Locally conformally Kähler structures on four-dimensional solvable Lie algebras

Abstract We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification.


Introduction
The aim of this note is to provide explicit examples of locally conformally Kähler structures on complex surfaces and higher dimensional manifolds, by classifying left-invariant lcK structures on four-dimensional solvable Lie groups.
A locally conformally Kähler (shortly, lcK) metric g on a complex manifold (X, J) is a Hermitian metric that locally admits a conformal change exp (−f )g| U making it Kähler. Equivalently, the associated ( , )-form Ω := g(J_, _) satis es dΩ = θ ∧ Ω where the Lee form θ loc = df is a closed -form. In other words, one gets a covering endowed with a Kähler metric on which the deck transformations group acts by holomorphic homotheties. One can refer to [9,16,28] and references therein for an open-ended account on lcK geometry: just to cite a few of the several contributions to lcK geometry in the last twenty years, see [1, 8, 10, 13, 14, 18, 20, 22-24, 26, 27, 29, 30, 34, 40]. With the only exception of some Inoue surfaces, every known compact complex surface admits lcK metrics by [10,13], see also [34] for a survey on the Vaisman question.
Among compact complex surfaces, one can describe complex tori, hyperelliptic surfaces, Inoue surfaces of type S , primary Kodaira surfaces, secondary Kodaira surfaces, and Inoue surfaces of type S ± as compact quotients of solvable Lie groups endowed with left-invariant complex structures by [21,Theorem 1]. Complex structures on four-dimensional Lie algebras are classi ed by [1,31,37,38], see also [32]. On the other hand, locally conformally Kähler metrics underlie locally conformally symplectic (lcs) structures, which are similarly de ned. Extending Ovando's results on four-dimensional symplectic Lie algebras [33], a classi cation of fourdimensional locally conformally symplectic Lie algebras is given with structure results in [6].
Locally conformally Kähler structures on four-dimensional reductive Lie algebras are studied in [1,Theorem 4.6]. In this note, we classify locally conformally Kähler structures on four-dimensional solvable Lie groups. (See [5] for a survey and results on invariant lcK structures on solvmanifolds.) The classi cation is up to linear equivalence, and complex automorphisms are speci ed. The results are summarized in Theorem 1.1 and Table 2. We use the classi cation of complex structures [32] and the classi cation of lcs structures [6] for four-dimensional solvable Lie algebras, and the computations have been performed with the help of the mathematical open-source software system Sage [35] (the authors will be happy to share the code with anyone who might be interested).
We are also interested in Vaisman structures on a -dimensional Lie algebra, that is, lcK structures whose Lee form is parallel with respect to the Levi-Civita connection of the Hermitian structure. Let us recall the following characterization result from [2]. Lemma 3.3]). Let g be a Lie algebra with an lcK structure (J, Ω, θ) and let A ∈ g be such that A ∈ (ker θ) ⊥ with respect to the compatible metric given by (J, Ω) and θ(A) = . Then (J, Ω, θ) is Vaisman if and only if the adjoint operator ad A is a skew-symmetric endomorphism of g.
Using this characterization, we determine, among all lcK structures for each -dimensional Lie algebra, the ones being Vaisman (see also [3] for a nice description of unimodular Lie algebras admitting Vaisman structures).
With the same aim as [32, page 56], hopefully the classi cation in Table 2 might be useful in the future to provide speci c examples and to solve open problems. In this direction we use our classi cation to exhibit explicit examples of Lie algebras in dimension higher than four admitting an lcK structure. More precisely, we adapt some constructions in [6,25] to the lcK case and, as an application, we can produce many examples in higher dimension starting from dimension four, as well as we give a geometric interpretation of some of the -dimensional structures in Table 2. In particular, the lcK extension discussed in Proposition 3.1 allows to recover lcK structures on Oeljeklaus-Toma manifolds [24]. Notation. Structure equations for Lie algebras are written using the Salamon notation: e.g. rh = ( , , − , ) means that we x a coframe (e , e , e , e ) for rh ∨ such that de = de = de = and de = −e ∧e . Complex structures and tensors are usually expressed in terms of the above coframe. For example, complex structures on g are de ned in terms of their dual J : g ∨ → g ∨ with the convention Jα = α(J − _). By lcs structure, we mean a non-symplectic structure, namely, the Lee form is assumed to be non-exact (actually non-zero).

Classi cation of lcK structures on four-dimensional Lie algebras
In this section, we summarize the classi cation of locally conformally Kähler structures on -dimensional Lie algebras up to linear equivalence. Here, by linear equivalent lcK structures (J , Ω , θ ) and (J , ω , θ ) on the Lie algebra g of dimension dim R g ≥ we mean that there is an automorphism A ∈ gl(g) of the Lie algebra such that J = A − • J • A and Ω = A * Ω = Ω (A_, A_); by the injectivity of Ω ∧ _ : ∧ g ∨ → ∧ g ∨ , we also get θ = A * θ .
• rh is the Lie algebra associated to primary Kodaira surfaces; • rr , is the Lie algebra associated to hyperelliptic surfaces; • r ,− ,δ with δ > is the Lie algebra associated to Inoue surfaces of type S ; • d is the Lie algebra associated to Inoue surfaces of type S + ; • d , is the Lie algebra associated to secondary Kodaira surfaces; • and the Lie groups associated to the other algebras do not admit compact quotients. See also [10,34] and references therein as for the problem of existence of lcK structures on compact complex surfaces, known as the Vaisman question.

. . gl
Consider the Lie algebra where e is a generator of R. Left-invariant complex structures are described in [1,Proposition 4.7]: they belong to two families, both depending on one parameter µ = µ + √ − µ ∈ C \ √ − R, and they are de ned by µ (e − e ) + µ µ e namely, with respect to the dual coframe, they are associated to the matrices and These two families are related by an automorphism of gl , namely, Then it is su cient to consider the family J ,µ . The only non-trivial automorphism of (gl , J ,µ ) is (1.1) The generic lcs structure is θ = θ e , Ω = ω e + ω e − ω θ e + ω e − ω θ e + ω θ e , Assuming Ω is J ,µ -invariant we have two cases. Indeed, the condition reduces to the system We consider rst when µ = and θ = −µ , when the rank of the above × matrix is zero. In this case, Ω is always J ,µ -invariant. The J ,µ -positivity of the lcs structure forces ω > , ω > , and ω ω − ω > . And the general lck structure is Applying the only non-trivial automorphism we can assume that ω ≥ ω and ω ≥ : see also [1,Theorem 4.9,item (ii)].
In the other case, when µ ≠ or θ ≠ −µ , the rank of the above × matrix is two. The compatibility of the lcs with the complex structure J ,µ forces ω = and ω = ω , with ω > , θ µ < . Summarizing, up to equivalence, the lcK structure on (gl , J ,µ ) is of the form, see [1, Theorem 4.9, item (i)], And there is no further reduction since the non-trivial automorphism xes the lck structure.
The automorphisms of (u , J a,b ) are of the form with the condition a + a = .
The generic lcs structure is θ = θ e , Ω = ω e + ω e + ω θ e + ω e − ω θ e + ω θ e with ω + ω + ω θ ≠ . By imposing the J a,b -invariance, we get the condition Assuming Ω to be J a,b -positive we obtain ω < , θ b > . In particular b ≠ −θ , and therefore the J a,binvariance implies that ω = ω = . Summarizing we reduce to the generic lcK structure, see [1,Theorem 4 (1.4) and no further reduction is possible since a possible automorphism xs the lcK structure. According to [32], there is only one complex structure up to linear equivalence. In terms of the frame for rh , it is given by specifying the (− On rh ∨ , we set the linear complex structure J ∈ End(rh ∨ ) by Jα := α(J − _). Then, in terms of the coframe above, we have Je = e , Je = −e , Je = e , Je = −e , that is, J is given by the matrix As in [6, Appendix 6.1 of the arXiv version, page 28], by requiring dθ = , dΩ − θ ∧ Ω = , and Ω ∧ Ω ≠ , we get that the generic (non-symplectic) lcs structure is of the form The generic complex automorphism of (rh ∨ , J) are given, with respect to the chosen coframe, by with the condition a + a ≠ .
First, we apply the automorphism with parameters a = , a = , a = θ θ , and a = − θ θ . This reduces the lcK structure to θ = θ e and Ω = − ω θ e + ω e , where ω < and θ < . Then we apply the automorphism with parameters a = − θ , the others zero, so to transform the generic lcK form in (we It is easy to see that such forms cannot be further reduced, since the generic automorphism transforms θ as −a e − a e + (−a − a )e , and correspondingly the coe cient of Ω along e as a + a + a + a σ .

Remark 2.1.
We determine now which of these lcK structures on rh are of Vaisman type. Let A = a e + a e + a e + a e . We determine a i such that θ(A) = and A ∈ (ker θ) ⊥ , that is, Ω(A, Jx) = for any x ∈ ker θ. In this case ker θ is generated by {e , e , e } and we obtain that A = −e ∈ Z(g) and ad A = . Therefore, it follows from Lemma 0.1 that all the lcK structures above are of Vaisman type.

Remark 2.2.
We observe that the Morse-Novikov cohomology with respect to θ = −e vanishes in any degree.
. rr , Consider the Lie algebra rr , = ( , − , , ) with the complex structure de ned as in terms of the chosen coframe. According to [6, Appendix 6.3 of the arXiv version, pages [31][32], the generic lcs structures fall in two di erent families.
We rst consider the case when the generic closed -form θ = θ e + θ e + θ e has θ = θ = . Then the generic lcs structure with Lee form θ is It is clear that such a form is never J-positive: indeed, with respect to the dual frame (e , e , e , e ), we have ω(e , Je ) = . Then, there is no lcK structure in this case.
Consider now the case θ + θ ≠ . The generic complex automorphisms (rr , , J) are given, with respect to the chosen coframe, by In particular, the complex automorphism transforms θ e + θ e + θ e to θ e + θ e − θ e . So, without loss of generality, we can assume ϑ ≠ . The generic lcs structure in this case reduces to and ω = − ω θ θ θ +θ . The J-positivity requires ω > and θ > . Then the generic lcK structure is and we obtain the lck structure where we denoted σ = θ +θ ω and δ = θ . This lck structure cannot be further reduced. Indeed, the generic automorphism transforms the coe cient of θ along e to −a σ δ , whence we chose a = . The coe cient of Ω along e is transformed to a : we then choose a = , getting the identity.

Remark 2.3.
We determine now which of these lcK structures on rr , are of Vaisman type. Let A = a e +a e + a e + a e . We determine a i such that θ(A) = and A ∈ (ker θ) ⊥ , that is, Ω(A, Jx) = for any x ∈ ker θ. In this case we obtain that A = δ σ e ∈ Z(g). Therefore, it follows from Lemma 0.1 that all the lcK structures above are of Vaisman type.
. rr , Consider the Lie algebra rr , = ( , − , − , ). Consider the complex structure associated, in the chosen coframe, to the matrix The generic lcs structures are the following: either There is no lcK structure with Lee form θ = −e . Indeed, the corresponding lcs structures are never Jpositive, since Ω(e , Je ) = .
We consider lcK structures with Lee form θ = − e . The J-invariance of Ω requires ω = and ω = . Therefore we are reduced to Ω = ω e + ω e . The J-positivity requires ω > and ω < . Finally, the generic lcK structure is The generic automorphisms of (rr , , J) are associated to For a = √ −ω and a = , we apply the automorphism to get the normal form There is no further reduction, since the generic linear complex automorphism transforms σe − e into σe − a + a e , xing the coe cient along e .
. rr , Consider the Lie algebra rr , = ( , − , , ), endowed with the complex structure associated to the matrix The generic lcs structure is It is clear that Ω is never J-positive: indeed ω(e , Je ) = . Then, there is no lcK structure in this case.
We rst consider the complex structure J associated to the matrix According to [6, Appendix 6.4 of the arXiv version, pages [36][37], the generic lcs structures are the following: There is no lcK structure with Lee form θ = θ e + θ e . Indeed, the corresponding lcs structures are never J -positive, since Ω(e , Je ) = ω = .
We consider lcK structures with Lee form θ = − γe . The J -invariance of Ω requires ω = and ω = . Therefore we are reduced to Ω = ω e + ω e . The J -positivity requires ω > and ω > . Finally, the generic lcK structure is The generic automorphisms of (rr ,λ , J ) are associated to For a = √ ω and a = , we apply the automorphism There is no further reduction, since the generic linear complex automorphism transforms σe + e into σe + a + a e , xing the coe cient along e . Next we consider the complex structure J associated to the matrix the only di erence in this case is that Ω(e , J e ) = −ω > , then ω < . In the same way as above we get the nal lcK form

. r r
Consider the Lie algebra r r = ( , − , , − ) with the complex structure de ned as in terms of the chosen coframe. A generic automorphism for (r r ∨ , J) is associated to the identity matrix or to As in [6, Appendix 6.5 of the arXiv version, pages [38][39], the generic lcs structures are either First we consider the case θ = θ e . The lcK condition yields the generic form The only complex automorphism xing the Lee form is the identity. Whence the above form is the generic lcK form up to linear equivalence. Remark 2.6. We determine now which of these lcK structures on r r with Lee form θ = −e are of Vaisman type. Let A = a e + a e + a e + a e . We determine a i such that θ(A) = and A ∈ (ker θ) ⊥ . In this case we obtain that A = −e + ω ω e − e + ω ω . Therefore From Lemma 0.1, it follows that none of the lcK structures above are of Vaisman type.
We now assume θ = θ e . Requiring Ω to be J-positive and J-invariant, we obtain In the same way as above there is no further reduction.

Remark 2.7.
We determine now which of these lcK structures on r r with Lee form θ = −e are of Vaisman type. Let A = a e + a e + a e + a e . We determine a i such that θ(A) = and A ∈ (ker θ) ⊥ . In this case we obtain that A = −e − e . Therefore From Lemma 0.1, it follows that none of the lcK structures above are of Vaisman type.
Finally if θ = θ e + θ e , requiring the lcK conditions we obtain Applying the automorphism (2.1) we can assume that σ ≤ τ.

Remark 2.8.
We determine now which of these lcK structures on r r with Lee form θ = σe +τe are of Vaisman type. Let A = a e + a e + a e + a e . We determine a i such that θ(A) = and A ∈ (ker θ) ⊥ . In this case we obtain that A = + τ σ+ στ+τ e + σ+ στ+τ e . Therefore From Lemma 0.1, it follows that none of the lcK structures above are of Vaisman type.
As in [6, Appendix 6.6 of the arXiv version, pages 43-45], the generic lcs structure is either According to [32] this Lie algebra admits several di erent complex structures given by in terms of the chosen coframe.
We study rst the complex structure J . The only complex automorphisms of (r , J ) are If the lcs structure is as in the rst case, then the generic lcK form is Applying the automorphism (2.2) we can assume θ > .
We now consider the second case for the lcs form.
Requiring Ω to be J -positive and J -invariant we obtain the lcK form Applying the automorphism (2.2) we can assume ω > . In the last case, the generic lcK form is There is no further reduction, since Ω is xed by a generic automorphism (2.2). Now we consider the second complex structure J . The complex automorphisms of (r , J ) are with a + a ≠ , and moreover, if (a, b) ≠ ( , ), then a = , a = .
By J -positivity, we are reduced to only one possibility for the lcs structure, namely, In the rst case θ = −( + α)e , we have that Ω is never J-positive: indeed Ω(e , Je ) = . Then, there is no lcK structure in this case. In the second case θ = − e , the J-invariance of Ω requires ω = and ω = and J-positive implies ω > and ω < . Finally the generic lcK structure is The generic automorphisms of (r ,α, , J) with α ≠ , are associated to For a = √ ω and a = , we apply the automorphism to get the lcK form There is no further reduction, since the generic linear complex automorphism transforms e + σe into a + a e + σe , xing the coe cient along e . According to [6, Appendix 6.10 of the arXiv version, page 52, and Appendix 6.11 of the arXiv version, page 55] the generic lcs structure for both Lie algebras is Requiring J-positive we obtain Ω(e , Je ) = −ω > , Ω(e , Je ) = ω > . Assuming Ω is J-positive, we obtain ω = ω = and ω = −ω = . Therefore the generic lcK structure for these Lie algebras is The generic complex automorphisms for these Lie algebras are associated to For a = √ ω and a = , we apply the automorphism There is no further reduction, since the generic linear complex automorphism transforms σe + e into σe + a + a e , xing the coe cient along e . . r ,γ,δ with δ > Consider the Lie algebra r ,γ,δ = ( , γ + δ , −δ + γ , ). According to [6, Appendix 6.12 of the arXiv version, page 56] the generic lcs structure is only when γ ≠ . According to [32] this Lie algebra admits two not equivalent complex structures. We consider rst the complex structure de ned as We impose now Ω to be J -invariant and J -positive and we reduce the generic lcs structure to only when γ ≠ . The generic automorphisms of (r ,γ,δ , J ) with α ≠ , are associated to For a = √ ω and a = , we apply the automorphism There is no further reduction, since the generic linear complex automorphism transforms σe + e into σe + a + a e , xing the coe cient along e . Next we consider the second complex structure given by in terms of the chosen coframe. Requiring Ω to be J -positive we get ω < , this is the only di erence with the case J . Also the complex automorphisms are the same. Taking the automorphism the generic lcK form for this Lie algebra reduces to According to [32] this Lie algebra admits two not equivalent complex structures de ned as Let us start with J . Requiring J to be positive we obtain ω < , in particular ω ≠ . Then the only possibility for a compatible lcs structure is Assuming Ω to be J -invariant and J -positive we obtain a generic lcK structure      θ = −e Ω = ω (e + e ) + ω e + ω e with ω < , ω < , −ω + ω ω > . The generic automorphisms of (d , J ) in the chosen coframe are associated to A generic automorphisim transforms θ = −e into a e − e hence a must be . Then for a = √ −ω , we apply the automorphism  to get the lcK form      θ = −e Ω = µ(e + e ) − e ∧ e + σe ∧ e with µ + σ < .
Finally applying the automorphism a = and a = − (if it is necessary) we can assume that µ ≥ . There is no further reduction, since the generic linear complex automorphism xes the coe cient along e and the sign of the coe cient along to e .

Remark 2.13. The generic A = a e + a e + a e + a e yields
Then ad A is skew-symmetric if and only if A = a e . But then θ(A) = . Therefore, by Lemma 0.1, there is no Vaisman structure among the above lcK structures.
Now we consider the complex structure J . We impose now Ω to be J -positive. In the rst and second case, there is no lcK structure because we need ω < . In the third case, we need ω −ω > and −ω −ω > , but J -invariance for Ω yields ω = . Then, there is no lcK structure for J . For a = √ ω , we apply the automorphism to get the lcK form There is no further reduction, since the generic linear complex automorphism transforms σe + e into σe + a e , xing the coe cient along e .
Finally we consider the case According to [32] d , admits three di erent complex structures associated to We consider rst the complex structure J . If θ = − e , then the associated lcs form Ω is never J -positive. Indeed, Ω(e , J e ) = ω > and Ω(e , J e ) = −ω = − ω > , wich is a contradiction. Therefore the only possibility is θ = θ e with θ ≠ − . Assuming Ω is J -invariant and J -positive we reduce to A generic automorphism for (d , , J ) is given by Applying the automorphism with a = and a = √ −ω we get There is no further reduction, since a generic automorphism applied to the Lee form θ = − e gives a e − e , then a must be . The only possible automorphisms between two lcK forms of this kind transform the coe cient along e into a e , then a = and the automorphism is the identity. According to [32] d ,λ admits two di erent complex structures associated to We consider rst the complex structure J . Requiring Ω(e , J e ) = ω > , we get that the only possibility for the Lee form is θ = (λ − )e . Assuming Ω is J -invariant and J -positive we reduce to the following generic lcK structure A generic automorphism for (d ,λ , J ) is given by We apply the automorphism with a = √ ω and we reduce to lcK form where σ = ω . And there is no further reduction since a generic automorphism xes the coe cient along to e . Now we focus on the complex structure J . If θ ≠ −( + λ)e , then associated -form Ω is never positive. Indeed, Ω(e , J e ) = ω = . We consider the case θ = −( +λ)e . Assuming Ω is J -invariant and J -positive we obtain that ω = ω = and we reduce the lcK form to Taking a = √ ω we get the lcK form and there is no further reduction since a possible automorphism applied to Ω xes the coe cient along to e .
Remark 2.14. Consider all the cases d ,λ for λ ≥ together. For the generic A = a e + a e + a e + a e , we get Therefore, ad A is skew-symmetric if and only if A = . By Lemma 0.1, we get that there is no Vaisman structure on d ,λ for any possible value of the parameter λ.
. d ,δ with δ ≥ Consider the Lie algebra d ,δ = ( δ + , − + δ , − + δ , ) with δ ≥ . According to [6, Appendix 6.15 of the arXiv version, page 63] the generic lcs structures are According to [32] d ,δ with δ ≥ admits two di erent complex structures associated to In the case δ > there are other two more non equivalent complex structures A generic automorphism for (d ,δ , J) with J ∈ {J , J , J , J } is given by (2.7) Notice that for any choice of the complex structure the J-invariance condition implies that ω = and ω = . Therefore the generic lcs structure reduces to and this lcs form is invariant by a generic automorphism given by (2.7). We consider rst the complex structure J . Assuming Ω is J -invariant and J -positive we get where µ = θ and σ = ω . As we mention above there is no further reduction. If we consider the complex structure J , in a very similar way we obtain where µ = θ and σ = ω . We now focus on the complex structure J (case δ > ), and we have that the generic lcK structures are where µ = θ and σ = ω . Finally, if J = J (case δ > ), then we obtain where µ = θ and σ = ω .

Remark 2.15.
In any of the above four cases, it follows from Lemma 0.1 that any lcK structure above is of Vaisman type if and only if δ = . Indeed, if A ∈ d ,δ such that θ(A) = and A ∈ (ker θ) ⊥ , then A = µ e . Therefore According to [32], h admits a complex structure associated to In both cases we obtain that Ω is not J-positive, since Ω(e , Je ) = . Therefore there is no lcK structure for this Lie algebra.

Applications
In this section, we show some applications of our classi cation of lcK structures in dimension . In particular, we adapt some constructions of lcs structures in [25] and [6] to the lcK case and, as an application, we can produce many examples in higher dimension, including lcK structures on Oeljeklaus-Toma manifolds, or give a geometric interpretation of some of the -dimensional structures in Table 2.

. LcK extensions
Let h be a Lie algebra equipped with an lcK structure (J, ·, · ), and let (ω, θ) be the underlying lcs structure. Let V be a vector space of dimension n with a Hermitian structure (J , ·, · ) and denote by ω the fundamental -form induced by (J , ·, · ). We consider a representation given by π(X) = − θ(X) Id +ρ(X) such that ρ(X) ∈ u(n) ⊂ sp(n, R) for all X ∈ h. According to [25], the Lie algebra g de ned by g = h π V admits an lcs structure (ω , θ ) given by

Remark 3.2. If the initial Lie algebra h is solvable, then ρ(h) is solvable. Since ρ(h) ⊂ u(n) and u(n) is a compact Lie algebra, then we obtain that ρ(h) is Abelian, and therefore it is contained in a maximal Abelian subalgebra of u(n). In particular ρ(h ) = , where h = [h, h] denotes the commutator ideal. Moreover, we may assume that J ∈ u(n) is in the same maximal Abelian subalgebra which contains ρ(h).
According to [25], we have that g is unimodular if and only if tr(ad h X ) = nθ(X) for all X ∈ h. Recall that given a Lie algebra h, the map χ : h → R de ned by χ(X) = tr(ad X ) is a Lie algebra homomorphism, and its kernel is called the unimodular kernel of the Lie algebra h. We have then the following result: Proposition 3.3. Let h be a Lie algebra with an lcK structure (ω, θ), and let (π, V) be a n-dimensional representation such that π(X) = − θ(X) Id +ρ(X) with ρ(X) ∈ u(n) for all X ∈ h. Then the Lie algebra g = h π V with the lcK structure (ω , θ ) as above is unimodular if and only if the unimodular kernel of h is equal to ker θ and tr(ad h A ) = n.
Note that in order to build unimodular examples we have to start with a non unimodular Lie algebra h. In particular we need that tr(ad h A ) = n = dim V ∈ N. This condition is enough when the commutator ideal h = [h, h] = ker θ.
Using Lemma 0.1, it is easy to see that the lcK structure constructed with Proposition 3.1 is not Vaisman. Indeed, the endomorphism ad g A : g → g is with ρ(A) skew-symmetric. Then ad g A cannot be skew-symmetric, and therefore, the lcK structure is not Vaisman. Moreover, we will prove next that these Lie algebras do not admit any Vaisman structure when n > : Proposition 3.4. Let g = h π V be the unimodular solvable Lie algebra built as above with dim V = n and n > . Then g does not admit any Vaisman structure.
Proof. Suppose that g admits a Vaisman structure (ω, θ). Then we know that the Morse-Novikov cohomology vanishes in any degree, and therefore ω = d θ η = dη − θ ∧ η for some -form η. Letω the be restriction of ω to V × V, then we have thatω is a symplectic form on V. Moreover,ω = −θ ∧ η since V is Abelian. If n > , theñ ω is degenerate.

. Examples arising from -dimensional lcK Lie algebras
We summarize which Lie algebras in Table 2 are not unimodular, and therefore can be used to construct new examples of unimodular Lie algebras of dimension higher or equal than six with an lcK structure. They are: rr , , rr , , rr ,γ , r r , r , r ,α, , r ,α,α , r ,γ,δ , d ,λ , and d ,δ .
Taking into account the lcK structures for each of these Lie algebras as exhibited in Table 2, it can be shown, using Proposition 3.3, that the only lcK Lie algebras which admit a unimodular lcK extension are the following, where the complex structure is written in the frame {e , e , e , e }: r r : Applying Proposition 3.1 to the Lie algebras r ,α,α and r ,γ,δ , we obtain Lie algebras of dimension greater than or equal to six, which are almost-Abelian Lie algebras. Recall that a Lie algebra is called almost-Abelian if it has an Abelian ideal of codimension one. Almost-Abelian Lie algebras admitting lcK structures were studied in [4], where the second-named author and Adrián Andrada proved that the associated Lie groups admit no lattices, whenever the dimension is greater than four. Remark 3.6. If we extend the Lie algebras d , , d ,λ and d ,δ by Proposition 3.1, we obtain almost-nilpotent Lie algebras of dimension greater than or equal to six with an lcK structure. We will explain in detail how to extend one of these Lie algebras in Example 3.7. Recall that a Lie algebra is called almost-nilpotent if it has a nilpotent ideal of codimension one. The existence of lattices in almost-nilpotent Lie groups was studied in [12].
We explain now how to extend one of the almost nilpotent cases: We consider the lcK structure on d ,δ given by We show that, for any n ∈ N, there is an lcK extension given by Proposition 3.1 of the Lie algebra d ,δ for a suitable choice of δ and µ in order to obtain a ( n + )-dimensional unimodular lcK Lie algebra g = d ,δ π R n for certain lcK representation π. It follows from Proposition 3.3 that g is unimodular if and only if δ = nµ.
We de ne next the representation π : d ,δ → gl( n, R) by π = − θ Id +ρ for some representation ρ : d ,δ → u(n). It follows from Remark 3. It is easy to verify that is an lcK structure on the ( n + )-dimensional Lie algebra g for any n ∈ N. Note that we can write g = Re (h × R n ), therefore g is an almost-nilpotent Lie algebra.

. OT Lie algebras as lcK extensions
Oeljeklaus-Toma manifolds (OT manifolds) are compact complex non-Käher manifolds which arise from certain number elds, and they can be considered as generalizations of the Inoue sufaces of type S . It was proved in [24] that certain OT manifolds (those of type (s, )) admit lcK metrics. According to [22], the OT manifolds are solvmanifolds. Moreover, it can be seen that the complex structure is induced by a left-invariant one on the corresponding simply connected solvable Lie group. These manifolds provided a counterexample to a conjecture made by Vaisman according to which the rst Betti number of a compact lcK manifold is odd (see [24]). We show in this subsection that the Lie algebras associated to these OT solvmanifolds of type (s, ) endowed with its lcK structure can be obtained using our construction given in Proposition 3.1. We recall the de nition of the Lie algebra associated to the ( n + )-dimensional Oeljeklaus-Toma solvmanifold of type (s, ) (see [22]), which we denote by g OT . The Lie brackets on g OT are given by in the basis {x , . . . , xn , y , . . . , yn , z , z } for some c i ∈ R. The complex structure on g OT is Jx i = y i and Jz = z . The lcK structure on g OT is given by in the dual basis {x , . . . , x n , y , . . . , y n , z , z }, where the associated Hermitian metric is de ned by g(·, ·) = ω(·, J·).
Next we show how to recover the -dimensional OT Lie algebra using Proposition 3.1 and Theorem 1.1. Let us consider the -dimensional Lie algebra r r with structure constants given by ( , − , , − ).
According to Table 2 this Lie algebra admits many non equivalent LCK structures up to Lie algebra complex automorphisms. The one we are interested now is The complex structure is given by J(e ) = e and J(e ) = e in terms of the coframe {e , e , e , e }. Taking into account Proposition 3.3 in order to obtain a -dimensional unimodular extension of this Lie algebra, we can simplify the lcK form to θ = e + e ω = e + e − e + e .
Let g be the -dimensional vector space g = r r ⊕ R . We de ne π : r r → gl( , R) by π(e ) = , π(e ) = , in the orthonormal basis {e , e } of R with J e = e . It is easy to check that π satis es conditions of Proposition 3.1, and therefore g = r r π R is a -dimensional unimodular Lie algebra admitting a lcK structure (ω, θ) given by θ = e + e ω = e + e − e + e + e .
Clearly, (g, ω , θ ) is isomorphic to the OT Lie algebra of dimension with lcK structure given by (3.3) with n = .
To generalize this case to higher dimensions we consider the non-unimodular n-dimensional Lie algebra aff(R) n with structure equations given by . , e n , f n }, with Hermitian metric g(·, ·) = ω(·, J·). Let g be the ( n + )-dimensional vector space g = aff(R) n ⊕ R . We extend the complex structure J in aff(R) n to g by Ju = u where {u , u } denotes an orthonormal basis of R . We de ne π : aff(R) n → gl( , R) by and π(f i ) = for i = , . . . , n and c i ∈ R. It is easy to check that π satis es Proposition 3.1, and therefore the unimodular Lie algebra g = r r π R n , admits a lcK structure, which we still denote by (ω, θ), given by It is clear that g is isomorphic to the ( n + )-dimensional Oeljeklaus-Toma Lie algebra with Lie bracket given by (3.2). Indeed, ϕ : g → g OT , ϕ(e i ) = x i , ϕ(f i ) = y i for i = , . . . , n and ϕ(u i ) = z i for i = , is a Lie algebra isomorphism which commutes with the complex structures and it also preserves the lcK forms.

. From coKähler to lcK
In [6], the rst-named author and G. Bazzoni and M. Parton observed that every lcs structure on -dimensional Lie algebras can be constructed either as a solution to the cotangent extension problem [6, Corollary 1.14], or as a mapping torus over a contact -dimensional Lie algebra [6, Theorem 1.4], or with a similar construction starting from a -dimensional cosymplectic Lie algebra [6, Proposition 1.8].
then in particular (η, ω) is a cosymplectic structure. For a cosymplectic structure, we denote by R the Reeb vector, determined by ι R ω = and ι R η = ; then one has the decomposition h * = η ⊕ R • , where R • denotes the annihilator of R , that coincides with the kernel of the map ω n− ∧ _. Recall also that (η, ξ , Φ) is called normal when Nij Φ + dη ⊗ ξ = . (cK5) When (η, ξ , Φ, g) is both cosymplectic and normal, then it is called coKähler. We refer to e.g. [11] for further details.
Proof. By [6, Proposition 1.8], we already know that g has a natural lcs structure. For the sake of completeness, we brie y recall the construction. On g = h D R, we de ne θ(X, a) := −αa, Ω := ω + η ∧ θ.
It su ces to show that Ω is actually lcK, that is, there is a natural integrable complex structure J on g such that JΩ = Ω. We set, see [36], see also [11,Section 6. Remark 3.9. The Lie algebra g is unimodular if and only if h is unimodular and Dη = −α(n − )η + ζ for some ζ ∈ R • . If h is unimodular then the lcK structure (Ω, ϑ) on g is not exact. Indeed, we recall the idea in [6]: unimodularity for g is equivalent to the generator of ∧ n g * being non-exact, that is equivalent to d ∧ n− g * = . Since ∧ n− g * = ω n− ∧ η ⊕ ∧ n− h * ∧ θ, we compute where φ ∈ ∧ n− h * , and we decomposed Dη = βη + ζ with ζ ∈ R • . The statement follows.
Next we show an example of a Lie algebra admitting a lcK structure in Table 2 constructed from adimensional coKähler Lie algebra. Recall from [17] that coKähler Lie algebras in dimension n + are in one-to-one correspondence with n-dimensional Kähler Lie algebras endowed with a skew-adjoint derivation B which commutes with its complex structure. Let (R , e ∧ e ) be a -dimensional Kähler Lie algebra, where Je = e in the orthonormal coframe {e , e }. Consider the derivation of R given by B(e ) = e and B(e ) = −e . Then the Lie algebra h = R B Rξ admits a coKähler structures (η, ξ , Φ, g) where g is the orthonormal extension of the Kähler metric in R , η is the dual -form of ξ and Φ = J in R and Φ(ξ ) = .
Let us consider now the derivation D : h → h given by D(ξ ) = , D(e ) = e and D(e ) = −e . Finally, the Lie algebra g = h D R admits a natural lcK structure according Proposition 3.8, and it is easy to see that this Lie algebra is isomorphic to the -dimensional Lie algebra r on Table 2.

Remark 3.10.
Concerning the construction in [6] as a mapping torus over a contact -dimensional Lie algebra [6,Theorem 1.4], we should mention that this construction, in the Hermitian case, corresponds to the known relation between lcK and Sasakian structures, see [3]. In particular, the subclass of Vaisman Lie algebras can be constructed in this way.
the support of Istituto Italiano di Cultura in Córdoba. The second-named author is supported by CONICET, SECyTUNC (Argentina) and the Research Foundation Flanders (Project G.0F93.17N)