{'fraction_non_alphanumeric': 0.0681894076003473, 'fraction_numerical': 0.010953048821211514, 'mean_word_length': 3.521844171532112, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 52, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Pseudo-K\"ahler Lie algebras with abelian complex structures

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which verifies certain compatibility conditions with respect to the associative product on U. The Riemannian and Ricci curvatures of the associated pseudo-K\"ahler metric are studied and a characterization of those Lie algebras which are Einstein but not Ricci flat is given. It is seen that all pseudo-K\"ahler Lie algebras can be inductively described by a certain method of double extensions applied to the associated complex asssociative commutative algebras.


Introduction
An abelian complex structure on a Lie algebra g is a linear map GL(g) such that J 2 = −id g and [Jx, Jy] = [x, y] for all x, y ∈ g. It is obvious that the Nijenhuis tensor for such a J vanishes and, therefore, the structure is integrable. The name abelian comes from the fact that the eigenspaces of J in the complexification g C of g are abelian Lie algebras. Note that they are a particular case of the so called nilpotent complex structures [11].
A pseudo-Kähler Lie algebra is a symplectic Lie algebra (g, ω) with a complex structure J which is skew-symmetric with respect to ω. Such a Lie algebra is equipped with the pseudo-Riemannian metric g defined by g(x, y) = ω(Jx, y) for all x, y ∈ g for which the complex structure is parallel. The aim of this paper is to study the structure and main properties of Lie algebras which are pseudo-Kähler for an abelian complex structure.
If an algebra g admits an abelian complex structure it must be 2-step solvable. One can easily verify that Lie algebras of dimension 2 have an abelian complex structure (actually, all complex structures on these algebras are abelian) and that every non-degenerate skew-symmetric bilinear form ω is a symplectic form compatible with the complex structure. So, 2-dimensional Lie algebras are the first examples of pseudo-Kähler Lie algebras with abelian complex structure. Recall that the unique non abelian 2dimensional Lie algebra is the algebra aff(R) of affine motions of R. An important role in the sequel will be played by the Lie algebras aff(A) constructed as the tensor product aff(A) = aff(R)⊗ R A of aff(R) with an associative commutative algebra A. A well-known example of these algebras is the underlying Lie algebra of the Kodaira-Thurston manifold, which is the algebra aff(A) when A is the 2-dimensiona nilpotent power algebra. All the algebras aff(A) carry abelian complex structures but in order to admit a compatible symplectic form the algebra A must verify further conditions. The existence of a symmetric bilinear B form on A such that the pair (A, B) is a symmetric algebra [12] guarantees that the corresponding Lie algebra aff(A) admits a pseudo-Kähler structure for an abelian complex structure; however, these examples do not exhaust the family of pseudo-Kähler Lie algebras with abelian complex structure.
In this paper we will show that every pseudo-Kähler Lie algebra with abelian complex structure is completely characterized by a pair (U, H) where U is a complex associative commutative algebra and H a sesquilinear hermitian form on U which verifies certain compatibility conditions. This characterization let us calculate some nice formulas for the Riemannian and the Ricci curvatures of the pseudo-Kähler metric and lead us to a complete description those algebras in our family for which the metric is Einstein but not Ricci flat. Moreover, in the last section we prove that all the pseudo-Kähler Lie algebras with abelian complex structure can be obtained by sucessive application of a method of double extension, which consist of a central extension and a generlized semi-direct product, on the associated pairs (U, H) starting from an algebra aff(A).
We will say that a complex structure J is abelian if [Jx, Jy] = [x, y] holds for all x, y ∈ g.
Definition 2.2 Let g be a complex or a real Lie algebra. A product structure on g is a linear map K ∈ gl(g) such that K 2 = id g , K = id g and [x, y] = K[Kx, y] + K[x, Ky] − [Kx, Ky] for all x, y ∈ g.
The product structure K provides a decomposition of the vector space g as the direct sum of the eigenspaces g + = ker(K − id g ), g − = ker(K + id g ). Actually, g + and g − are subalgebras of g. When both subalgebras g ± have the same dimension, the product structure is said to be a paracomplex structure on g.
We will say that a product structure K is abelian if [Kx, Ky] = −[x, y] holds for all x, y ∈ g. In such case one easily verifies that the subalgebras g ± are actually abelian.
Remark 2.1 One can also use Definition 2.1 to define the notion of complex structure for complex Lie algebras, considering J to be C-linear. However, this is not very interesting since for a complex Lie algebra g there is one-to-one correspondence between such C-linear complex structures and product structures on g. Actually, a straightforward computation shows that a C-linear complex mapping J on a complex Lie algebra g is a complex structure if and only if the map K = iJ is a product structure, where i stands for the imaginary unit. Further, J is abelian if and only if K is so.
The following result is very well-known (see, for instance, [3]): Lemma 2.1 Every Lie algebra admitting an abelian complex structure is 2-step solvable. Let A be a commutative associative algebra. The vector space A ⊕ A with the product defined by , for a, b ∈ A, is an abelian complex structure on aff(A).

2.
The underlying real algebra of a complex Lie algebra with an abelian product structure When (g, K) is a complex Lie algebra with a C-linear abelian product structure, the underlying real Lie algebra g R is naturally endowed with the abelian complex structure defined by the R-linear map J = iK.

Definition 2.3
We say that a Lie algebra g admits a symplectic structure if it admits a non-degenerate scalar 2-cocycle ω. The pair (g, ω) is said to be a symplectic Lie algebra. Two symplectic Lie algebras (g 1 , ω 1 ), (g 2 , ω 2 ) are said to be symplectomorphic if there exists a Lie algebras homomorphism ϕ : g 1 → g 2 such that ω 1 (x, y) = ω 2 (ϕx, ϕy) for all x, y ∈ g 1 . Remark 2.2 A symplectic Lie algebra (g, ω) is naturally endowed with a structure of left-symmetric algebra compatible with the Lie structure; this is to say, the product defined by ω(x · y, z) = −ω(y, [x, z]) for x, y, z ∈ g verifies [x, y] = x · y − y · x and [x, y] · z = x · (y · z) − y · (x · z) for all x, y, z ∈ g.
Geometrically, this means that each Lie group with Lie algebra g can be equiped with the torsion free flat left-invariant connection defined by ∇ ω x y = x · y for x, y ∈ g.
Definition 2.4 Let (g, ω) be a symplectic Lie algebra and J a complex structure on g such that ω(Jx, Jy) = ω(x, y) for all x, y ∈ g. The triple (g, ω, J) will be called a pseudo-Kähler Lie algebra.
If the complex structure is abelian we will say that the algebra g is endowed with an abelian pseudo-Kähler structure. The pseudo-Riemannian metric g defined on g by g(x, y) = ω(Jx, y) will be called the pseudo-Kähler metric of (g, ω, J).
The following lemma will be useful in the next sections.
Lemma 2.2 Let (g, ω, J) be a pseudo-Kähler Lie algebra where J is abelian. The left-symmetric product defined by ω on g verifies J(x · y) = (Jx) · y for all x, y ∈ g.
Proof. If x, y, z ∈ g then we have which proves the result.
Remark 2.3 One may think that for every associative and commutative algebra A the corresponding Lie algebra aff(A) always admits an abelian pseudo-Kähler structure. However the result is not true and one may find associative commutative algebras A such that aff(A) does not even admit a symplectic form. For instance, if A is the 3-dimensional associative algebra spanned by a 1 , a 2 , a 3 with non-trivial products a 1 a 1 = a 2 a 2 = a 3 , then the algebra aff(A) is spanned by E 1 , . . . , E 6 where E j = (a j , 0) and E 3+j = (0, a j ) for 1 ≤ j ≤ 3 and their only non-trival brackets are [E 1 , The first identity shows that ω(E 6 , E 2 ) = ω(E 6 , E 3 ) = ω(E 6 , E 5 ) = 0 and the second one implies ω(E 6 , E 1 ) = ω(E 6 , E 4 ) = 0. Therefore ω(E 6 , x) = 0 for all x ∈ aff(A), which proves that the 2-cocycle cannot be non-degenerate.
We recall the following definition [12]: Definition 2.5 An associative algebra A over K is said to be a symmetric algebra if A is equipped with a non-degenerate symmetric bilinear form B : A × A → K that satisfies B(ab, c) = B(a, bc) for a, b, c ∈ A.
It should be noticed that, according to the results given in [9], one can construct a symmetric associative commutative algebra by applying the method of T * -extension to an arbitrary associative commutative algebra.

Examples 2.2
The following examples of pseudo-Kähler Lie algebras with abelian complex structures show that the class of such algebras is a wide one.
1. The Lie algebra aff(A) of a symmetric associative commutative algebra.
If (A, B) is a commutative associative symmetric algebra then the algebra aff(A) with the abelian complex structure defined above and the symplectic form given by is pseudo-Kähler. In this case we will say that aff(A) is equipped with the standard pseudo-Kähler structure.
A simple calculation shows that the left-symmetric product defined by ω on aff(A) is given by which is, actually, associative.
2. The underlying real algebra of a complex Lie algebra with an abelian para-Kähler structure If (g, K, ω) is a complex Lie algebra with an abelian para-Kähler structure, then the underlying real Lie algebra g R is pseudo-Kähler if one considers the complex structure J = iK and the symplectic form ω R = Re(ω) where Re(ω) stands for the real part of ω.
It should be noticed that the real dimension of each of those algebras is 4k for some k ∈ N.
3. An example with non associative left-symmetric product Let us consider the 6-dimensional real Lie algebra g linearly spanned by {E 1 , . . . , E 6 } with the non-trivial brackets Notice that g is the Lie algebra denoted by n 7 in [3]. The skew-symmetric bilinear map ω : g×g → R defined by and ω(E j , E k ) = 0 for the other cases in which j < k, turns to be a symplectic form on g and, further, the map J ∈ gl(g) such that J(E j ) = E 3+j , J(E 3+j ) = −E j , for j = 1, 2, 3, is an abelian complex structure compatible with ω. Therefore, (g, ω, J) is pseudo-Kähler.
However, the left-symmetric product defined by ω is not associative. Actually, one can easily verify that, for instance, Note that this algebra provides an example of pseudo-Kähler Lie algebra with abelian complex structure which is not of the types of the two previous examples.
A complex structure J on a real Lie algebra g provides a decomposition of the complexification g C as the direct sum of the vector spaces g 1,0 = ker(J C − iid g ), g 0,1 = ker(J C + iid g ), where J C is the C-linear map on g C defined by J C (x 1 + ix 2 ) = Jx 1 + iJx 2 for x 1 , x 2 ∈ g. One can easily verify that g 1,0 and g 0,1 are complex subalgebras of g C and, obviously, they have the same dimension. Hence, the complex Lie algebra g C is naturally endowed with a paracomplex structure. Further, if the complex structure J is abelian, then the subalgebras g 1,0 and g 0,1 are abelian and, thus, the paracomplex structure on g C is also abelian [6]. Further, if the real Lie algebra is pseudo-Kähler, then its complexification results to be para-Kähler. The following result will be used in next section. We omit its proof since it is nothing but a straghtforward calculation. Lemma 2.3 If (g, ω, J) be a pseudo-Kähler Lie algebra with abelian complex structure then its complexification g C is naturally endowed with the abelian para-Kähler structure defined by the complex symplectic , where x, x ′ , y, y ′ ∈ g and the abelian paracomplex structure K(x + iy) = −Jy + iJx. Further, the left-symmetric product defined by ω C on g C is obtained from the left-symmetric product on g as follows: In order to fix notations, we will recall some well-known definitions on linear algebra. In the sequel, for a complex number α ∈ C, we will denote by Re(α) and Im(α) respectively its real and imaginary part and α will mean its complex conjugate.
Let V be a complex vector space. A semilinear map τ : V → V is a R-linear map such that τ (αv) = ατ (v) for all α ∈ C and v ∈ V . A sesquilinear form is a R-bilinear map H : V × V → C which is C-linear in the left component and semilinear in the right one. Further, a sesquilnear form H is said to be hermitian if H(v 1 , v 2 ) = H(v 2 , v 1 ) holds for all v 1 , v 2 ∈ V . Note that, even though we will only consider non-degenerate hermitian forms, we shall not impose H to be definite.
Let (V, H) be a complex vector space with a non-degenerate hermitian form. If W is a complex subspace of V , then we will denote by W ⊥ its H-orthogonal subspace, this is to say Remark 2.4 All through the paper we will consider traces of R-linear maps. Even for a C-linear map, the trace considered will be the trace of the corresponding R-linear map on the underlying real vector space.
A simple calculation shows that for a semilinear map τ on a complex vector space one always has trace(τ ) = 0.

Compatible complex associative commutative algebras
As we will see below, every pseudo-Kähler Lie algebra with abelian complex structure is obtained from a pair (U, H) where U is a complex associative commutative algebra and H a non-degenerate hermitian form on U which verifies some compatibility conditions with respect to the associative product on U . We start this section with the construction of the associated pseudo-Kähler Lie algebras for a given pair (U, H). Definition 3.1 We will say that a complex associative commutative algebra U endowed with a nondegenerate hermitian sesquilinear form H is compatible with an abelian pseudo-Kähler structure or, shortly, APK-compatible if and only if for every x, y, z ∈ U it holds that where R z stands for the multiplication by z in U .
Proposition 3.1 Let U be a complex associative commutative algebra and H a non-degenerate hermitian form such that the pair (U, H) is APK-compatible. Let g U denote the underlying real vector space of U and define [·, ·] : g U × g U → g U , J : g U → g U and ω : g U × g U → R as follows: for all x, y ∈ U . The pair (g U , [·, ·]) is a real Lie algebra and (ω, J) provides an abelian pseudo-Kähler structure on g U .
Proof. First note that R * αx = αR * x and, obviously, R * x+y = R * x + R * y for all x, y ∈ U and α ∈ C. This shows that the bracket is R-bilinear. Furhter, Jacobi identity follows immediately from the equalities which can be easily derived from the condition given in equation (1) and the definition of the bracket.
Since H is non-degenerate so must be ω because for every x, y ∈ U one has Further, ω is obviously skew-symmetric, because H is hermitian, and for all x, y, z ∈ U we get and, therefore, its imaginary part also vanishes, which proves that ω is a symplectic form.
Finally, it is clear that J turns to be an abelian complex structure on g and that ω(Jx, Jy) = ω(x, y). Thus (g, ω, J) is pseudo-Kähler.

Remark 3.1
The following facts concerning the pseudo-Kähler Lie algebra contructed above should be noticed: (1) The pseudo-Kähler metric is given by g(x, y) = ω(Jx, y) = Re(H(x, y)) for all x, y ∈ g U .
(2) The left-symmetric product defined by ω on g U is as follows: (3) It is straightforward to show that when U = U 1 ⊕ U 2 is a direct sum of H-orthogonal ideals then g U is also the ω-orthogonal direct sum of the ideals g U1 and g U2 .
The following result shows that every abelian pseudo-Kähler Lie algebra of type aff(A) can be viewed as the Lie algebra associated to a compatible pair (U, H) and provides a necessary and sufficient condition to recognize such algebras among those constructed by APK-compatible pairs. Proposition 3.2 If (A, B) is a symmetric associative commutative algebra then aff(A) with the standard pseudo-Kähler structure is holomorphically symplectomorphic to the abelian pseudo-Kähler Lie algebra constructed by the APK-compatible pair (U, H) defined by the complexification U = A C and the hermitian form given by Conversely, if for an abelian pseudo-Kähler Lie algebra g U defined by an APK-compatible pair (U, H) there exists a semilinear involution σ : U → U such that hold for all x, y, z ∈ U , then g U is holomorphically symplectomorphic to aff(A) with its standard pseudo-Kähler structure, where A is the real form of U defined by σ endowed with a bilinear form B obtained, up to constant, by the restriction of H to A.
Proof. Suppose first that (A, B) is a symmetric associative commutative algebra and define H on U = A C as in the statement. Notice that H is non-degenerate if and only if B is so. A simple calculation shows that R * a+ib = R a−ib for all a, b ∈ A. Thus, the condition (1) follows immediately from the associativity and commutativity of U . If ψ : aff(A) → g U is defined by ψ(a, b) = −(a + ib)/2, one easily sees that ψ commutes with the complex structures and that ImH( Moreover, ψ is an isomorphism of real Lie algebras since it is clearly R-linear and for Conversely, if (U, H) is an APK-compatible algebra admitting a semilinear involution σ verifying the properties given above, then U admits the associative commutative real form From the first identity we get that the form B defined on A by B(a, b) = − 1 4 H(a, b) is a real-valued symmetric bilinear form on A and, from the second, that B(ab, c) = B(a, bc) for a, b, c ∈ A. Therefore, (A, B) is a symmetric associative commutative algebra and, according to the first part of the proposition, aff(A) is holomorphically symplectomorphic to the Lie algebra constructed with the APK-compatible pair Since U = A C , we only need to prove thatH = H. But this follows at once since bothH and H are hermitian and coincide on the real form A of U .
for all x, y, z ∈ U .
Proof. Let us consider u, x, y, z ∈ U . Since the product in U is associative and commutative, we have Lemma 3.4 Let (U, H) be an APK-compatible pair and let ann(U ) denote the annihilator of U and R * U U the linear C-span of all the elements R * x y with x, y ∈ U . The following holds: (a) The vector space R * U U is the H-orthogonal subspace of ann(U ) and it is an ideal of the associative commutative algebra U .
Proof. It is clear that H(R * x y, x 0 ) = 0 for all x, y ∈ U if and only if H(y, xx 0 ) = 0, this is to say, x 0 ∈ ann(U ). In order to prove that R * U U is an ideal, take x, y, z ∈ U and x 0 ∈ ann(U ). We then have which proves that xR * y z ∈ ann(U ) ⊥ = R * U U . This completes the proof of (a). Now, if ann(U ) = {0} then R * U U = ann(U ) ⊥ = U and hence it only remains to see that it coincides with U 2 . Let us take an element x ∈ (U 2 ) ⊥ . For all y, z, t ∈ U we then have This clearly implies that xR * y z = 0 for all y, z ∈ U but, since U = R * U U , one obtains x ∈ ann(U ) = {0}. Hence, the H-orthogonal subspace of U 2 is null, which shows that U = U 2 .
Proposition 3.5 Let (U, H) be an APK-compatible pair and (g U , ω, J) the corresponding pseudo-Kähler Lie algebra with abelian complex structure.
Proof. If ann(U ) = {0} we have, according to the lemma above, that U = R * U U . We can then define a map σ : It is obvious that σ 2 = id U and that Further, using Lemma 3.3, we get that R * x (yz) = z (R * x y) holds for all x, y, z ∈ U and, therefore, Bearing in mind Proposition 3.2 it only remains to show that the identities H(σ(x), σ(y)) = H(x, y) and H(xy, z) = H(x, σ(y)z) are verified for all x, y, z ∈ U . But this follows at once because from equation (1) we have which yields the desired identity.
In the cases where ann(U ) ∩ (ann(U )) ⊥ = {0} something similar occurs, as we prove in the following proposition which also clarifies the structure of the corresponding real symmetric associative commutative algebra.
Proposition 3.6 Let (U, H) be an APK-compatible pair. If ann(U ) ∩ (ann(U )) ⊥ = {0} then g U is holomorphically symplectomorphic to an algebra aff(A) endowed with the standard pseudo-Kähler structure for a certain symmetric form B.
Moreover, the algebra A decomposes as a B-orthogonal sum of ideals where p is the number of simple ideals in the semisimple part of A and each A i is a unital algebra.
Proof. If ann(U ) ∩ (ann(U )) ⊥ = {0} then U = ann(U ) ⊕ (ann(U )) ⊥ , which is an orthogonal sum of ideals. The pseudo-Kähler Lie algebra constructed with the algebra ann(U ) is clearly abelian and, therefore, holomorphically symplectomorphic to aff(A 0 ) for a real algebra with A 2 0 = {0}. The idealŨ = (ann(U )) ⊥ obviously verifies ann(Ũ ) = {0} and according to Proposition 3.5 gŨ is holomorphically symplectomorphic to aff(Ã) for some real symmetric associative commutative algebra (Ã,B). From ann(Ũ ) = {0} we immediately get that ann(Ã) is also null. This implies thatÃ cannot be nilpotent and hence it has a non trivial semisimple part. SinceÃ is commutative, its semisimple part is the direct sum of several copies of R and C with their usual field structure. Let p be the number of those simple ideals and let us choose on each of them an idempotent element e i . Consider A i = R eiÃ . It is clear that A i is an ideal ofÃ for all i ≤ p with unity e i , that A i A j = {0} and that they are mutually orthogonal since when i = j we have B(e i a, e j b) = B(a, e i e j b) = 0 for all a, b ∈Ã. To see that Obviously, A 2 +· · ·+A p ⊂ M 1 and M 1 has trivial annhilator. We can proceed with M 1 in the same way, taking its Pierce decomposition relative to the idempotent e 2 . Repeating sucessively the same argument, we obtain thatÃ = A 1 ⊕ · · · ⊕ A p ⊕ M p where M p = {a ∈Ã : e i a = 0, for all i ≤ p} is a nilpotent ideal ofÃ. If we suppose that M p is not null, then we may find an element a ∈ ann(M p ), a = 0. But this would imply a ∈ ann(Ã), a contradiction. Hence M p = {0} andÃ = A 1 ⊕ · · · ⊕ A p .
Finally, if we take the orthogonal sum A = A 0 ⊕Ã one easily proves that ann(A) = A 0 and that aff(A) = aff(A 0 ) ⊕ aff(Ã), and the result follows.

Remark 3.2 A well-known result on unitary symmetric algebras implies that and the restriction
The main result of this section shows that every abelian pseudo-Kähler Lie algebra is associated to certain APK-compatible complex associative commutaive algebra.
Theorem 3.7 Let (g, ω, J) be a pseudo-Kähler Lie algebra with abelian complex structure. There exists an APK-compatible pair (U, H) such that g and g U are holomorphically symplectomorphic.
Proof. If g is pseudo-Kähler, then (g C , ω C , K) is a para-Kähler complex Lie algebra for ω C and K defined as in Lemma 2.3. Note that g 1,0 = ker(J C − iid) = ker(K + id) and, according to [6,Lemma 4.1], it is a complex associative commutative algebra. Let us then consider Recall that the associative product on U is nothing but the restriction of the left-symmetric product defined on g C by ω C , this is to say, for x, y, z, t ∈ g we have Now we can define the non-degenerate hermitian form H : U × U → C as follows: and, for x, y ∈ g, let us put x + iy = x − iy. Since ω C (ξ 1 , ξ 2 ) = ω C (ξ 1 , ξ 2 ) and [ξ 1 , ξ 2 ] = [ξ 1 , ξ 2 ] for all ξ 1 , ξ 2 ∈ g C , for the left-symmetric product defined by ω C one also has ξ 1 · ξ 2 = ξ 1 · ξ 2 because holds for all ξ ∈ g C . Further, one easily verifies that g C = U ⊕ U verifies for all u, u ′ ∈ U (which can also be deduced from the results on para-Kähler algebras given in [6]). Thus, for all u, u ′ , u ′′ ∈ U , we have Therefore we get that R * u u ′ = u ′ · u for all u, u ′ ∈ U . Since U is commutative and (g C , ·) is left-symmetric we get We have, thus, proved that (U, H) is an APK-compatible pair.
To see that g and g U are holomorphically symplectomorphic, we will consider the map ψ : g → g U given by ψ(x) = 1 2 (x − iJx) for all x ∈ g. It is clear that ψ(Jx) = iψ(x), that ω(x, y) = Im(H(ψ(x), ψ(y)) and that ψ is R-linear. Let us prove that ψ[x, y] = [ψ(x), ψ(y)] gU . But, if we take x, y ∈ g, then Since ψ is obviously invertible, it is an isomorphism of real Lie algebras.
We shall use that every Lie algebra with abelian pseudo-Kähler structure is of the form g U for some APK-compatible pair (U, H) to see when such a Lie algebra is unimodular. We first prove the following lemma: Proof. We will proceed by induction on k. The result is clear for k = 2 because and this shows that if (R x + R * x ) k−1 verifies the condition, so does (R x + R * x ) k . Proposition 3.9 Let g be a pseudo-Kähler Lie algebra with abelian complex structure and (U, H) an APK-compatible pair such that g = g U . The following conditions are equivalent: To see that g is nilpotent whenever U is so, let us consider x, y ∈ U . Recall that R * [x,y] = R y R * x − R x R * y and hence we have and, therefore, ad(x) k+1 (y) = (−1) k (R x + R * x ) k [x, y] for all k ≥ 1. If we have that U n = {0} then the Lemma above implies that (R x + R * x ) 2n = 0 and, thus, ad(x) 2n+1 = 0, which proves the nilpotency of g. It is obvious that if g is nilpotent then it is unimodular and, thus, it only remains to show that for a unimodular g we get U nilpotent. Note that the map G x = ad(x) + R * x is semilinear and, therefore, traceless. This shows that trace(ad(x)) = −trace(R * x ) = −trace(R x ). Consequently, if g is unimodular, then we have trace(R x ) = 0 for all x ∈ U . But this implies that U is nilpotent since, otherwise, there must exist an idempotent element e ∈ U and one easily proves using the Peirce decomposition of U that trace(R * e ) = d where d is the real dimension of the ideal eU .

Curvature of the pseudo-Kähler metric
We will now calculate the curvatures of the pseudo-Riemannian metric of a Lie algebra with abelian pseudo-Kähler structure. We shall give two different approaches: a first one using the left-symmetric product defined by the symplectic form, and a second one based on the description in terms of APKcompatible pairs. Let us consider a Lie algebra with abelian pseudo-Kähler structure (g, ω, J) and denote by (g, ·) the left-symmetric algebra structure defined by ω. Proposition 4.1 Let (g, J, ω) be a pseudo-Kähler Lie algebra with abelian complex structure and g the pseudo-Kähler metric. The Levi-Civita connection and the Riemannian curvature tensor of g are respectively given for x, y, z ∈ g by If K denotes the Killing form of g, then the Ricci curvature tensor Ric(x, y) = trace R(x, −)y is given by Ric(x, y) = trace(ad(Jy · Jx)) − K(x, y) , x, y ∈ g.
Proof. For x, y, z ∈ g the Levi-Civita connection is given by and therefore, since J commutes with right multiplications and [x, y] = [Jx, Jy], we have for all x, y ∈ g. Now, the curvature tensor is given by for all x, y, z ∈ g.
In order to compute the Ricci form, let L x denote the left multiplication by an element x ∈ g with the left-symmetric product induced by ω. Then we have x ′ · Jx = (ad(x)J + L Jx )x ′ for all x ′ ∈ g and it follows that R(x, z)y = −L Jy Jad(x)z + (ad(x)J + L Jx )L y Jz − L y·Jx Jz.
Recalling that JL y = L Jy , we get Ric(x, y) = −trace(L Jy Jad(x)) + trace(ad(x)L Jy J) + trace(L Jx L y J) − trace(L y·Jx J) Now, the result follows immediately from the fact that for all z ∈ g, the map L z is the adjoint map with respect to ω of −ad(z) and that a linear map and its ω-adjoint have the same trace.   (A, B) be a real symmetric associative commutative algebra and let us consider on aff(A) the standard pseudo-Kähler structure. We had already seen that the left-symmetric product defined by the symplectic form was given by (a, b)·(a ′ , b ′ ) = −(aa ′ , ba ′ ) for all a, b, a ′ , b ′ ∈ A. Consequently, the levi-Civita connection and the Riemannian curvature tensor are

Note that this clearly implies that the metric is flat if and only if A 3 = {0}
Besides, we have that from where it is clear that Then, the metric is Ricci flat if and only if trace(R aa ′ ) = 0 for all a, a ′ ∈ A. But this only occurs if and only if A is nilpotent because, otherwise, there exists an idempotent e ∈ A and one always has trace(R ee ) = dim(eA) ≥ 1.
Flatness or Ricci flatness are easier to study if one describes the Riemannian and the Ricci curvature in terms of the products of an APK-compatible pair. We have the following: Proposition 4.2 Let g U denote the pseudo-Kähler Lie algebra with abelian complex structure defined by an APK-compatible pair (U, H). The Levi-Civita connection, the Riemannian curvature tensor and the Ricci curvature are given for x, y ∈ U by . Proof. According to (2) in Remark 3.1, and Proposition 4.1, we get that the Levi-Civita connection is In order to compute the Ricci curvature, let us consider for each x ∈ g the semilinear map G x : U → U defined by G x (y) = R * y x. It follows that Since R x G y and G xy are semilinear, they are traceless and as a consequence we get x for x, y ∈ U , and one obviously deduces that R(x, y) = 0. Conversely, if the metric is flat we have x R x which gives the desired identity. (c2) The left-symmetric product defined by ω is, actually, associative.
(c3) The derived ideal [g, g] is contained in ann(U ).
If one of the conditions above is fulfilled, then the other two are equivalent.
Proof. Recall that, as in the proof of the corollary above, the metric is flat if and only if 3R x R * y = R * y R x for all x, y ∈ U . On the other hand, the left-symmtric product is associative if and only if Finally, [g, g] ⊂ ann(U ) is equivalent to R x R * y = 0 for all x, y ∈ U because we have for all x, y, z ∈ U . Now, it suffices to realize that the combination of two of these conditions imply R x R * y = R * x R y = 0 and, then, the remaining condition is also verified.
, [10]. If g is a pseudo-Kähler Lie algebra constructed with an APK-compatible pair (U, H) and (g, ·) denotes the left-symmetric algebra defined by the symplectic form then we have and, accordingly, From these two equations we immediately get that (g, ·) is Novikov if and only if R z R * y = R * y R z . This proves that, for our algebras, the Novikov condition is equivalent to the associativity of the left-symmetric product. Proof. Let (U, H) be an APK-compatible pair for g. A simple computation shows that for all x, y, z ∈ U one has [z, [x, y]] = −(R z + R * z )[x, y] and, therefore, if g is 2-nilpotent, then for x, y, z ∈ U one has which shows that [g, g] lies in the annhilator of U and the result is just a consequence of the proposition above.
The following proposition, in which we give necessary and sufficient conditions on the APK-compatible pair to yield a 2-step nilpotent Lie algebra and to assure, in such case, that the metric is flat, clarifies the conditions of the Corollary. Proof. As we had seen in the proofs of Proposition 4.4 and Corollary 4.5, when g U is 2-step nilpotent one has R x R * y = 0 for all x, y ∈ U . But this is equivalent to R * U U ⊂ (R * U U ) ⊥ and, since R * U U = (ann(U )) ⊥ , it follows that (ann(U )) ⊥ ⊂ ann(U ). Recalling that we also had [z, [x, y]] = −(R z + R * z )[x, y] for all x, y, z ∈ U , we arrive at and, thus, we get H(xzt, y) = H(t, R * z R * x y) = 0, showing that A 3 = {0}. The converse follows at once from the identity [z, [x, y] Since R x R * y = 0 for all x, y ∈ U , the flatness of the pseudo-Kähler metric is equivalent to R * x R y = 0 for all x, y ∈ U . But this is nothing but the condition H(U 2 , U 2 ) = {0}.

Example 4.2
We had seen that if A is a nilpotent symmetric associative commutative algebra such that A 3 = {0}, then the standard pseudo-metric on aff(A) is not flat. But one can also construct non-flat 2-step nilpotent algebras. For example, let us consider the pair (U, H) where the associative algebra U is the C-span of {u 0 , u 1 , u 2 , u 3 , u 4 } with the non-trivial products u 4 u 3 = u 3 u 4 = u 4 u 4 = u 2 and the hermitian form such that H(u j , u k ) = 1 if j + k = 4 and H(u j , u k ) = 0 whenever j + k = 4. Obviously, Therefore, R x R * y = 0 for all x, y ∈ U . This proves that (U, H) is APK-compatible and that the Lie algebra g U is 2-step nilpotent. However, one has H(U 2 , U 2 ) = 0, which shows that the metric cannot be flat.
Although there exist nilpotent Lie algebras with non flat abelian pseudo-Kähler structures, all of them are Ricci flat as it was shown in a more general context in [14, Lemma 6.3] and can be directly proved: Corollary 4.7 Every unimodular pseudo-Kähler Lie algebra with abelian complex structure is Ricci flat.
Proof. We had seen that for pseudo-Kähler Lie algebras with abelian complex structures unimodularity and nilpotency are equivalent. Further, this means that if (U, H) is an associated APK-compatible pair, then U is nilpotent. This clearly implies that R x is nilpotent, hence traceless, for all x ∈ U and, thus, Bearing in mind the formula given for the Ricci curvature and the explicit calculation done in the case of the algebras aff(A), one could think that Ricci flatness is only possible for nilpotent algebras. However, this is not the case. If we consider the pair (U, H) where U is the complex span of {u 0 , u 1 , u 2 , u 3 } with the only non trivial products u 1 u 3 = u 3 u 1 = u 1 , u 2 3 = u 3 and the hermitian form defined by H(u 0 , u 3 ) = H(u 1 , u 2 ) = 1 and H(u i , u j ) = 0 in the remaining pairs {i, j}, then one easily sees that U is not nilpotent but R x R * y = R * x R y = 0 for all x, y ∈ U . Hence, the pseudo-Kähler metric is flat.
We shall finish this section con a characterization of the non Ricci flat Einstein case. Recall that a pseudo-Riemannian metric g is Einstein whenever there exists γ ∈ R such that g = γRic.
Proposition 4.8 If (g, ω, J) is a Lie algebra with abelian pseudo-Kähler structure and the pseudo-Kähler metric is Einstein but not Ricci flat then g = aff(A), endowed with the standard abelian pseudo-Kähler structure, where A is a semisimple associative commutative algebra and the symmetric form on A is (up to a scalar) the trace form of the regular representation.
Proof. Let us consider an APK-compatible pair (U, H) such that g = g U . If the metric is Einstein, then Ric(x, y) = −2trace(R y R * x ) must be non-degenerate and, hence, ann(U ) must vanish. According to Proposition 3.5, g is holomorphically symplectomorphic to an algebra aff(A) for some symmetric associative commutative algebra (A, B). But, in this case, we had seen in the Example 4.1 that Bearing in mind the construction of the the pseudo-Kähler metric, we have for all a, b ∈ A that B(a, b) = ω((a, 0), (0, b)) = g((0, −a), (0, b)) = γRic((0, −a), (0, b)) = γtrace(R ab ).
Since such form is non-degenerate if and only if A is semisimple, the result follows.

Inductive construction of pseudo-Kähler Lie algebras with abelian complex structure
In [17] and [13], the authors describe the method of symplectic double extensions by one-dimensional Lie algebras to construct new symplectic Lie algebras from a given one. Actually, they prove that every nilpotent symplectic Lie algebra can be obtained by a series of double extensions starting from the zero algebra. The same constructions (slightly modified) have been used in [6] to give an inductive description of all complex para-Kähler Lie algebras with abelian paracomplex structure. However, the method is not applicable in the pseudo-Kähler case unless one uses double extensions by planes instead of by lines. Since every pseudo-Kähler Lie algebra with abelian complex structure is associated to an APKcompatible pair, we will define an inductive method of construction for such pairs. The method is again based on two extensions of a given associative commutative algebra, in a similar way as it is done in [7] for symplectic associative commutative algebras.
All the associative algebras of this section will be, except if the contrary is explicitly said, complex algebras. We first recall some definitions: Definition 5.1 Let U be an associative commutative algebra and ϕ : U × U → C a symmetric bilinear form such that ϕ(xy, z) = ϕ(x, yz) for all x, y, z ∈ U . Then the vector space U ⊕ Cu 0 with the product is then an associative commutative algebra which we will call the (one-dimensional) central extension of U by means of ϕ.
It can be easily seen that Cv ⊕ U is an associative commutative algebra if and only if δ(xy) = (δx)y for all x, y ∈ U and δ 2 − ε δ = R x0 where ε ∈ C is such that v 2 = εv. In this case we will say that the algebra Cv ⊕ U is the generalized semidirect product of U and Cv by means of (δ, x 0 ).
Our aim is to prove that, essentially, all APK-compatible pairs may be obtained by a double extension given by a central extension and a generalized semidirect product from another APK-compatible pair. We first give conditions to assure that a double extension of that type on an APK-compatible pair leads to another APK-compatible pair.
Proposition 5.1 Let (U, H) be an APK-compatible pair, ε ∈ C, D ∈ gl(U ) such that D(xy) = (Dx)y for all x, y ∈ U and D 2 − εD = R a0 for some a 0 ∈ U and consider a semilinear mapping τ : U → U such that H(x, τ y) = H(τ x, y). Suppose that that there exist b 0 ∈ U such that and that for all x ∈ U the following conditions hold Let us construct a new algebraŨ = Cv 0 ⊕ U ⊕ Cu 0 with a commutative product ⋆ such that u 0 annihilatesŨ , v 0 ⋆ v 0 = εv 0 + a 0 + α 0 u 0 for some α 0 ∈ C and hold for all x, y ∈ U . If we define a non-degenerate hermitian formH onŨ bỹ Proof. Let us first show thatŨ is actually an associative commutative algebra constructed by performing a central extension on U and then a semidirect product to the extended algebra. It is straightforward to prove that the map ϕ : U × U → C given by ϕ(x, y) = H(x, τ y) is bilinear and symmetric. Moreover, for all x, y, z ∈ U we have ϕ(xy, z) = H(xy, τ z) = H(x, R * y τ z) = H(x, τ (yz)) = ϕ(x, yz).
This shows that U ′ = U ⊕ Cu 0 is the central extension of U by means of ϕ. Now, if we consider δ ∈ gl(U ′ ) given by δ(x + αu 0 ) = Dx + H(x, b 0 )u 0 for x ∈ U, α ∈ C, then we have that is verified for all x, y ∈ U . This shows that δ fulfills the necessary conditions to construct a generalized semidirect product of U ′ and Cv 0 . Further, one immediatly sees that the product onŨ is nothing but the product on such generalized semidirect product.
It only remains to prove, thus, that the condition (1) of the definition of an APK-compatible pair is verified. Let us denote byR ξ the multiplication by ξ ∈Ũ inŨ . It is obvious thatR * u0 = 0 since u 0 ∈ ann(Ũ ) and rather long but direct calculations yield: )u 0 and, therefore, they are equal if and only if DD * = R b0 and Db 0 = τ b 0 . If we now take x, y ∈ U , then we get x and hence, bearing in mind that τ R * x = R x τ , we obtain for all z ∈ U , H(xD * y, z) = H(y, zτ (x)) = H(R * z y, τ x) = H(x, τ R * z y) = H(x, zτ (y)) = H(yD * x, z). Since we also have Dτ = τ D * , we deduce that x ⋆R * v0 y = y ⋆R * v0 x. From the conditions DR * x = R τ x and τ 2 = R b0 we easily get that x ⋆R * Finally, when x, y, z ∈ U we have x ⋆R * y z = x ⋆ R * y z = xR * y z + H(x, τ R * y z)u 0 = zR * y x + H(R * y x, τ z)u 0 = z ⋆ R * y x = z ⋆R * y x, because (U, H) is APK-compatible and τ R * y = R y τ . The result now follows at once since the cases involving u 0 are trivial because u 0 andR * ξ u 0 are in the annihilator ofŨ for all ξ ∈Ũ . Proof. First notice that ann(U ) ∩ R * U U = {0} since R * U U = (ann(U )) ⊥ . If z ∈ ann(U ) then, for all x, y ∈ U , we have yR * x z = zR * x y = 0, which proves that ann(U ) ∩ R * U U is stable by all R * x , x ∈ U . Since R * x R * y = R * xy and αR * x + βR * y = R * αx+βy for all α, β ∈ C, x, y ∈ U , the set L = {R * x : x ∈ U }, where R * x holds for the restriction of R * x to ann(U )∩R * U U , is a complex abelian Lie subalgebra of gl(ann(U )∩R * U U ). The existence of u 0 and a function λ : U → C verifying the condition is then guaranteed by Lie's theorem and the semilinearity of λ is directly deduced from αR * x + βR * y = R * αx+βy .
Theorem 5.3 Every APK-compatible pair (Ũ ,H) such that ann(Ũ )∩(ann(Ũ )) ⊥ = {0} is an APK double extension of another APK-compatible pair (U, H) by a one-dimensional algebra. As a consequence, every APK-compatible pair (Ũ ,H) is either the pair corresponding to an algebra of type aff(A) with its standard pseudo-Kähler structure or can be obtained by a series of APK double extensions starting from the pair associated to an algebra aff(A).
Proof. Let us apply the lemma above and consider u 0 ∈ ann(Ũ ) ∩R * UŨ , u 0 = 0 such thatR * ξ u 0 = λ(ξ)u 0 for all ξ ∈Ũ . It is clear thatH(u 0 , u 0 ) = 0 sinceR * UŨ = (ann(Ũ )) ⊥ . AsH is non degenerate, we may find an element v 0 ∈Ũ such thatH(v 0 , v 0 ) = 0 andH(u 0 , v 0 ) = 1. This gives a decomposition of vector spacesŨ = Cu 0 ⊕ U ⊕ Cv 0 where U = (Cu 0 + Cv 0 ) ⊥ . The subset I = (Cu 0 ) ⊥ = Cu 0 ⊕ U is actually an ideal ofŨ because for all x ∈ I and ξ ∈Ũ one has where ⋆ denotes the product onŨ . Note that the restriction H ofH to U × U is non degenerate and that U is naturally endowed with a structure of associative commutative algebra if we consider the product given by the projection to U of the product in I. Moreover, if for all x, y ∈ U we denote xy ∈ U the projection of x ⋆ y to U , we have x ⋆ y = xy + ϕ(x, y)u 0 , where ϕ is certain symmetric bilinear form on U such that ϕ(xy, z) = ϕ(x, yz). Let τ : U → U be the unique semilinear mapping such ϕ(x, y) = H(x, τ y) for all x, y ∈ U , which is guaranteed because H is non-degenerate. It can be easily shown that H(x, τ y) = H(τ x, y) and τ R x = R * x τ because ϕ is symmetric and associative. Recall that I = Cu 0 ⊕ U is an ideal ofŨ and, therefore, if we denote by D ∈ gl(U ) the projection on U of the restriction ofR v0 to U , we can describe the product inŨ by u 0 ∈ ann(Ũ ) and v 0 ⋆ v 0 = εv 0 + a 0 + α 0 u 0 , v 0 ⋆ x = Dx + φ(x)u 0 , x ⋆ y = xy + H(x, τ y)u 0 , x, y ∈ U for some some linear form φ : U → C and certain ε, α 0 ∈ C, a 0 ∈ U . The non degeneracy of H let us find b 0 ∈ U such that φ(x) = H(x, b 0 ) for all x ∈ U . The associativity ofŨ implies that D(xy) = (Dx)y, τ Dx = R * x b 0 hold for all x, y ∈ U and that τ a 0 = D * b 0 − εb 0 , D 2 − ε D = R a0 and, since the product ⋆ is given as the one in Proposition 5.1, one sees by simply reversing the arguments of its proof that the condition ξR * ν ξ ′ = ξ ′R * ν ξ for ξ, ξ ′ , ν ∈Ũ implies that (U, H) is an APK-compatible pair and, further, that all the conditions on τ, D, a 0 and b 0 to have an APK double extension are fulfilled.
The second part of the statement follows by applying sucessively the first part. If (Ũ ,H) is not the pair corresponding to an algebra aff(A) then, according to Proposition 3.6, ann(Ũ ) ∩ (ann(Ũ )) ⊥ = {0} and it is an APK double extension of certain pair (U 1 , H 1 ). The same argument can be then applied to U 1 and so on. (1) The process to view an APK-compatible pair as a series of double extensions stops when we arrive at an algebra in which the annhilator does not intersect its orthogonal. This means that the symmetric real associative commutative algebra A of the second part of the theorem can be taken to be one of the algebras considered in Proposition 3.6. In particular, every nilpotent APK-compatible pair (U, H) such that U 2 = {0} is obtained by a sequence of APK double extensions starting from an algebra with zero multiplication.
(2) When g U is the Lie algebra constructed with an APK-compatible pair (U, H), the signature of the pseudo-Kähler metric is twice the one of H and when (Ũ ,H) is an APK double extension of a compatible pair (U, H) then the signature ofH is given by sig(H) = sig(H) + 1. This shows that if the pseudo-Kähler metric is definite positive, then g cannot be an APK double extension and must actually be an algebra aff(A) for an associative commutative symmetric algebra (A, B) as those in Proposition 3.6, this is to say, an orthogonal sum A = ann(A) ⊕ A 1 ⊕ · · · ⊕ A p where each ideal A i is of the form e i A for some idempotent e i . When the Kähler metric of aff(A) is definite positive, also B must be definite (negative). If the nilradical N i of A i is non-zero, x ∈ N i and k ∈ N, k ≥ 2 is such that x 2k = 0, x 2k−2 = 0 then we have 0 = B(x 2k , x 2 ) = B(x k+1 , x k+1 ) and hence x k+1 = 0, which is obviously a contradiction with x 2k−2 = 0 if k ≥ 3 and also if k = 2 because x 3 = 0 implies B(x 2 , x 2 ) = B(x 3 , x) = 0. Thus N i = {0} and then e i A must be simple. This proves that each ideal e i A is isomorphic to either R or C. But a simple calculation shows that C does not admit a definite form of the type defined in Definition 2.5. Thus, one immediately obtains the well-known result (see, for instance, [4,Theorem 4.1]) that a Kähler Lie algebra with abelian complex structre is an orthogonal sum of several copies of aff(R) and an even dimensional abelian algebra.