φ(e_{0}) = ad(e_{0}) ∈ Der 𝔢(1, 1) is as in (6). Let e′_{0} = e_{0} − ae_{1} + ce_{2} − de_{3}, then

[e′_{0}, e_{1}] = 0, [e′_{0}, e_{2}] = 0, [e′_{0}, e_{3}] = (a + b)e_{3};

therefore, 𝔤 ≅ ℝ × 𝔢(1, 1) = ℝ × 𝔯_{3,−1} or 𝔤 ≅ 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) depending on a + b = 0 or a + b ≠ 0, respectively.

1.6. Case 𝔳 = 𝔥_{3}.

Assume that 𝔤 ≅ ℝe_{0} ⋉_{φ} 𝔥_{3} where φ(e_{0}) = ad(e_{0}) is given by

( \begin{matrix}A & {0 \atop 0} \\ b c & trA \end{matrix})

(see (7)). We may assume that b = c = 0. In fact, setting e′_{0} = e_{0} − ce_{1} + be_{2} it turns out that ad(e′_{0}) is given by

( \begin{matrix}A & {0 \atop 0} \\ 0 0 & trA \end{matrix}) .  (11)

Assume first that A has two real eigenvalues γ, β; then A takes the form

i) A = (\begin{matrix}γ & 0 \\ 0 & β \end{matrix}) ,  or  ii) A = (\begin{matrix}γ & 1 \\ 0 & γ \end{matrix}) .

Observe that in all cases, once we change the basis e_{1}, e_{2} to e′_{1}, e′_{2}, we must set e′_{3} = [e′_{1}, e′_{2}] in order to obtain a Lie algebra isomorphism.

Case i)\begin{cases}γ = β = 0, then 𝔤 ≅ ℝ × 𝔥_{3}; \\ γ = −β ≠ 0, then 𝔤 ≅ 𝔡_{4}; \\ γ + β ≠ 0, then 𝔤 ≅ 𝔡_{4,λ}, λ = \frac{γ}{γ + β}; \end{cases} Case ii)\begin{cases}γ = 0, then 𝔤 ≅ 𝔫_{4}; \\ γ ≠ 0, then 𝔤 ≅ 𝔥_{4}. \end{cases}

We show that 𝔡_{4,λ} ≅ 𝔡_{4,1−λ}. This follows by changing the basis e_{i}, 0 ⩽ i ⩽ 3, to the basis e′_{i}, 0 ⩽ i ⩽ 3, where:

e′_{0} = e_{0}, e′_{1} = e_{2}, e′_{2} = e_{1}, e′_{3} = −e_{3}.

Therefore, we may assume that λ ⩾ 1/2.

In case ii), γ ≠ 0, in order to show that 𝔤 ≅ 𝔥_{4} one has to start with e′_{0} = \frac{1}{γ} e_{0},

then take e′_{1}, e′_{2} ∈ span{e_{1}, e_{2}} such that

ad(e′_{0}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})

with respect to {e′_{1}, e′_{2}} and e′_{3} = [e′_{1}, e′_{2}].

If A has no real eigenvalues, then ad(e′_{0}) takes the form

(\begin{matrix}λ & 1 & 0 \\ −1 & λ & 0 \\ 0 & 0 & 2λ \end{matrix}) ,

and we conclude that 𝔤 ≅ 𝔡′_{4,λ}. Hence, we have shown so far that any four dimensional solvable Lie algebra is isomorphic to one of those listed in the statement of the theorem. It remains to show that they are pairwise non isomorphic.

1.7. Isomorphism classes

In Table 1, we list the four dimensional solvable Lie algebras according to their commutator. After that, we proceed to distinguish them up to isomorphism.

• [𝔤, 𝔤] = ℝ: ℝ × 𝔥_{3} is nilpotent but ℝ × 𝔯_{3,0} is not, therefore they are not isomorphic.

• [𝔤, 𝔤] = ℝ^{2}, 𝔷 = {0}: Both 𝔞𝔣𝔣(ℝ)×𝔞𝔣𝔣(ℝ) and 𝔡_{4,1} are completely solvable^{1} and therefore not isomorphic to 𝔞𝔣𝔣(ℂ), which is not completely solvable. The unimodular kernel of 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) (resp. 𝔡_{4,1}) is 𝔯_{3,−1} (resp. 𝔥_{3}), hence 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) is not isomorphic to 𝔡_{4,1}.

1Recall that a solvable Lie algebra 𝔤 is completely solvable when ad(x) has real eigenvalues for all x ∈ 𝔤.

\begin{matrix}[𝔤, 𝔤] & 𝔤 \\ \{0\} & ℝ^{4} \\ ℝ & ℝ × 𝔥_{3} , ℝ × 𝔯_{3,0} \\ ℝ^{2}, 𝔷 = \{0\} & 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ), 𝔞𝔣𝔣(ℂ), 𝔡_{4,1} \\ ℝ^{2}, 𝔷 ≠ \{0\} & ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ} (|λ| ⩽ 1, λ ≠ 0), ℝ × 𝔯′_{3,λ} (λ ⩾ 0), 𝔯_{4,0}, 𝔫_{4} \\ ℝ^{3} & 𝔯4 , 𝔯4,λ (λ ≠ 0) , 𝔯4,μ,λ (μλ ≠ 0, −1 ⩽ μ ⩽ λ ⩽ 1), 𝔯{′ \atop 4,μ,λ} (μ > 0) \\ 𝔥_{3} & 𝔡4, 𝔡4,λ (λ ≠ 1, λ ⩾ 1/2), 𝔡{′ \atop 4,λ} (λ ⩾ 0), 𝔥4 \end{matrix} Table 1:

• [𝔤, 𝔤] = ℝ^{2}, 𝔷 ≠ {0}: If 𝔤 = ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ} (|λ| ⩽ 1, λ ≠ 0) or ℝ × 𝔯′_{3,λ} (λ ⩾ 0) then 𝔷 ∩ [𝔤, 𝔤] = {0}, while 𝔷 ∩ [𝔤, 𝔤] ≠ {0} when 𝔤 = 𝔯_{4,0} or 𝔫_{4}. Also 𝔤 = ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ} (|λ| ⩽ 1, λ ≠ 0) and ℝ × 𝔯′_{3,λ} (λ ⩾ 0) are not pairwise isomorphic since 𝔯_{3}, 𝔯_{3,λ} and 𝔯′_{3,λ} are 3-dimensional non isomorphic Lie algebras. On the other hand, 𝔫_{4} is nilpotent but 𝔯_{4,0} is not, hence they are not isomorphic.

• [𝔤, 𝔤] = ℝ^{3}: 𝔯_{4}, 𝔯_{4,λ} (λ ≠ 0), 𝔯_{4,μ,λ}, 𝔯′_{4,μ,λ}. In this case, it follows from Lemma 1.4 that any pair of Lie algebras belonging to different families can not be isomorphic. The last family consists of non completely solvable Lie algebras.

The fact that two Lie algebras 𝔯_{4,λ}, λ ≠ 0, and 𝔯_{4,λ′}, λ′ ≠ 0, are isomorphic if and only if λ = λ′ follows by applying Lemma 1.4.

Let us show that if 𝔯_{4,μ,λ}, −1 < μ ⩽ λ ⩽ 1, μλ ≠ 0, is isomorphic to 𝔯_{4,μ′,λ′}, −1 < μ′ ⩽ λ′ ⩽ 1, μ′λ′ ≠ 0, then μ = μ′ and λ = λ′. From Lemma 1.4, there exists γ ≠ 0 such that the sets of eigenvalues {1, μ, λ} and {γ, γμ′, γλ′} must coincide. If γ = 1 the desired assertion follows from μ ⩽ λ and μ′ ⩽ λ′. If γ = μ then either γμ′ = 1 or γλ′ = 1, hence μ′ = 1 or λ′ = 1, therefore γ = 1 and again this implies μ = μ′, λ = λ′. The case γ = λ is proved in a similar way.

Let us show that if 𝔯_{4,−1,λ}, −1 ⩽ λ < 0, is isomorphic to 𝔯_{4,−1,λ′}, −1 ⩽ λ′ < 0, then λ = λ′. We apply Lemma 1.4 again to obtain that there exists γ ≠ 0 such that {1, −1, λ} and {γ, −γ, γλ′} must coincide. We cannot have γ = −1, since this would imply λ = −λ′, a contradiction, since both, λ and λ′ are negative. If γ = λ, then −γ = 1 and −1 = γλ′ = λλ′ > 0, a contradiction. Thus γ = 1 and λ = λ′.

If 𝔯_{4,μ,λ}, −1 < μ ⩽ λ ⩽ 1, μλ ≠ 0, were isomorphic to 𝔯_{4,−1,λ′}, −1 ⩽ λ′ < 0, then Lemma 1.4 would imply that that there exists γ ≠ 0 such that {1, μ, λ} = {γ, −γ, γλ′}. If γ = 1 then μ = −1, which is impossible. On the other hand, γ = μ implies −γ = 1 or γλ′ = 1, hence μ = γ = −1, a contradiction. The case γ = λ is similar; therefore, the above Lie algebras are not isomorphic.

Assume now that 𝔯′_{4,μ,λ}, μ > 0, is isomorphic to 𝔯′_{4,μ′,λ′}, μ′ > 0, we must show that μ = μ′ and λ = λ′. We apply Lemma 1.4 again to obtain that there exists γ ≠ 0 such that μ = γμ′ and λ ± i = γ(λ′ ± i). It follows from the second equality that γ = ±1, and the first equality implies γ = 1, since both μ and μ′ are positive. Therefore, μ = μ′ and λ = λ′, as claimed.

• [𝔤, 𝔤] = 𝔥_{3}: The Lie algebras 𝔡_{4}, 𝔡_{4,λ} (λ ⩾ 1/2, λ ≠ 1), and 𝔡′_{4,λ}, 𝔥_{4} are distinguished by 𝔤/𝔷([𝔤, 𝔤]), as the following table shows:

\begin{matrix}𝔤 & 𝔡_{4} & 𝔡_{4,λ}, \begin{cases}λ ⩾ 1/2 \\ λ ≠ 1 \end{cases} & 𝔡{′ \atop 4,λ}, λ ⩾ 0 & 𝔥_{4} \\ 𝔤/𝔷([𝔤, 𝔤]) & 𝔯_{3,−1} & 𝔯_{3,−1+1/λ} & 𝔯′_{3,λ} & 𝔯_{3} \end{matrix}

Remarks. (i) In [DS] it was proved that 𝔡′_{4,λ}, λ ⩾ 0, are all non-isomorphic. Observe that 𝔤_{λ} in [DS] corresponds to 𝔡′_{4,1/λ} for λ ≠ 0 (resp. 𝔡_{4,1/2} for λ = 0). (ii) We observe that 𝔞𝔣𝔣(ℂ) is the Lie algebra of the group of affine motions of the complex line, which is isomorphic to the complexification of 𝔞𝔣𝔣(ℝ) looked upon as a real Lie algebra. [It should be noted that the Lie algebra Der 𝔢(2) (see Lemma 1.2) is isomorphic to 𝔞𝔣𝔣(ℂ).] Also, 𝔯_{4,1,1} is the Lie algebra of a solvable Lie group which acts simply and transitively on the real hyperbolic space ℝH^{4} and 𝔡_{4,1/2} is the Lie algebra of a solvable Lie group which acts simply and transitively on the complex hyperbolic space ℂH^{2}.

2. Product structures on four dimensional solvable Lie algebras

2.1. Basic definitions

An almost product structure on a Lie algebra 𝔤 is a linear endomorphism E : 𝔤 ⟶ 𝔤 satisfying E^{2} = Id (and not equal to ±Id). It is said to be integrable if

E[x, y] = [Ex, y] + [x, Ey] − E[Ex, Ey] for all x, y ∈ 𝔤.  (12)

An integrable almost product structure will be called a product structure. An almost product structure on 𝔤 gives rise to a decomposition of 𝔤 into

𝔤 = 𝔤_{+} ⊕ 𝔤_{−}, E|𝔤_{+} = Id, E|𝔤_{−} = −Id.  (13)

The integrability of E is equivalent to 𝔤_{+} and 𝔤_{−} being subalgebras. When dim 𝔤_{+} = dim 𝔤_{−}, the product structure E is called a paracomplex structure.

Three Lie algebras (𝔤, 𝔤_{+}, 𝔤_{−}) form a double Lie algebra if 𝔤_{+} and 𝔤_{−} are Lie subalgebras of 𝔤 and 𝔤 = 𝔤_{+} ⊕ 𝔤_{−} as vector spaces. This will be denoted by 𝔤 = 𝔤_{+} ▷◁ 𝔤_{−}. Observe that a double Lie algebra (𝔤, 𝔤_{+}, 𝔤_{−}) gives a product structure E : 𝔤 ⟶ 𝔤 on 𝔤, where E|𝔤_{+} = Id and E|𝔤_{−} = −Id. Conversely, a product structure on the Lie algebra 𝔤 gives rise to a double Lie algebra (𝔤, 𝔤_{+}, 𝔤_{−}), where 𝔤_{±} is the eigenspace associated to the eigenvalue ±1 of E. The notion of double Lie algebra is a natural generalization of that of semidirect product. We will denote 𝔤 = 𝔤_{+} ⋉ 𝔤_{−} the semidirect product of 𝔤_{+} and 𝔤_{−} where 𝔤_{−} is an ideal of 𝔤, that is, there is a split exact sequence

0 ⟶ 𝔤_{−} ⟶ 𝔤 ⟶ 𝔤_{+} ⟶ 0.

Product structures or, equivalently, double Lie algebras, were used in several contexts (see [AS], [LW]). Important examples of double Lie algebras are Manin triples and complex product structures.

𝔤  ℝ^{2} ▷◁ ℝ^{2}  𝔞𝔣𝔣(ℝ) ▷◁ ℝ^{2}  𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ)
ℝ^{4}  ⟨e_{0}, e_{1}⟩ × ⟨e_{2}, e_{3}⟩  no  no
𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  {⟨e_{1} + e_{3}, e_{2}⟩ \atop ▷◁ ⟨e_{0}, e_{1}⟩}  ⟨e_{0}, e_{3}⟩ × ⟨e_{1}, e_{2}⟩
ℝ × 𝔥_{3}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩  no  no
ℝ × 𝔯_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  ⟨e_{1}, e_{2}⟩ ▷◁ ⟨e_{0}, e_{3}⟩  no
ℝ × 𝔯_{3,λ}, λ ≠ 0 ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  ⟨e_{1}, e_{2}⟩ ⋉ ⟨e_{0}, e_{3}⟩  {⟨e_{0} + e_{1}, e_{2}⟩ \atop ▷◁ ⟨e_{1} − λe_{0}, e_{3}⟩}
ℝ × 𝔯_{3,0}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  ⟨e_{1}, e_{2}⟩ × ⟨e_{0}, e_{3}⟩  no
ℝ × 𝔯′_{3,λ}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  no  no
𝔫_{4}  ⟨e_{0}, e_{3}⟩ ▷◁ ⟨e_{1}, e_{2}⟩  no  no
𝔞𝔣𝔣(ℂ)  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩ {⟨e_{0}, e_{2}⟩ \atop ▷◁ ⟨e_{0} − e_{3}, e_{1} + e_{2}⟩}  no
𝔯_{4}  no  ⟨e_{0}, e_{1}⟩ ▷◁ ⟨e_{2}, e_{3}⟩  no
𝔯_{4,λ}, λ ≠ 0  no  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  {⟨e_{0}, e_{1}⟩ \atop ▷◁ ⟨e_{0} + λe_{3}, e_{2}⟩}
𝔯_{4,0}  ⟨e_{0}, e_{2}⟩ ▷◁ ⟨e_{1}, e_{3}⟩  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  no
𝔯_{4,μ,λ}  no  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  {⟨e_{0} − e_{1}, e_{2}⟩ \atop ▷◁ ⟨e_{0} + e_{1}, e_{3}⟩}
𝔯′_{4,μ,λ}  no  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  no
𝔡_{4}  no  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  {⟨e_{0} + e_{2}, e_{1} − e_{3}⟩ \atop ▷◁ ⟨e_{0} − e_{2}, e_{1} + e_{3}⟩}
𝔡_{4,λ}, λ ≠ 1  no  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  {⟨e_{0}, e_{3}⟩ ▷◁ ⟨e_{0} + λe_{2}, \atop (1 − λ)e_{1} + λe_{3}⟩}
𝔡_{4,1}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩  {⟨e_{0}, e_{1}⟩ \atop ▷◁ ⟨e_{0} + e_{2}, e_{3}⟩}
𝔡′_{4,λ}  no  no  no
𝔥_{4}  no  ⟨e_{0}, e_{1}⟩ ▷◁ ⟨e_{2}, e_{3}⟩  {⟨e_{0}, e_{3}⟩ \atop ▷◁ ⟨e_{0} − e_{2}, e_{1} − e_{3}⟩}

Table 2: Paracomplex structures on four dimensional solvable Lie algebras

2.2. Paracomplex structures

It is the main goal of this subsection to determine all 4-dimensional solvable Lie algebras admitting paracomplex structures. We will give realizations of the Lie algebras obtained in Theorem 1.5 as double Lie algebras with subalgebras of dimension 2 when such a structure exists (see Table 2), or prove the non existence otherwise. It turns out that among all four dimensional solvable Lie algebras there is only one family, whose commutator ideal is 𝔥_{3}, not admitting any paracomplex structure (Theorem 2.7). Since there are only two non-isomorphic two-dimensional Lie algebras: ℝ^{2} and 𝔞𝔣𝔣(ℝ), the possible decompositions 𝔤_{+} ▷◁ 𝔤_{−} are ℝ^{2} ▷◁ ℝ^{2}, ℝ^{2} ▷◁ 𝔞𝔣𝔣(ℝ) and 𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ).

By simple computations one can verify that the decompositions given in Table 2 satisfy the required properties. We prove below the non existence results.

Proposition 2.1. Let 𝔤 be a Lie algebra with an abelian commutator ideal 𝔤′ of codimension 1. Then any abelian subalgebra of dimension n > 1 is contained in 𝔤′.

Proof. In this case there is e_{0} ∈ 𝔤 such that ad(e_{0}) is an isomorphism of 𝔤′. Let 𝔥 be an abelian subalgebra of 𝔤, dim 𝔥 > 1, and let x, y ∈ 𝔥 linearly independent. If x = a_{0}e_{0} + x′, y = b_{0}e_{0} + y′ with a_{0}, b_{0} ∈ ℝ and x′, y′ ∈ 𝔤′, then

0 = [x, y] = [e_{0}, a_{0}y′ − b_{0}x′].

This implies that a_{0}y′ − b_{0}x′ = 0, that is, a_{0}y − b_{0}x = 0 and hence a_{0} = b_{0} = 0. Therefore, x, y ∈ 𝔤′, as asserted.

The previous result together with Table 1 imply

Corollary 2.2. The Lie algebras 𝔯_{4}, 𝔯_{4,λ} (λ ≠ 0), 𝔯_{4,μ,λ}, 𝔯′_{4,μ,λ} do not admit a decomposition of type ℝ^{2} ▷◁ ℝ^{2}.

Proposition 2.3 ([P]). If 𝔤 is a Lie algebra which admits a decomposition 𝔤 = 𝔤_{+} ▷◁ 𝔤_{−} with 𝔤_{+} and 𝔤_{−} abelian subalgebras, then 𝔤 is 2-step solvable (i.e., 𝔤′ is abelian).

Proof. If 𝔤 = 𝔤_{+} ▷◁ 𝔤_{−} with 𝔤_{+} and 𝔤_{−} abelian then [(x_{1}, x_{2}), (y_{1}, y_{2})] is determined by [(x_{1}, 0), (0, y_{2})] = (α(x_{1}, y_{2}), β(x_{1}, y_{2})) where α and β denote the components on 𝔤_{+} and 𝔤_{−} respectively. Since the bracket on 𝔤 satisfies the Jacobi identity one obtains

1. α(x_{1}, β(y_{1}, z_{2})) = α(y_{1}, β(x_{1}, z_{2})),

2. β(α(z_{1}, y_{2}), x_{2}) = β(α(z_{1}, x_{2}), y_{2}),

3. β(x_{1}, β(y_{1}, z_{2})) = β(y_{1}, β(x_{1}, z_{2})),

4. α(α(z_{1}, y_{2}), x_{2})) = α(α(z_{1}, x_{2}), y_{2})).

Now, using the above relations one can show that

α(α(x_{1}, y_{2}), β(u_{1}, v_{2})) = α(α(u_{1}, v_{2}), β(x_{1}, y_{2}))

and

β(α(x_{1}, y_{2}), β(u_{1}, v_{2})) = β(α(u_{1}, v_{2}), β(x_{1}, y_{2})).

But the above relations immediately imply

[[(x_{1}, 0), (0, y_{2})], [(u_{1}, 0), (0, v_{2})]] = 0

and the assertion follows.

The above proposition together with Table 1 imply

Corollary 2.4. The Lie algebras 𝔡_{4}, 𝔡_{4,λ} (λ ≠ 1), 𝔡′_{4,λ}, 𝔥_{4} do not admit a decomposition of type ℝ^{2} ▷◁ ℝ^{2}.

Lemma 2.5. The Lie algebras ℝ^{4}, ℝ×𝔥_{3}, 𝔫_{4} and ℝ×𝔯′_{3,λ} do not contain 𝔞𝔣𝔣(ℝ) as a subalgebra. Hence, these Lie algebras do not admit decompositions of type 𝔞𝔣𝔣(ℝ) ▷◁ ℝ^{2} or 𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ).

Proof. Since ℝ^{4}, ℝ × 𝔥_{3} and 𝔫_{4} are nilpotent, they cannot have subalgebras isomorphic to 𝔞𝔣𝔣(ℝ). Let us show next that the same holds for 𝔤 := ℝ × 𝔯′_{3,λ}. In fact, assume that there exist x, y ∈ 𝔤 such that [x, y] = y. Then y ∈ 𝔤′. If x ∈ ⟨e_{0}, e_{2}, e_{3}⟩ then y = 0, thus assume that x = e_{1} + u. So [x, y] = [e_{1}, y] = y implies that y = 0 since ad(e_{1}) has no real eigenvalues in 𝔤′.

Proposition 2.6. The Lie algebras ℝ × 𝔯_{3}, ℝ × 𝔯_{3,0}, 𝔞𝔣𝔣(ℂ), 𝔯_{4}, 𝔯_{4,0} and 𝔯′_{4,μ,λ} do not admit a decomposition of type 𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ).

Proof. Let 𝔤 := ℝ × 𝔯_{3} and 𝔥 a subalgebra of 𝔤 isomorphic to 𝔞𝔣𝔣(ℝ). Then 𝔥 has a basis of the form {e_{1} +u, e_{2}} with u ∈ ⟨e_{0}, e_{3}⟩. Thus, any decomposition of ℝ×𝔯_{3} of the form 𝔥_{1}+𝔥_{2} with 𝔥_{1} ≃ 𝔞𝔣𝔣(ℝ) ≃ 𝔥_{2} is not direct since e_{2} ∈ 𝔥_{1}∩𝔥_{2}. If 𝔤 = ℝ×𝔯_{3,0}, then dim 𝔤′ = 1 and therefore the assertion follows.

Assume next that 𝔤 ≅ 𝔞𝔣𝔣(ℂ). Every subalgebra of 𝔤 isomorphic to 𝔞𝔣𝔣(ℝ) is of the form ⟨e_{0} +u, v⟩ with u, v ∈ ⟨e_{2}, e_{3}⟩, thus it is contained in the subspace spanned by {e_{0}, e_{2}, e_{3}}. Therefore, 𝔞𝔣𝔣(ℂ) is not of type 𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ).

If 𝔤 is either 𝔯_{4}, 𝔯_{4,0} or 𝔯′_{4,μ,λ}, one can show that e_{1} ∈ 𝔤 belongs to any Lie subalgebra of 𝔤 isomorphic to 𝔞𝔣𝔣(ℝ) and thus 𝔤 cannot be decomposed as 𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ). We give a proof of this fact in the case 𝔤 = 𝔯_{4}. Let 𝔲 = ⟨u, v⟩ be a Lie subalgebra of 𝔯_{4} isomorphic to 𝔞𝔣𝔣(ℝ), with [u, v] = v. If u = ∑_{i=0}^{3} a_{i}e_{i}, v = ∑_{i=0}^{3} b_{i}e_{i} with a_{i}, b_{i} ∈ ℝ, i = 0, …, 3, b_{0} = 0 since v ∈ [𝔯_{4}, 𝔯_{4}], then we have

\begin{cases}a_{0}b_{1} + a_{0}b_{2} = b_{1}, \\ a_{0}b_{2} + a_{0}b_{3} = b_{2}, \\ a_{0}b_{3} = b_{3} \end{cases}  which implies \begin{cases}b_{1}(a_{0} − 1) = −a_{0}b_{2}, \\ b_{2}(a_{0} − 1) = −a_{0}b_{3}, \\ b_{3}(a_{0} − 1) = 0. \end{cases}

If a_{0} − 1 ≠ 0, then b_{3} = 0 and thus b_{2}(a_{0} − 1) = 0, which implies b_{2} = 0. From this, we have b_{1}(a_{0} − 1) = 0, and therefore b_{1} = 0, i.e. v = 0, a contradiction. Hence, a_{0} = 1 and then b_{2} = b_{3} = 0. Also b_{1} ∈ ℝ\{0} is arbitrary, and we may take b_{1} = 1. So,

u = e_{0} + a_{2}e_{2} + a_{3}e_{3}, v = e_{1},

hence, e_{1} ∈ 𝔯_{4}, as asserted. The proofs of the remaining cases are similar.

Theorem 2.7. If 𝔤 is a four dimensional solvable Lie algebra then 𝔤 does not admit any paracomplex structure if and only if 𝔤 is isomorphic to 𝔡′_{4,λ} for some λ ⩾ 0.

Proof. We first show that if 𝔤 is a Lie algebra in the family 𝔡′_{4,λ} with λ ⩾ 0 then 𝔤 does not admit a paracomplex structure. Let 𝔲 be a 2-dimensional Lie subalgebra of 𝔤 with a basis {u, v}, where u = ∑_{i=0}^{3} a_{i}e_{i}, v = ∑_{i=0}^{3} b_{i}e_{i} with a_{i}, b_{i} ∈ ℝ, i = 0, …, 3 Case 1: [u, v] = 0 and hence 𝔲 ≅ ℝ^{2}. In this case we get that

\begin{cases}λ(a_{0}b_{1} − a_{1}b_{0}) + a_{0}b_{2} − a_{2}b_{0} = 0, \\ λ(a_{0}b_{2} − a_{2}b_{0}) − a_{0}b_{1} + a_{1}b_{0} = 0, \\ 2λ(a_{0}b_{3} − a_{3}b_{0}) + a_{1}b_{2} − a_{2}b_{1} = 0. \end{cases}

From the first two equations we arrive at (a_{0}b_{2} − a_{2}b_{0})(λ^{2} + 1) = 0, and therefore a_{0}b_{2} − a_{2}b_{0} = 0, which in turn implies a_{0}b_{1} − a_{1}b_{2} = 0. Summing up, we have

(a_{0}, a_{1}, a_{2}) × (b_{0}, b_{1}, b_{2}) = (−2λ(a_{0}b_{3} − a_{3}b_{0}), 0, 0)

and hence

\begin{cases}2λa_{0}(a_{0}b_{3} − a_{3}b_{0}) = 0, \\ 2λb_{0}(a_{0}b_{3} − a_{3}b_{0}) = 0. \end{cases}

We have two cases:

(i) λ = 0. Then (a_{0}, a_{1}, a_{2}) × (b_{0}, b_{1}, b_{2}) = (0, 0, 0) and therefore (b_{0}, b_{1}, b_{2}) = β(a_{0}, a_{1}, a_{2}), with β ∈ ℝ. Since u and v are linearly independent, we must have b_{3} − βa_{3} ≠ 0. Thus, we obtain that

e_{3} = \frac{1}{b_{3} − βa_{3}} (v − βu)

(ii) λ ≠ 0. If we suppose a_{0}b_{3} − a_{3}b_{0} ≠ 0, we arrive at a contradiction; thus a_{0}b_{3} − a_{3}b_{0} = 0 and (a_{0}, a_{1}, a_{2}) × (b_{0}, b_{1}, b_{2}) = (0, 0, 0). As in the previous case, we have that e_{3} ∈ 𝔲.

Case 2: [u, v] = v and hence 𝔲 ≅ 𝔞𝔣𝔣(ℝ). In this case we obtain that b_{0} = 0 and

\begin{cases}λa_{0}b_{1} + a_{0}b_{2} = b_{1}, \\ λa_{0}b_{2} − a_{0}b_{1} = b_{2}, \\ 2λa_{0}b_{3} + a_{1}b_{2} − a_{2}b_{1} = b_{3}. \end{cases}  (14)

Let us observe first that a_{0} ≠ 0, since otherwise from (14) we obtain that b_{1} = b_{2} = b_{3} = 0, i.e. v = 0, a contradiction. Combining now the first two equations from (14), we arrive at

b_{1}b_{2} ((λa_{0} − 1)^{2} + a_{0}^{2}) = 0.

Since clearly (λa_{0} − 1)^{2} + a_{0}^{2} ≠ 0, we have that b_{1}b_{2} = 0. It is easily seen that this implies b_{1} = b_{2} = 0. Hence, we need only consider now the equation 2λa_{0}b_{3} = b_{3}, with b_{3} ≠ 0.

(i) λ = 0. In this case, we obtain that b_{3} = 0, a contradiction. Thus, 𝔡′_{4,0} does not have any Lie subalgebra isomorphic to 𝔞𝔣𝔣(ℝ). (ii) λ ≠ 0. Here, since b_{3} ≠ 0, we have a_{0} = \frac{1}{2λ} and u and v are given by

u = \frac{1}{2λ}e_{0} + a_{1}e_{1} + a_{2}e_{2}, v = e_{3}.

Note that e_{3} ∈ 𝔲.

In all cases, e_{3} ∈ 𝔤 belongs to any 2-dimensional Lie subalgebra of 𝔤, and hence this Lie algebra cannot be decomposed as a direct sum (as vector spaces) of two 2-dimensional Lie subalgebras.

The theorem follows by observing that the remaining Lie algebras possess paracomplex structures (see Table 2).

We give next a characterization of the four dimensional solvable Lie algebras which can be decomposed as a semidirect product of two dimensional subalgebras.

2.3. Semidirect extensions of ℝ^{2} Assume that 𝔤 contains ℝ^{2} as an ideal and that the short exact sequence

0 → ℝ^{2} → 𝔤 → 𝔤/ℝ^{2} → 0

splits. The next result gives a list of the Lie algebras with this property.

Proposition 2.8. Let 𝔤 be a four dimensional solvable Lie algebra.

(i) If there is a split exact sequence

0 → ℝ^{2} → 𝔤 → ℝ^{2} → 0

then 𝔤 ≅ ℝ^{4}, 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ), ℝ × 𝔥_{3}, ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ}, ℝ × 𝔯′_{3,λ}, 𝔞𝔣𝔣(ℂ) or 𝔡_{4,1}. (ii) If there is a split exact sequence

0 → ℝ^{2} → 𝔤 → 𝔞𝔣𝔣(ℝ) → 0

then 𝔤 ≅ ℝ × 𝔯_{3,λ}, 𝔯_{4,λ}, 𝔯_{4,μ,λ}, 𝔯′_{4,μ,λ}, 𝔡_{4} or 𝔡_{4,λ}.

Proof. (i) Table 2 exhibits decompositions of 𝔤 as a semidirect product ℝ^{2} ⋉ ℝ^{2} in case 𝔤 ≅ ℝ^{4}, 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ), ℝ × 𝔥_{3}, ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ}, ℝ × 𝔯′_{3,λ}, 𝔞𝔣𝔣(ℂ) or 𝔡_{4,1}. It follows from Corollaries 2.2 and 2.4 that 𝔯_{4}, 𝔯_{4,λ}, 𝔯′_{4,λ}, λ ≠ 0, 𝔯_{4,μ,λ}, 𝔯′_{4,μ,λ}, 𝔡_{4}, 𝔡_{4,λ}, λ ≠ 1, 𝔡′_{4,λ} and 𝔥_{4} do not admit such a decomposition. It remains to consider the case 𝔤 ≅ 𝔫_{4} or 𝔯_{4,0}. Assume that 𝔤 = 𝔞 ⋉ 𝔟 with 𝔞 ≅ 𝔟 ≅ ℝ^{2}. Then 𝔤′ ⊂ 𝔟, hence 𝔤′ = 𝔟 since in both cases 𝔤′ = ℝ^{2} (Table 1).

Consider next the case 𝔤 ≅ 𝔫_{4}, hence 𝔟 = ⟨e_{2}, e_{3}⟩ and 𝔞 = ⟨x, y⟩ with x = ae_{0} + be_{1} + u, y = ce_{0} + de_{1} + v, ad − bc ≠ 0, u, v ∈ 𝔟. We calculate

[x, y] = (ad − bc)e_{2} + [e_{0}, av − cu]

which is non zero since the second summand on the right hand side is a multiple of e_{3}. This contradicts the fact that 𝔞 ≅ ℝ^{2}. Therefore, 𝔫_{4} does not decompose as ℝ^{2} ⋉ ℝ^{2}.

The case 𝔤 ≅ 𝔯_{4,0} is similar. We have 𝔟 = ⟨e_{1}, e_{2}⟩ and 𝔞 = ⟨x, y⟩ with x = ae_{0} + be_{3} + u, y = ce_{0} + de_{3} + v, ad − bc ≠ 0, u, v ∈ 𝔟. We calculate

[x, y] = [e_{0}, av − cu] + (ad − bc)e_{2}

which is non zero since the first summand on the right hand side is a multiple of e_{1}. This contradicts the fact that 𝔞 ≅ ℝ^{2} and part (i) of the proposition follows.

(ii) If 0 → ℝ^{2} → 𝔤 → 𝔞𝔣𝔣(ℝ) → 0 splits, then there is a subalgebra 𝔥 of 𝔤 isomorphic to 𝔞𝔣𝔣(ℝ) such that 𝔤 = 𝔥 ⋉ ℝ^{2}. Set

ρ : 𝔥 → 𝔤𝔩(2, ℝ), ρ(u) = ad(u)|_{ℝ^{2}}, u ∈ 𝔥.

Then ρ is a Lie algebra homomorphism. Let 𝔥 = ⟨x, y⟩, [x, y] = y. If ρ ≡ 0 then 𝔤 ≅ 𝔞𝔣𝔣(ℝ) × ℝ^{2} = ℝ × 𝔯_{3,0}. If dim Im ρ = 1, then 0 = [ρ(x), ρ(y)] = ρ([x, y]) = ρ(y) and ρ(x) is given as follows:

(\begin{matrix}μ & 0 \\ 0 & λ \end{matrix}) , λ ≠ 0,  (\begin{matrix}λ & 1 \\ 0 & λ \end{matrix})  or  (\begin{matrix}α & β \\ −β & α \end{matrix}) , β ≠ 0.

The first possibility gives 𝔤 ≅ ℝ × 𝔯_{3,λ} in case μ = 0 and 𝔤 ≅ 𝔯_{4,μ,λ} if μ ≠ 0. The second possibility yields 𝔤 ≅ 𝔯_{4,λ} and the last one gives 𝔤 ≅ 𝔯′_{4,1/β,α/β}.

If dim Im ρ = 2, then ρ(x), ρ(y) are linearly independent and since 𝔤′ is nilpotent and y ∈ 𝔤′, we may assume that

ρ(y) = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) .

It follows from [ρ(x), ρ(y)] = ρ(y) that ρ(x) takes the following form:

ρ(x) = (\begin{matrix}α + 1/2 & β \\ 0 & α − 1/2 \end{matrix}) .

We can take β = 0 by replacing x with x − βy. Let us denote by 𝔤_{α} the Lie algebra corresponding to

ρ(x) = (\begin{matrix}α + 1/2 & 0 \\ 0 & α − 1/2 \end{matrix}) , ρ(y) = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) .

The following table gives the possibilities for 𝔤_{α} according to the parameter α:

\begin{matrix}α & 𝔤_{α} \\ −1/2 & 𝔡_{4} \\ 1/2 & 𝔡_{4,1} \\ α ∈ (−1/2, 1/2) ∪ (1/2, 3/2] & 𝔡_{4,λ}, λ = \frac{2}{2α + 1} \\ α ∈ (−∞, −1/2) ∪ (3/2, ∞) & 𝔡_{4,λ}, λ = \frac{α − 1/2}{α + 1/2} \end{matrix}

This completes the proof of the proposition.

2.4. Semidirect extensions of 𝔞𝔣𝔣(ℝ)

Assume that 𝔤 contains 𝔞𝔣𝔣(ℝ) as an ideal and that the short exact sequence

0 → 𝔞𝔣𝔣(ℝ) → 𝔤 → 𝔤/𝔞𝔣𝔣(ℝ) → 0

splits. The next result states that 𝔤 is a direct product, that is, 𝔤 is isomorphic to ℝ^{2} × 𝔞𝔣𝔣(ℝ) or 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ). The precise statement is the following:

Proposition 2.9. Let 𝔤 be a four dimensional solvable Lie algebra.

(i) If there is a split exact sequence

0 → 𝔞𝔣𝔣(ℝ) → 𝔤 → ℝ^{2} → 0

then 𝔤 ≅ ℝ × 𝔯_{3,0} = ℝ^{2} × 𝔞𝔣𝔣(ℝ).

(ii) If there is a split exact sequence

0 → 𝔞𝔣𝔣(ℝ) → 𝔤 → 𝔞𝔣𝔣(ℝ) → 0

then 𝔤 ≅ 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ).

Proof. Let 𝔞𝔣𝔣(ℝ) = ⟨z, w⟩ with [z, w] = w, then the algebra of derivations is given as follows:

Der 𝔞𝔣𝔣(ℝ) = \{(\begin{matrix}0 & 0 \\ a & b \end{matrix}) , a, b ∈ ℝ\}

\begin{aligned}ℝ ▷◁ 𝔥 & 𝔤 \\ ℝ ▷◁ ℝ^{3} & ℝ^{4}, ℝ × 𝔥_{3}, ℝ × 𝔯_{3}, ℝ × 𝔯_{3,λ}, ℝ × 𝔯′_{3,λ}, 𝔫_{4}, 𝔯_{4}, 𝔯_{4,λ}, 𝔯_{4,μ,λ}, 𝔯′_{4,μ,λ} \\ ℝ ▷◁ 𝔥_{3} & ℝ × 𝔥_{3} , 𝔫_{4}, 𝔯_{4,0}, 𝔡_{4}, 𝔡_{4,λ} (λ ⩾ 1/2), 𝔡′_{4,λ} (λ ⩾ 0), 𝔥_{4} \\ ℝ ▷◁ 𝔯_{3} & ℝ × 𝔯_{3}, 𝔯_{4}, 𝔯_{4,λ}, (λ ≠ 0), 𝔡_{4,1} \\ ℝ ▷◁ 𝔯_{3,0} & ℝ × 𝔯_{3,λ}, 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ), 𝔯_{4,0}, 𝔡_{4}, 𝔡_{4,1} \\ ℝ ▷◁ 𝔯′_{3,0} & ℝ × 𝔯′_{3,0}, 𝔞𝔣𝔣(ℂ) \\ ℝ ▷◁ 𝔯_{3,λ} & ℝ × 𝔯_{3,λ}, 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) (λ = −1), 𝔞𝔣𝔣(ℂ) (λ = 1), 𝔯_{4,λ}, 𝔯_{4,μ,λ}, \\ & 𝔥_{4} (λ = 2), 𝔡_{4,λ}, 𝔡_{4,1−λ} \\ ℝ ▷◁ 𝔯′_{3,λ} & ℝ × 𝔯′_{3,λ}, 𝔞𝔣𝔣(ℂ), 𝔯′_{4,μ,λ} \end{aligned} Table 3:

with respect to {z, w}. If 0 → 𝔞𝔣𝔣(ℝ) → 𝔤 → 𝔤/𝔞𝔣𝔣(ℝ) → 0 splits, then there is a subalgebra 𝔥 of 𝔤 isomorphic to 𝔤/𝔞𝔣𝔣(ℝ) such that 𝔤 = 𝔥 ⋉ 𝔞𝔣𝔣(ℝ). Set

ρ : 𝔥 → Der 𝔞𝔣𝔣(ℝ), ρ(u) = ad(u)|_{𝔞𝔣𝔣(ℝ)}, u ∈ 𝔥.

Then ρ is a Lie algebra homomorphism.

(i) In this case 𝔥 = ℝ^{2}, so the image of ρ is an abelian subalgebra of Der 𝔞𝔣𝔣(ℝ), hence it is one dimensional. Let ℝ^{2} = ⟨x, y⟩, 𝔞𝔣𝔣(ℝ) = ⟨z, w⟩. We may assume that

ρ(y) = 0. Let ρ(x) = (\begin{matrix}0 & 0 \\ a & b \end{matrix}). If a = b = 0 the assertion follows. If b ≠ 0 we may

assume that b = 1 and we can reduce to a = 0 by changing z to z − aw. Hence, we may assume that the only non zero brackets are [x, w] = w, [z, w] = w and therefore 𝔤 = ⟨x − z, y⟩ × ⟨z, w⟩ where ⟨x − z, y⟩ ≅ ℝ^{2}. If b = 0, a ≠ 0, we may assume that a = 1, therefore 𝔤 = ⟨x + w, y⟩ × ⟨z, w⟩ where ⟨x + w, y⟩ ≅ ℝ^{2} and the desired assertion follows.

(ii) We have 𝔥 = 𝔞𝔣𝔣(ℝ) = ⟨x, y⟩, [x, y] = y, and the following possibilities for ρ:

ρ(x) = (\begin{matrix}0 & 0 \\ a & 1 \end{matrix}) , ρ(y) = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})  or  ρ(x) = (\begin{matrix}0 & 0 \\ a & b \end{matrix}) , ρ(y) = 0.

We show next that, in both cases, 𝔤 ≅ 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ).

If the first possibility occurs, take ⟨x − z + aw, y + w⟩ and ⟨z − aw, w⟩. These are complementary ideals isomorphic to 𝔞𝔣𝔣(ℝ), hence 𝔤 ≅ 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ).

In the second case, take ⟨x − bz + aw, y⟩ and ⟨z, w⟩, which are ideals isomorphic to 𝔞𝔣𝔣(ℝ), and the desired assertion follows.

2.5. Product structures of type ℝ ▷◁ 𝔥

We exhibit in Table 3 realizations of the Lie algebras obtained in Theorem 1.5 as double Lie algebras where 𝔥 is a three dimensional subalgebra. Note that the problem of finding such a decomposition is equivalent to the determination of the three dimensional subalgebras.

• ℝ ▷◁ ℝ^{3}: The Lie algebras ℝ×𝔥_{3}, ℝ×𝔯_{3}, ℝ×𝔯_{3,λ}, ℝ×𝔯′_{3,λ}, 𝔫_{4}, 𝔯_{4}, 𝔯_{4,λ}, 𝔯_{4,μ,λ} and 𝔯′_{4,μ,λ} were obtained in Theorem 1.5 as semidirect extensions of ℝ^{3}.

• ℝ ▷◁ 𝔥_{3}: The Lie algebras ℝ × 𝔥_{3} , 𝔫_{4}, 𝔡_{4}, 𝔡_{4,λ} (λ ⩾ 1/2), 𝔡′_{4,λ} (λ ⩾ 0), and 𝔥_{4} were obtained in Theorem 1.5 as semidirect extensions of 𝔥_{3}. On the other hand,

𝔯_{4,0} ≅ ⟨e_{1}⟩ ▷◁ ⟨e_{0}, e_{2}, e_{3}⟩.

• ℝ ▷◁ 𝔯_{3}:

𝔯_{4} ≅ ⟨e_{3}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{2}⟩,
𝔯_{4,λ} ≅ ⟨e_{1}⟩ ▷◁ ⟨e_{0}, e_{2}, e_{3}⟩,  λ ≠ 0,
𝔡_{4,1} ≅ ⟨e_{0}⟩ ⋉ ⟨e_{0} + e_{2}, e_{1}, e_{3}⟩.

• ℝ ▷◁ 𝔯_{3,0}:

𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) ≅ ⟨e_{0}⟩ ⋉ ⟨e_{1}, e_{2}, e_{3}⟩,
𝔯_{4,0} ≅ ⟨e_{3}⟩ ⋉ ⟨e_{0}, e_{1}, e_{2}⟩,
𝔡_{4} ≅ ⟨e_{2}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{3}⟩,
𝔡_{4,1} ≅ ⟨e_{1}⟩ ▷◁ ⟨e_{0}, e_{2}, e_{3}⟩.

• ℝ ▷◁ 𝔯′_{3,0}: 𝔞𝔣𝔣(ℂ) ≅ ⟨e_{0}⟩ ⋉ ⟨e_{1}, e_{2}, e_{3}⟩.

• ℝ ▷◁ 𝔯_{3,λ}: 𝔞𝔣𝔣(ℝ)×𝔞𝔣𝔣(ℝ) was obtained in Theorem 1.5 as a semidirect extension of 𝔯_{3,−1}.

𝔥_{4} = ⟨e_{2}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{3}⟩ ≅ ℝ ▷◁ 𝔯_{3,2},  λ = 2,
𝔞𝔣𝔣(ℂ) = ⟨e_{1}⟩ ⋉ ⟨e_{0}, e_{2}, e_{3}⟩ ≅ ℝ ⋉ 𝔯_{3,1},  λ = 1,
𝔯_{4,λ} ≅ ⟨e_{3}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{2}⟩,
𝔯_{4,μ,λ} ≅ ⟨e_{2}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{3}⟩,
𝔯_{4,μ,λ} ≅ ℝ ▷◁ 𝔯_{4,μ} = ⟨e_{3}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{2}⟩,
𝔡_{4,λ} ≅ ⟨e_{2}⟩ ▷◁ ⟨e_{0}, e_{1}, e_{3}⟩, 𝔡_{4,1−λ} ≅ ⟨e_{1}⟩ ▷◁ ⟨e_{0}, e_{2}, e_{3}⟩.

• ℝ ▷◁ 𝔯′_{3,λ}:

𝔞𝔣𝔣(ℂ) ≅ ⟨e_{0}⟩ ⋉ ⟨λe_{0} − e_{1}, e_{2}, e_{3}⟩,
𝔯′_{4,μ,λ} ≅ ⟨e_{1}⟩ ▷◁ ⟨e_{0}, e_{2}, e_{3}⟩.

3. Applications: Manin triples and complex product structures

3.1. Manin triples on 4-dimensional solvable Lie algebras

An important example of double Lie algebras are Manin triples [LW]. We recall that a Manin triple is a double Lie algebra (𝔤, 𝔤_{+}, 𝔤_{−}) with an invariant metric, that is, a non degenerate symmetric bilinear form ( , ) which satisfies:

([x, y], z) + (y, [x, z]) = 0 for all x, y, z ∈ 𝔤

such that 𝔤_{+} and 𝔤_{−} are isotropic subalgebras. In particular 𝔤 = 𝔤_{+} ▷◁ 𝔤_{−}, where 𝔤_{+} and 𝔤_{−} have the same dimension. Thus, Manin triples are special cases of paracomplex structures.

The next proposition makes use of [BK] and the results of the previous section to obtain that there is only one four dimensional solvable Lie algebra giving rise to Manin triples.

Proposition 3.1. Let (𝔤, 𝔤_{+}, 𝔤_{−}) be a Manin triple such that 𝔤 is a non abelian four dimensional solvable Lie algebra. Then 𝔤 is isomorphic to 𝔡_{4} with the invariant metric given by:

α = (e_{0}, e_{3}) = (e_{1}, e_{2}), α ≠ 0,

where the isotropic subalgebras 𝔤_{+} and 𝔤_{−} are given as follows:

(i) 𝔤_{+} = ⟨e_{0} + μe_{2}, e_{1} − μe_{3}⟩, 𝔤_{−} = ⟨e_{0} + νe_{2}, e_{1} − νe_{3}⟩ with μ ≠ ν; or (ii) 𝔤_{+} = ⟨e_{0} + μe_{2}, e_{1} − μe_{3}⟩, 𝔤_{−} = ⟨e_{2}, e_{3}⟩: μ, ν ∈ ℝ.

Proof. According to [BK] a non abelian solvable Lie algebra which admits an invariant metric is isomorphic either to 𝔡′_{4,0} or 𝔡_{4}. It was proved in Theorem 2.7 that 𝔡′_{4,0} does not admit paracomplex structures. Thus, we need to investigate the possible paracomplex structures on 𝔡_{4}. It is easy to see that the metric on 𝔡_{4} given by:

(e_{0}, e_{3}) = (e_{1}, e_{2}) = α, with α ≠ 0,

is invariant. Any two-dimensional isotropic non abelian subalgebra of 𝔡_{4} is isometrically isomorphic to:

⟨e_{0} + μe_{2}, e_{1} − μe_{3}⟩,

where the isometric isomorphism is given by ϕ(e_{1}) = e_{2}, ϕ(e_{i}) = −e_{i}, i = 0, 3. On the other hand, any two-dimensional isotropic abelian subalgebra is isometrically isomorphic to:

⟨e_{2}, e_{3}⟩.

It follows from 2.4 that 𝔡_{4} does not admit a decomposition of type ℝ^{2} ▷◁ ℝ^{2}. If both 𝔤_{+} and 𝔤_{−} are isomorphic to 𝔞𝔣𝔣(ℝ) then we are led to case (i). In case 𝔤_{+} ≅ 𝔞𝔣𝔣(ℝ) and 𝔤_{−} ≅ ℝ^{2} we obtain case (ii), and the proposition follows.

3.2. Complex product structures on four dimensional solvable Lie alge-

bras In this subsection we determine all four dimensional solvable Lie algebras which

admit a complex product structure (see Table 4), using the classification of complex structures on this class of Lie algebras given in [SJ, O1] together with the results in §2.2. We give in this way an alternative proof of a result by Blazić and Vukmirović ([BV]), where complex product structures were referred to as para-hypercomplex structures.

We recall that a complex structure on a Lie algebra 𝔤 is an endomorphism J : 𝔤 → 𝔤 such that J^{2} = − Id and

J[x, y] = [Jx, y] + [x, Jy] + J[Jx, Jy]

Lie algebra  Complex structure  Paracomplex structure
𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ)  Je_{0} = e_{3}, Je_{1} = e_{2}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
ℝ × 𝔥_{3}  Je_{0} = −e_{3}, Je_{1} = e_{2}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
ℝ × 𝔯_{3,0}  Je_{3} = e_{0}, Je_{1} = e_{2}  ⟨e_{1}, e_{3}⟩ ⋉ ⟨e_{0}, e_{2}⟩
ℝ × 𝔯_{3,1}  Je_{0} = e_{1}, Je_{2} = e_{3}  ⟨e_{1}, e_{3}⟩ ⋉ ⟨e_{0}, e_{2}⟩
𝔞𝔣𝔣(ℂ)  Je_{0} = e_{2}, Je_{1} = e_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
𝔯_{4,1}  Je_{0} = e_{3}, Je_{1} = e_{2}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
𝔯_{4,λ,λ} , λ ≠ 0  Je_{0} = e_{1}, Je_{2} = e_{3}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩
𝔯_{4,μ,1} , μ ≠ 0, ±1  Je_{0} = e_{2}, Je_{1} = e_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
𝔯′_{4,μ,λ}  Je_{0} = e_{1}, Je_{2} = e_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
Je_{0} = e_{1}, Je_{2} = −e_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
𝔡_{4}  Je_{0} = −e_{1}, Je_{2} = e_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
{Je_{0} = e_{3} − e_{1}, Je_{1} = e_{0} − e_{2}, \atop Je_{2} = e_{3}}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩
𝔡_{4,1}  Je_{0} = e_{1}, Je_{2} = −e_{3}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩
𝔡_{4,1/2}  Je_{0} = e_{3}, Je_{1} = e_{2}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
Je_{0} = e_{3}, Je_{1} = −e_{2}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
Je_{0} = e_{1}, Je_{2} = −2e_{3}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩
𝔡_{4,λ} , λ ≠ 1, 1/2  Je_{0} = (1 − λ)e_{2}, Je_{1} = e_{3}  ⟨e_{0}, e_{1}⟩ ⋉ ⟨e_{2}, e_{3}⟩
Je_{0} = −λe_{1}, Je_{2} = e_{3}  ⟨e_{0}, e_{2}⟩ ⋉ ⟨e_{1}, e_{3}⟩
𝔥_{4}  Je_{0} = 4e_{2}, Je_{1} = 4e_{3}  ⟨e_{0}, e_{1}⟩ ▷◁ ⟨e_{2}, e_{3}⟩

Table 4: Complex product structures

for all x, y ∈ 𝔤. A complex product structure on a Lie algebra 𝔤 is a pair {J, E} where J is a complex structure and E is a product structure on 𝔤 such that JE = −EJ. This is equivalent to having a splitting of 𝔤 as 𝔤 = 𝔤_{+} ⊕ 𝔤_{−}, where 𝔤_{+} and 𝔤_{−} are Lie subalgebras of 𝔤 such that 𝔤_{−} = J𝔤_{+}. From this it follows that E is, in fact, a paracomplex structure on 𝔤.

At this point we refer the reader to Table 4. Remarks.

a) The Lie algebras ℝ×𝔯′_{3,λ} admit complex structures (see [SJ]) and paracomplex structures (see Table 2). Nevertheless, they do not admit any complex product structure. To show this, we state the following result, which is proved in [AD]: Proposition 3.2. Let {J, E} be a complex product structure on the Lie algebra 𝔤 and let (𝔤, 𝔤_{+}, 𝔤_{−}) be the associated double Lie algebra. Then the following assertions are equivalent:

(i) J is an abelian complex structure, i.e., [Jx, Jy] = [x, y] for all x, y ∈ 𝔤. (ii) The Lie subalgebras 𝔤_{+} and 𝔤_{−} are abelian; (iii) E is an abelian product structure, i.e., [Ex, Ey] = −[x, y] for all x, y ∈ 𝔤.

It is known that the Lie algebra ℝ × 𝔯′_{3,λ} does not admit any abelian complex structure (see [SJ]). However, from Lemma 2.5 this Lie algebra admits only abelian paracomplex structures and thus, from the previous proposition, there is no complex product structure on ℝ × 𝔯′_{3,λ}.

b) The Lie algebra 𝔞𝔣𝔣(ℂ) admits other complex structures, given by:

J_{α,β}e_{0} = \frac{α}{β}e_{0} + \frac{α^{2} + β^{2}}{β} e_{1}, J_{α,β}e_{2} = e_{3}

with α ∈ ℝ, β ∈ ℝ \ {0}. However, there is no paracomplex structure on 𝔞𝔣𝔣(ℂ) which anticommutes with J_{α,β}. Let us show this last assertion. It is known from Proposition 2.6 that 𝔞𝔣𝔣(ℂ) does not admit decompositions of type 𝔞𝔣𝔣(ℝ) ▷◁ 𝔞𝔣𝔣(ℝ). Also, since the complex structure J_{α,β} is not abelian, any complex product structure on 𝔞𝔣𝔣(ℂ) induces a decomposition of type 𝔞𝔣𝔣(ℝ) ▷◁ ℝ^{2}. Let 𝔥 be a subalgebra of 𝔞𝔣𝔣(ℂ) isomorphic to 𝔞𝔣𝔣(ℝ); then 𝔥 has a basis u = e_{0} + a_{2}e_{2}, v = b_{2}e_{2} + b_{3}e_{3}. Then J_{α,β}u = \frac{α}{β}e_{0} + \frac{α^{2}+β^{2}}{β} e_{1} − a_{3}e_{2} + a_{2}e_{3}, J_{α,β}v = −b_{3}e_{2} + b_{2}e_{3}. Then we must have [Ju, Jv] = 0 and from this we obtain the system

\begin{cases}\frac{α^{2}+β^{2}}{β} b_{2} + \frac{α}{β}b_{3} = 0 \\ \frac{α}{β}b_{2} − \frac{α^{2}+β^{2}}{β} b_{3} = 0 \end{cases}

It is easy to see that the only solution of this system is b_{2} = b_{3} = 0, i.e., v = 0, a contradiction. Therefore, there are no product structures on 𝔞𝔣𝔣(ℂ) which anticommute with J_{α,β}.

c) The Lie algebras 𝔡′_{4,λ} admit complex structures (see [O1]) but, according to Theorem 2.7, they do not admit any paracomplex structure. Hence, they do not carry complex product structures.

d) Table 4 shows examples of complex product structures on ℝ × 𝔥_{3} and 𝔯_{4,1,1}. On the other hand, all equivalence classes of complex product structures on these Lie algebras were determined in [AS], section 6.2.

Appendix I - Matrix realizations

We exhibit below matrix realizations of the indecomposable Lie algebras listed in Theorem 1.5, where indecomposable means that they do not split as a direct product of lower dimensional Lie algebras. All matrices have real coefficients.

\begin{matrix}𝔫_{4}: & (\begin{matrix}0 & x & 0 & w \\ 0 & 0 & x & y \\ 0 & 0 & 0 & z \\ 0 & 0 & 0 & 0 \end{matrix}) \\ 𝔞𝔣𝔣(ℂ): & (\begin{matrix}x & z & y \\ −z & x & w \\ 0 & 0 & 0 \end{matrix}) \\ 𝔯_{4}: & (\begin{matrix}x & x & 0 & y \\ 0 & x & x & z \\ 0 & 0 & x & w \\ 0 & 0 & 0 & 0 \end{matrix}) \\ 𝔯_{4,λ}: & (\begin{matrix}x & 0 & 0 & y \\ 0 & λx & x & z \\ 0 & 0 & λx & w \\ 0 & 0 & 0 & 0 \end{matrix}) \\ \begin{matrix}𝔯_{4,μ,λ} : \\ μλ ≠ 0, −1 < μ ⩽ λ ⩽ 1 \\ or & −1 = μ ⩽ λ < 0 \end{matrix} & (\begin{matrix}x & 0 & 0 & y \\ 0 & μx & 0 & z \\ 0 & 0 & λx & w \\ 0 & 0 & 0 & 0 \end{matrix}) \\ 𝔯′_{4,μ,λ} :_{μ > 0} & (\begin{matrix}μx & 0 & 0 & y \\ 0 & λx & x & z \\ 0 & −x & λx & w \\ 0 & 0 & 0 & 0 \end{matrix}) \\ 𝔡_{4}: & \{ \begin{matrix}( & \begin{matrix}0 & x & z \\ 0 & w & y \\ 0 & 0 & 0 \end{matrix} & ) \\ ( & \begin{matrix}w & 0 & 0 & x \\ 0 & −w & 0 & y \\ −\frac{1}{2}y & \frac{1}{2}x & 0 & z \\ 0 & 0 & 0 & 0 \end{matrix} & ) \end{matrix} \\ 𝔡_{4,λ} :_{λ ⩾ \frac{1}{2}} & \{ \begin{matrix}( & \begin{matrix}w & x & z \\ 0 & (1 − λ)w & y \\ 0 & 0 & 0 \end{matrix} & ) \\ ( & \begin{matrix}λw & 0 & 0 & x \\ 0 & (1 − λ)w & 0 & y \\ −\frac{1}{2}y & \frac{1}{2}x & w & z \\ 0 & 0 & 0 & 0 \end{matrix} & ) \end{matrix} \\ 𝔡′_{4,λ} :_{λ ⩾ 0} & (\begin{matrix}λw & w & 0 & x \\ −w & λw & 0 & y \\ −\frac{1}{2}y & \frac{1}{2}x & 2λw & z \\ 0 & 0 & 0 & 0 \end{matrix}) \\ 𝔥_{4}: & (\begin{matrix}\frac{1}{2}w & w & 0 & x \\ 0 & \frac{1}{2}w & 0 & y \\ −\frac{1}{2}y & \frac{1}{2}x & w & z \\ 0 & 0 & 0 & 0 \end{matrix}) \end{matrix}

Homology, Homotopy and Applications, vol. 7(1), 2005 31

Appendix II - Comparison with previous classifications

In this section we carry out a comparison with various results which can be found in the literature. Our main goal is to establish a correspondence between the description obtained by other authors and the Lie algebras appearing in Theorem 1.5.

3.3.

We start by comparing our results with the ones obtained by Dozias as appearing in [Ve], Table 1.1, p. 180.

\begin{matrix}𝔤_{4,1} & 𝔤_{4,2} & 𝔤_{4,3} & 𝔤_{4,4} & 𝔤_{4,5}(α, β) & 𝔤_{4,6}(α) & 𝔤_{4,7} & 𝔤_{4,8}(α, β) & 𝔤_{4,9}(0) \\ 𝔡_{4,0} & 𝔞𝔣𝔣(ℂ) & 𝔫_{4} & 𝔯_{4,0} & 𝔯_{4,α,β} & 𝔯_{4,α} & 𝔯_{4} & 𝔯′_{4,α,β} & 𝔡_{4} \end{matrix}

\begin{matrix}𝔤_{4,9}(α) , α ≠ 0 & 𝔤_{4,10} & 𝔤_{4,11}(α) \\ 𝔡_{4,1−1/α} & 𝔥_{4} & 𝔡′_{4,α} \end{matrix}

3.4.

We recall below the classification given by Mubarakzyanov [Mu] and then we establish the correspondence with the algebras appearing in Theorem 1.5.

Notation

in [Mu]  Lie bracket relations
𝔤_{4,1}  [e_{2}, e_{4}] = e_{1}  [e_{3}, e_{4}] = e_{2}
𝔤_{4,2}  [e_{1}, e_{4}] = αe_{1} [e_{2}, e_{4}] = e_{2}  [e_{3}, e_{4}] = e_{2} + e_{3}
𝔤_{4,3}  [e_{1}, e_{4}] = e_{1}  [e_{3}, e_{4}] = e_{2}
𝔤_{4,4}  [e_{1}, e_{4}] = e_{1}  [e_{2}, e_{4}] = e_{1} + e_{2}  [e_{3}, e_{4}] = e_{2} + e_{3}
𝔤_{4,5}  [e_{1}, e_{4}] = e_{1}  [e_{2}, e_{4}] = βe_{2}  [e_{3}, e_{4}] = γe_{3}  -1⩽ γ ⩽ β ⩽ 1,
γβ ≠ 0
𝔤_{4,6}  [e_{1}, e_{4}] = αe_{1} [e_{2}, e_{4}] = pe_{2} − e_{3} [e_{3}, e_{4}] = e_{2} + pe_{3} α ≠ 0, p⩾ 0
𝔤_{4,7}  [e_{2}, e_{3}] = e_{1}  [e_{1}, e_{4}] = 2e_{1}  [e_{2}, e_{4}] = e_{2}  [e_{3}, e_{4}] = e_{2} + e_{3}
𝔤_{4,8}  [e_{2}, e_{3}] = e_{1}  [e_{1}, e_{4}] = (1 + h)e_{1} [e_{2}, e_{4}] = e_{2}  [e_{3}, e_{4}] = he_{3},
|h| ⩽1
𝔤_{4,9} [e_{2}, e_{3}] = e_{1} [e_{1}, e_{4}] = 2pe_{1} [e_{2}, e_{4}] = pe_{2} − e_{3} [e_{3}, e_{4}] = e_{2} + pe_{3},
p⩾0
𝔤_{4,10}  [e_{1}, e_{3}] = e_{1}  [e_{2}, e_{3}] = e_{2}  [e_{1}, e_{4}] = −e_{2}  [e_{2}, e_{4}] = e_{1} + e_{3}

The correspondence is as follows:

\begin{matrix}𝔤_{4,1} & 𝔤_{4,2} & 𝔤_{4,3} & 𝔤_{4,4} & 𝔤_{4,5} & 𝔤_{4,6} & 𝔤_{4,7} & 𝔤_{4,8} & 𝔤_{4,9} & 𝔤_{4,10} \\ 𝔫_{4} & 𝔯_{4,α} & 𝔯_{4,0} & 𝔯_{4} & 𝔯_{4,β,γ} & 𝔯′_{4,α,p} & 𝔥_{4} & 𝔡_{4}, 𝔡_{4,1/1+b} & 𝔡′_{4,a} & 𝔞𝔣𝔣(ℂ) \end{matrix}

3.5.

In [PSWZ] invariants of real Lie algebras of dimension at most five are given. In particular, a list of four dimensional solvable Lie algebras, based on that of [Mu], is shown in Table I, p. 988. The relation with Theorem 1.5 is :

\begin{matrix}A_{4,1} & A_{4,2}^{a} & A_{4,3} & A_{4,4} & A_{4,5}^{a,b} & A_{4,6}^{a,b} & A_{4,7} & A_{4,8} \\ 𝔫_{4} & 𝔯_{4,a} & 𝔯_{4,0} & 𝔯_{4} & 𝔯_{4,a,b} & 𝔯′_{4,a,b} & 𝔥_{4} & 𝔡_{4} \end{matrix}

\begin{matrix}A_{4,9}^{b} & A_{4,10} & A_{4,11}^{a} & A_{4,12} \\ 𝔡_{4,1/1+b} & 𝔡′_{4,0} & 𝔡′_{4,a} & 𝔞𝔣𝔣(ℂ) \end{matrix}

3.6.

The classification of complex structures on four dimensional Lie algebras was carried out by Snow in [SJ] and by Ovando in [O1]. To achieve this classification a description is given in [SJ], p. 400, of four dimensional solvable Lie algebras when the commutator ideal has dimension 1 or 2. We compare below the list given by Snow with the one obtained in Theorem 1.5.

\begin{matrix}S1 & S2 & S3 & S4 & S5_{d} , d ≠ 0 & S6 \\ ℝ × 𝔥_{3} & ℝ^{2} × 𝔞𝔣𝔣(ℝ) & 𝔯_{4,0} & 𝔫_{4} & ℝ × 𝔯_{3,d} & ℝ × 𝔯_{3} \end{matrix}

\begin{matrix}S7_{0,c} , c > 0 & S7_{1,c} , 4c > 1 & S8 & S9 \\ ℝ × 𝔯′_{3,0} & ℝ × 𝔯′_{3,\sqrt{4c−1}} & 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) & 𝔡_{4,1} \end{matrix}

ℝ × 𝔯_{3,d}  𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ)  𝔞𝔣𝔣(ℂ)^{S10_{d,d} , d ≠ 0 S10_{d,c}, c ≠ d, d ≠ 0 S11_{d,c} , d^{2} − 4c < 0, d = 0, 1}

The above correspondence shows that some of the families appearing in [SJ] become a single Lie algebra. Also, there exist isomorphisms between different families. We give below the proof of these statements.

• S7 We recall from [SJ] the definition of the Lie algebra S7_{d,c} , d^{2} − 4c < 0, d = 0 or 1 with basis x, y, z, w:

[x, y] = w, [x, w] = −cy + dw.

Observe that if d = 0 then c > 0 and ad(x)_{|𝔤′} has eigenvalues ±ic. We can

take a real basis of 𝔤′ such that ad(x)_{|𝔤′} takes the form (\begin{matrix}0 & c \\ −c & 0 \end{matrix}). Changing

x by x/c we see that S7_{0,c} ≅ ℝ × 𝔢(2) for all c > 0. If c = 1, then ad(x)_{|𝔤′} has eigenvalues 1/2 ± iλ/2, where λ = \sqrt{4c − 1}. Taking x′ = x/2, there

exists a real basis of 𝔤′ such that ad(x′)_{|𝔤′} takes the form (\begin{matrix}1 & λ \\ −λ & 1 \end{matrix}), hence

S7_{1,c} ≅ ℝ × 𝔯′_{3,λ} for all c such that 1 − 4c < 0.

• S10 Consider next the Lie algebra S10_{d,c} , c, d ∈ ℝ, d ≠ 0:

[x, y] = y,  [x, w] = dw,  [z, y] = y,  [z, w] = cw.  (15)

If c = d, then changing z by x − z, we see that S10_{d,d} ≅ ℝ × 𝔯_{3,d} for all d ≠ 0. If c ≠ d, let x′, y′, z′, w′ be the basis of S10_{1,0} satisfying (15) and x, y, z, w the corresponding basis of S10_{d,c}. Define a linear map ψ : S10_{d,c} → S10_{1,0} by

ψ(x) = x′ +(d−1)z′,  ψ(y) = w′,  ψ(z) = x′ +(c−1)z′,  ψ(w) = y′.

It turns out that ψ is a Lie algebra isomorphism for all c ≠ d and therefore S10_{d,c} ≅ S10_{1,0} ≅ 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ), where the last isomorphism follows by changing x′ to x′ − z′.

• S11 Consider the Lie algebra S11_{d,c} , d^{2} − 4c < 0, d = 0, 1:

[x, y] = y,  [x, w] = w,  [z, y] = w,  [z, w] = −cy + dw.

If d = 0, then ad(z/c)_{|𝔤′} has eigenvalues ±i and there exists a real basis of 𝔤′

such that ad(z/c) = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}), hence S11_{0,c} ≅ 𝔞𝔣𝔣(ℂ) for all c > 0.

If d = 1, then ad(z)_{|𝔤′} has eigenvalues 1/2 ± iλ/2, where λ = \sqrt{4c − 1}. Taking z′ = z/2, there exists a real basis of 𝔤′ such that ad(z′)_{|𝔤′} takes the form

(\begin{matrix}1 & λ \\ −λ & 1 \end{matrix}). Changing z′ to z′′ = (z′ − x)/λ, so that ad(z′′) = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}), we

conclude that S11_{1,c} ≅ 𝔞𝔣𝔣(ℂ) for all c such that 4c > 1.

Finally, in case the commutator ideal is three dimensional, we establish the correspondence with Table 1 in [O1], p. 22.

A1_{λ_{1},λ_{2}}, λ_{1} ≠ λ_{2} ∈ ℝ\\{0, 1\} A1_{λ,\overline{λ}}, Imλ ≠ 0 A2_{λ}, λ ∈ ℝ\\{0, 1\} A3_{λ}, λ ∈ ℝ\\{0, 1\}
𝔯_{4,λ_{1},λ_{2}}  𝔯′_{4,1/Im λ,Re λ/Im λ}  𝔯_{4,λ,λ}  𝔯_{4,λ}

\begin{matrix}A4 & A5 & A6 & H1 & H2 & H3 & H4 & H5_{λ} , λ ∈ ℝ\\{0, 1\} & H6_{λ} , λ ∈ ℂ\ℝ \\ 𝔯_{4,1,1} & 𝔯_{4,1} & 𝔯_{4} & 𝔡_{4} & 𝔡′_{4,0} & 𝔡_{4,1/2} & 𝔥_{4} & 𝔡_{4,λ} & 𝔡′_{4,−1/Im λ} \end{matrix}

Appendix III - Some known results related to 4-dimensional geometry

Using the characterization of homogeneous manifolds of negative curvature given by Heintze in [H] we can conclude that the following four dimensional Lie algebras do admit metrics with negative curvature:

• 𝔯_{4,μ,λ}, 0 < μ ⩽ λ ⩽ 1,

• 𝔯′_{4,μ,λ}, μ > 0, λ > 0,

• 𝔡_{4,λ}, 1/2 ⩽ λ < 1,

• 𝔡′_{4,λ}, λ > 0,

• 𝔥_{4}. Concerning non positive sectional curvature, we can mention a result appearing in [Dru], where it is proved that a left invariant metric with non positive curvature on a four dimensional solvable Lie group either has geometric rank one or it comes from an inner product on 𝔞𝔣𝔣(ℝ) × 𝔞𝔣𝔣(ℝ) or ℝ × 𝔯_{3,1}, up to scaling.

We understand that the classification of rank one four dimensional homogeneous spaces of non positive curvature is not known. On the other hand, Jensen classified in [J] the four dimensional Lie algebras admitting Einstein metrics:

• ℝ × 𝔯_{3,1},

• 𝔯_{4,1,1},

• ℝ × 𝔯′_{3,0},

• 𝔯′_{4,λ,λ}, λ > 0.

• 𝔡_{4,λ}, λ ⩾ 1/2,

Among these, it follows from [Al] that there are only two four dimensional Lie algebras admitting Einstein metrics of non positive curvature: 𝔯_{4,1,1} and 𝔡_{4,1/2}. Concerning left invariant anti-self-dual metrics on four dimensional Lie groups, it was proved in [DS] (Theorem 1.6) that if a four dimensional Lie group admits such a metric, then its Lie algebra is one of the following:

• 𝔡_{4,1/2},

• 𝔡′_{4,λ}, λ > 0. It is proved in [F2] that 𝔡_{4,2} is the only four dimensional solvable Lie algebra admitting an almost Kähler structure whose Ricci tensor is invariant with respect to the almost complex structure.

The classification of complex structures on four dimensional solvable Lie algebras was carried out by Snow in [SJ], when the commutator subalgebra has dimension one or two, and by Ovando in [O1], when the commutator subalgebra is three dimensional. The classification of hypercomplex structures was obtained in [B1].

Concerning the existence of symplectic structures, it is shown in [FG] that the solvmanifold obtained as a quotient of E(1, 1), the simply connected Lie group with Lie algebra 𝔢(1, 1), by a lattice, admits a symplectic structure but no complex structure. The classification of symplectic structures on four dimensional Lie algebras is done in [O2], where the cohomology of all four dimensional solvable Lie algebras is computed.

The hyper-Kähler metrics conformal to left invariant metrics metric on four dimensional Lie groups were determined in [B2]. It turns out that the solvable Lie groups appearing in this list are those with Lie algebra ℝ^{4}, 𝔞𝔣𝔣(ℂ), 𝔯_{4,1,1} or 𝔡_{4,1/2}. It was proved in [F1] that the cotangent bundle of a Lie group with Lie algebra 𝔞𝔣𝔣(ℂ) or 𝔯_{4,1,1} also admits a metric conformal to a hyper-Kähler metric.

The determination of hypersymplectic structures on four dimensional Lie algebras was carried out in [An]. According to this, the only Lie algebras admitting such a structure are ℝ^{4}, ℝ × 𝔥_{3}, 𝔯_{4,−1,−1} and 𝔡_{4,2}.

References

[Al] D. Alekseevsky, Homogeneous Riemannian spaces of negative curvature, Math. USSR Sbornik 25(1) (1975), 87–109.

[An] A. Andrada, Hypersymplectic 4-dimensional Lie algebras, preprint. Available at arXiv:math.DG/0310462

[AD] A. Andrada, I. Dotti, Double products and hypersymplectic structures on ℝ^{4n}, preprint. Available at arXiv:math.DG/0401294

[AS] A. Andrada, S. Salamon, Complex product structures on Lie algebras, to appear in Forum Math.

[ACFM] L. C. de Andrés, L. Cordero, M. Fernández, J. J. Mencía, Examples of four-dimensional compact locally conformal Kähler solvmanifolds, Geom. Dedicata 29, No.2, 227–232 (1989).

[AFGM] L. C. de Andrés, M. Fernández, A. Gray, J.Mencía, Moduli spaces of complex structures on compact four dimensional nilmanifolds, Boll. Unione Mat. Ital., VII Ser., A 5, No.3, 381–389 (1991).

[B1] M. L. Barberis, Hypercomplex structures on four dimensional Lie groups, Proc. Amer. Math. Soc. 125 (1997), 1043–1054.

[B2] M.L. Barberis, Hyper-Kähler metrics conformal to left invariant metrics on four dimensional Lie groups, Math. Phys., Analysis and Geometry 6 (1), (2003), 1–8.

[BK] H. Baum, I. Kath, Doubly extended Lie groups - Curvature, holonomy and Parallel spinors, Differential Geometry and its Applications 19(2003), 253-280.

[BV] N. Blazić, S. Vukmirović, Four-dimensional Lie algebras with a parahypercomplex structure, available at arXiv: math.DG/0310180

[CFG] V. Cruceanu, P. Fortuny and P.M. Gadea, A survey on paracomplex geometry, Rocky Mountain J. Math. 26(1) (1996), 83-115.

[DS] V. De Smedt, S. Salamon, Anti-self dual metrics on Lie groups, Contemp. Math. 308 (2002), 63–75.

[D] J. Dozias, Sur les algèbres de Lie résolubles réelles de dimension inférieure ou égale à 5, Thèse de 3 cycle, novembre 1963, Faculté des Sciences de Paris.

[Dru] M.J. Druetta, Visibility and rank one in homogeneous spaces of K ⩽ 0, Trans. AMS 304 (1987), 307–321.

[FG] M. Fernández and A. Gray, Compact symplectic solvmanifolds not admitting complex structures, Geom. Dedic. 34 (1990), 295–299.

[F1] A. Fino, Cotangent bundles of 4-dimensional hypercomplex Lie groups, Manuscr. Math. 109(4) (2002), 527–541.

[F2] A. Fino, Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor, preprint math.DG/0308014.

[GOV] V. Gorbatsevich, A. Onishchik, E. Vinberg, Structure of Lie groups and Lie algebras, English transl. in Encycl. Math Sc. 41, Springer-Verlag, Berlin, Heidelberg, 1994.

[H] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann, 211 (1974), 23–34.

[J] G.R. Jensen, Homogeneous Einstein metrics of dimension four, J. Differential Geom. 3 (3) (1969), 309–349.

[LM] M. de León and D. Martín de Diego, Almost product structures in mechanics, Diff. Geom. Appl., Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic, Masaryk University, Brno 1996, 539–548.

[LW] Lu, J.-H. and Weinstein, A.: Poisson Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom. 31 (1990), 501–526.

[Mi] J. Milnor, Curvatures of left invariant metrics on Lie groups , Advances in Mathematics, 21 (1976), 293–329.

[Miq] V. Miquel, Some examples of Riemannian almost-product manifolds, Pac. J. Math. 111 (1984), 163-178.

[Mu] G.M. Mubarakzyanov, On solvable Lie algebras, Izv. Vyssh. Uchebn. Zaved., Mat 32 (1) (1963), 114–123 (Russian). Zbl 166,42.

[N] A.M. Naveira, A classification of Riemannian almost product manifolds, Rendiconti di Matematica (3) 1983 Vol. 3, Serie VII, 577–592.

[O1] G. Ovando, Invariant complex structures on solvable real Lie groups, Manuscripta Math., 103, (2000), 19–30.

[O2] G. Ovando, Complex, symplectic and Kähler structures on four dimensional Lie groups, to appear in Rev. Un. Mat. Argentina.

[PSWZ] J. Patera, R.T. Sharp, P. Winternitz, H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Math. Physics, 17, (1976), 986–994.

[P] P. Petravchuk, Lie algebras decomposable as a sum of an abelian and a nilpotent subalgebra, Ukr. Math. J. 40(3) (1988), 385–388.

[SJ] J. E. Snow, Invariant complex structures on four dimensional solvable real Lie groups, Manuscripta Math, 66 (1990), pag 397–412.

[Ve] M. Vergne, Construction de représentations irréductibles des groupes de Lie résolubles, in P. Bernat et al., Représentations des groupes de Lie résolubles, Monographies de la Société Mathématique de France 4, Dunod, Paris, 1972.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2005/n1a2/v7n1a2.(dvi,ps,pdf)

A. Andrada andrada@mate.uncor.edu
M. L. Barberis barberis@mate.uncor.edu
I. G. Dotti idotti@mate.uncor.edu
G. P. Ovando ovando@mate.uncor.edu

CIEM, FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, (5000) Córdoba, Argentina