The classification of left-invariant para-Kähler structures on simply connected four-dimensional Lie groups
aa r X i v : . [ m a t h . S G ] F e b The classification of left-invariant para-K¨ahler structures onsimply connected four-dimensional Lie groups
M. W. Mansouri and A. Oufkou a Universit´e Ibno TofailFacult´e des Sciences. K´enitra-Maroce-mail: [email protected]@uit.ac.ma
Abstract
We give a complete classification of left invariant para-K¨ahler structures on four-dimensionalsimply connected Lie groups up to an automorphism. As an application we discuss some curva-tures properties of the canonical connection associated to these structures as flat, Ricci flat andexistence of Ricci solitons.
Keywords:
Symplectic Lie algebras, para-K¨ahler structures, Ricci soliton
1. Introduction and main results
An almost para-complex structure on a 2 n -dimensional manifold M is a field K of endomor-phisms of the tangent bundle T M such that K = Id T M and the two eigendistributions T ± M : = ker ( Id ± K ) have the same rank. An almost para-complex structure K is said to be integrable ifthe distributions T ± M are involutive. This is equivalent to the vanishing of the Nijenhuis tensor N K defined by N K ( X , Y ) = [ X , Y ] + [ KX , KY ] − K [ KX , Y ] − K [ X , KY ] , for vector fields X , Y on M . In such a case K is called a para-complex structure. A para-K¨ahlerstructure on a manifold M is a pair ( h ., . i , K ) where h ., . i is a pseudo-Riemannian metric and K isa parallel skew-symmetric para-complex structure. If ( h ., . i , K ) is a para-K¨ahler structure on M ,then ω = h ., . i ◦ K is a symplectic structure and the ± − eigendistributions T ± M of K are two in-tegrable ω -Lagrangian distributions. Due to this, a para-K¨ahler structure can be identified with abi-Lagrangian structure ( ω, T ± M ) where ω is a symplectic structure and T ± M are two integrableLagrangian distributions. Moreover the Levi-Civita connection associate to neutral metric h ., . i coincides with the canonical connection associate to bi-Lagrangian structure (the unique sym-plectic connection with parallelizes both foliations [9]). For a survey on paracomplex geometrysee [6] and for background on bi-Lagrangian structures and their associated connections, thesurvey [7] is a good starting point and contains further references (See as well [1] and [5]).Suppose now that M is a Lie group G and ω , h ., . i and K are left invariant. If we denote by g the Lie algebras of G , then ( h ., . i , K ) is determined by is restrictions to the Lie algebra g . In thissituation, ( g , h ., . i e , K e ) or ( g , ω e , K e ) is called a para-K¨ahler Lie algebra (e is unit of G ), in the Preprint submitted to Elsevier March 2, 2021 est of this paper a para-K¨ahler Lie algebra will be noted ( g , ω, K ). Recall that two para-K¨ahlerLie algebras ( g , ω , K ) and ( g , ω , K ) are said to be equivalent if there exists an isomorphismof Lie algebras T : g −→ g such as T ∗ ω = ω and T ∗ K = K . Para-K¨ahler (bi-Lagrangian)structures on Lie algebras in general have been studied, for example, in [2], [3] and [4]. In [8],there is a study the existences of bi-Lagrangian structures on symplectic nilpotent Lie algebrasof dimension 2,4 and 6. A first classification of para-K¨ahler structures on four-dimensional Liealgebras was obtained by Calvaruso in [6]. Another classification based on the classification ofsymplectic Lie algebras is proposed by Smolentsev and Shagabudinova in [12]. Benayadi andBoucetta provide in [3] a new characterization of para-K¨ahler Lie algebras using left symmetricbialgebras inroduced by Bai in [2]. Based on this characterization we propose in this paper,the classification of para-K¨ahler structures on four-dimensional Lie algebras. Notice that ourclassification is more complete and precise than the other classifications existing in the literature. Notations : For { e , e , e , e } a basis of g , we denote by { e , e , e , e } the dual basis on g ∗ and e i j the two-form e i ∧ e j , ˙ e i j is the symmetric two-form e i ⊙ e j and E i j is the endomorphism whichsends e j to e i and vanishes on e k for k , j .The para-K¨ahler Lie algebras ( g , h ., . i , K ) is necessarily symplectic Lie algebra ( g , ω ). It is wellknown that a symplectic four-dimensional Lie algebra is necessarily solvable. The classificationof symplectic four-dimensional Lie algebras ( g , ω ) is given by the following Table (see [10]).Case No vanishing brackets ω rh [ e , e ] = e e + e rr , [ e , e ] = e e + e rr , − [ e , e ] = e , [ e , e ] = − e e + e rr ′ , [ e , e ] = − e , [ e , e ] = e e + e r τ [ e , e ] = e , [ e , e ] = e e + µ e + e r ′ [ e , e ] = e , [ e , e ] = e , e + e [ e , e ] = e , [ e , e ] = − e n [ e , e ] = e , [ e , e ] = e e + e r , [ e , e ] = e , [ e , e ] = e e ∓ e r , − [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e − e e + e r , − ,β [ e , e ] = e , [ e , e ] = − e , [ e , e ] = β e e + e r ,α, − α [ e , e ] = e , [ e , e ] = α e , [ e , e ] = − α e e + e r ′ , ,δ [ e , e ] = e , [ e , e ] = − δ e , [ e , e ] = δ e e ∓ e d , [ e , e ] = e , [ e , e ] = e , e − e [ e , e ] = e e − e + e d , [ e , e ] = e , [ e , e ] = e , e − e [ e , e ] = e , [ e , e ] = − e e ∓ e d ,λ [ e , e ] = e , [ e , e ] = e , e − e [ e , e ] = λ e , [ e , e ] = (1 − λ ) e d ′ ,δ [ e , e ] = e , [ e , e ] = δ e − e , ∓ ( e − δ e )[ e , e ] = δ e , [ e , e ] = e + δ e h [ e , e ] = e , [ e , e ] = e , ∓ ( e − e )[ e , e ] = e , [ e , e ] = e + e Table 1: Symplectic four-dimensional Lie algebras( µ ≥ − ≤ β < − < α < δ > λ ≥ , λ , , Theorem 1.1.
Let ( g , ω, K ) be a four-dimensional para-K¨ahler Lie algebra. Then ( g , ω, K ) isisomorphic to one of the following Lie algebras with the given para-K¨ahler structures: Lie algebra rh For ω = e + e K = E − E − E + E − E − E K = E − E + E − E K = − E + E − E + E Lie algebra rr , For ω = e + e K = − E + E − E + E K = E + xE − E + E − E Lie algebra rr , − For ω = e + e K = E + E ± E − E − E K = − E − E ± E + E + E K = E + E + xE − E − E Lie algebra r r For ω = e + µ e + e , ( µ > K = − E + E + E + E − E − E K = − E + E + E − E K = E − E − E − E + E + E For ω = e + e K = E − E − E + E − E + E K = − E + xE + E + xE + E + xE + xE − E + x E K = − E + E − E + E K = − E + E + E + xE − E K = E − E − E − E + E + E K = E + xE − E + E + yE − E K = E + xE + xE − E + xE + E + xE − E Lie algebra r ′ For ω = e + e K = E + xE − E − x E + E − E K = − E − E + E + E = xE + yE + (1 − x ) E − yE − yE + xE + yE + (1 − x ) E + (1 + x ) E + yE − xE − yE − yE + (1 + x ) E + yE − xE K = E + xE + E − E − E Lie algebra r , For ω = e ± e K = − E + E − E + E K = E − E + E − E Lie algebra r , − For ω = e + e K = E + xE − E − E + E K = − E + E + E − E Lie algebra r , − ,β ( − < β < For ω = e + e K = E + E − E − E + E K = E − E − E − E + E K = E − E − E + E K = E − E + xE + x E K = E − E + E − E K = − E + E + E + E − E K = − E − E + E + E − E Lie algebra r , − , − For ω = e + e K = − E + xE + E + E − E + E + E K = − E + E + E − E + E K = − E − E + E − E + E K = − E + E − E + E K = E − E − E − E + E K = E + E − E − E + E K = E − E − E + xE + E K = − E + E + E − E K = E + E − E + E − E K = E − E − E + E − E Lie algebra r ,α, − α ( − < α < )For ω = e + e K = − E + E + E − E + E = − E + E − E − E + E K = ∓ ( − E + E − E + xE + E ) K = E + E − E − E K = E + E + E − E − E K = − E − E + E + E + E K = E + E − E − E − E K = E − E − E + E − E K = E − E + E + E − E K = − E − E − E + E + E K = − E − E + E + E Lie algebra d , For ω = e − e K = E + E − E − E + E K = E − E − E − E + E K = E − E − E + xE + E K = E − E + E − E K = − E + E + E + E − E K = − E − E + E + E − E K = E + E − E + xE + E − xE − E K = − E + E + E − E + E K = − E − E + E − E + E K = − E + E − E + xE + E For ω = e − e + e K = E + xE − E − E + E K = − E + xE + E + E − E Lie algebra d , For ω = e − e K = E + E − E − E + E K = E − E − E − E + E K = E − E − E + E K = E − E + E + xE − E For ω = e − e K = − E − E + x E + E − xE + E K = − E − E − E + xE + E + E + E K = E + E − E + xE − E + E − E = E − E + xE + xE + E + xE + xE − E K = − E + E − E + xE + E K = E − E − xE + E + xE − E K = − E + xE + E − xE − E − xE − xE + E For ω = e + e K = − E − E + x E + E − xE + E K = − E + E − E + xE + E K = E − E + xE + E − xE − E Lie algebra d , For ω = e − e K = E − E − E + xE + E K = E − E + E − E Lie algebra d ,λ ( λ > , λ , , )For ω = e − e K = E + E − E − E + E K = E − E − E − E + E K = − E + E + E − E + E K = − E − E + E − E + E K = E − E − E + xE + E K = − E + E − E + xE + E K = − E + xE + E + E − E K = E + xE − E + E − E Lie algebra h For ω = ± ( e − e ) K = E − E − E + E K = − E + E + E − E Corollary 1.1.
The symplectic Lie algebras rr ′ , , n , r ′ , ,δ and d ′ ,δ does not admit a para-K¨ahlerstructure. The paper is organized as follows. Section 2 contains the basic results which are essential to theclassification of four-dimensional para-K¨ahler Lie algebras (proof of the Theorem 1.1). Theorem2.1 and Theorem 2.2 are the key steps in this proof. Section 3 is devoted to some curvatureproperties of four-dimensional para-K¨ahler metrics. Section 4 contains the tables of Theorems2.2 and the isomorphisms tables used in the proof of Theorem 1.1.The software Maple 18 r has been used to check all needed calculations.6 . Proof of The Theorem 1.1 In this section we begin with a reminder of the new approach introduced by Benayadi andBoucetta in [3], which characterizes the para-K¨ahler Lie algebras.Recall that, a para-K¨ahler Lie algebra ( g , h ., . i , K ) is carries a Levi-Civita product, the productcharacterized by Koszul’s formula:2 h u . v , w i = h [ u , v ] , w i + h [ w , u ] , v i + h [ w , v ] , u i . The subalgebras g = ker( K − Id g ) and g − = ker( K + Id g ) have the following properties, g and g − are isotropic with respect to h ., . i , Lagrangian with respect to ω and checking that g = g ⊕ g − , moreover the restriction of the Levi-Civita product on g and g − induces a left symmetricstructures. i.e. for any u , v , w ∈ g (resp. g − ), ass ( u , v , w ) = ass ( v , u , w )where ass ( u , v , w ) = ( u . v ) . w − u . ( v . w ). In particular, g and g − are left symmetric algebras.For any u ∈ g − , let u ∗ denote the element of ( g ) ∗ given by u ∗ ( v ) = h u , v i . The map u u ∗ realizes an isomorphism between g − and ( g ) ∗ . Thus, we can identify ( g , h ., . i , K ) relative to thephase space ( g ⊕ ( g ) ∗ , h ., . i , K ), where h ., . i and K are given by: h u + α, v + β i = α ( u ) − β ( v ) and K ( u + α ) = u − α. Both g and ( g ) ∗ carry a left symmetric algebra structure. For any u ∈ g and for any α ∈ ( g ) ∗ ,we denote L u : g → g and L α : ( g ) ∗ → ( g ) ∗ as the left multiplication by u and α , respectively,i.e., for any v ∈ g and any β ∈ ( g ) ∗ , L u v = u . v and L α β = α.β. The Levi-Civita product (and the Lie bracket) on g is determined entirely by their restrictions to( g ) ∗ and g : For any u ∈ g and for any α ∈ g ) ∗ , u .α = L tX α and α. u = − L t α X . Conversely, let U be a finite dimensional vector space and U ∗ is its dual space. We suppose thatboth U and U ∗ have the structure of a left symmetric algebra. We extend the products on U and U ∗ to U ⊕ U ∗ for any X , Y ∈ U and for any α , β ∈ U ∗ , by putting( X + α ) . ( Y + β ) = X . Y − L t α Y − L tX β + α.β. (1)We say that two left symmetric products on U and U ∗ is Lie-extendible if the commutator of theproduct on U ⊕ U ∗ given by (1) is a Lie bracket. In this case we have the following theorem: Theorem 2.1. [3] Let ( U , . ) and ( U ∗ , . ) be two Lie-extendible left symmetric products. Then, ( U ⊕ U ∗ , h ., . i , K ) , endowed with the Lie algebra bracket associated with the product given by (1) is a para-K¨ahler Lie algebra. Where ω , h ., . i and K are given by: ω ( u + α, v + β ) = β ( u ) − α ( v ) , h u + α, v + β i = α ( u ) + β ( v ) and K ( u + α ) = u − α. Moreover, all para-K¨ahler Lie algebras are obtained in this manner.
Let now U be a 2-dimensional vector space and U ∗ its dual space and let { e , e } , { e , e } be abasis of U and U ∗ . We base on the previous theorem and the classification of real left-symmetricalgebras in dimension 2 listed below (see Theorem 1.2. of [11]),7 ,α : e . e = e , e . e = α. e b : e . e = e , e . e = e . e b : e . e = e , e . e = e + e b + : e . e = e , e . e = − e , e . e = − e b ,α α , e . e = e , e . e = (1 − α ) e , e . e = e b − : e . e = − e , e . e = − e , e . e = − e c : Trivial left-symmetric algebra c : e . e = e c + : e . e = e , e . e = e , e . e = e , e . e = e c : e . e = e c − : e . e = e , e . e = e , e . e = e , e . e = e c : e . e = e , e . e = e , e e = e Remark 1. b stands for algebras with non-commutative associated Lie algebra and c stands foralgebras with commutative associated Lie algebra . Theorem 2.2.
Let ( g , h ., . i , K ) be a four-dimensional para-K¨ahler Lie algebra. Then there existsa basis { e , e , e , e } of g such that ω = e + e , h ., . i = ˙ e + ˙ e and K = E + E − E − E and the non vanishing Lie brackets as listed in the Table 4 and 5.Proof. We will give the proof in the case B since all cases should be handled in a similar way.In that case the left-symmetric product in U is given by e . e = e , e . e = e + e and let e . e = a e + b e e . e = a e + b e e . e = a e + b e e . e = a e + b e be an arbitrary product in U ∗ , let’s look for products in U ∗ which satisfy the Jacobi identity H [[ e i , e j ] , e k ] = ≤ i < j < k ≤
4, where H denotes the cyclic sum.The identity H [[ e , e ] , e ] = H [[ e , e ] , e ] = b + a + a = a = b + a = a = b + a = b = − a − a , the identity H [[ e , e ] , e ] = H [[ e , e ] , e ] = a a − a a − a d − a − a d = a ( a + a ) = a = ( a − a ) d + d ( a − d ) = a + (2 a − a + d ) a − a ( a − d ) = a = a + d + a = a = a = a =
0, and d =
0. Then the product in U ∗ is given by e . e = e (who is indeed a left-symmetric product) and the Lie bracket in U ⊕ U ∗ is given by[ e , e ] = − e , [ e , e ] = xe − e − e , [ e , e ] = − e . roof. of the Theorem 1.1 .The Theorem 2.2 confirms that for each Lie algebra g of the tables 4 and 5 there exist a base B = ( e , e , e , e ) such that the para-k¨ahler structure is given by ω = e + e and K = E + E − E − E ant the Lie brackets depend on some parameters. In Tables 6 and 7 we build a family of isomor-phisms (depending on the values of parameters) from g ( B i , j or C i , j ) onto a four-dimensional Liealgebra, (say A ) of the Table 1. Each isomorphism is given by the passage matrix P from B to B = ( f , f , f , f ). The image by P of the para-K¨ahler structure ( ω, K ) is given by the matricesof its component in the bases B and B ∗ by t P ◦ ω ◦ P = ω i and P − ◦ K ◦ P = K i . In this way we collect all the possible para-k¨ahler structures ( ω i , K i ) on A . Thereafter, we proceedto the classification in A (up to automorphism).We will give the proof in the case rr , since all cases should be handled in a similar way. Wewill show that the Lie algebra rr , admits two non-equivalent para-K¨ahler structures. Note thatin this case the non vanishing Lie bracket is[ f , f ] = f the symplectic form is ω = f + f and the automorphisms is T = a , a , a , a , a , a , a , a , . The groups of automorphisms of four dimensional Lie algebras were given in [10].From Table 6 and Table 7, rr . is obtained four times.1. The transformation: f = − e , f = e , f = e , f = e gives an isomorphism from C , to rr , and the para-K¨ahler structure obtained on rr , is ω = f − f and K = − E + E − E + E .
2. The transformation: f = − e , f = − ye + e , f = e , f = e gives an isomorphismfrom C , with x = rr , and the para-K¨ahler structure obtained on rr , is ω = − f + f and K = E − yE − E + E − E .
3. The transformation: f = − e , f = e , f = e , f = e gives an isomorphism from C , with x = , y = rr , and the para-K¨ahler structure obtained on rr , is ω = − f + f and K = E − E + E − E .
4. The transformation: f = e − e , f = e , f = e , f = e gives an isomorphism from C , with x = , y = rr , and the para-K¨ahler structure obtained on rr , is ω = − f + f + f and K = E − E + E − E . rr , support ω as a unique symplectic structure (up to automorphism), thereforethere are four families of automorphisms T i , i ∈ { , ..., } such that, T ∗ i ω i = ω for i ∈ { , ..., } , adirect calculation gives us T = a , a , a , − a , a , a , a , , T = a , − a , a , + a , a , a , a , T = a , a , a , − a , a , a , a , , T = a , − − a , a , + a , a , a , a , . Thus we obtain four para-K¨ahler structures on rr , given by ( ω , K i ), i ∈ , ..., K i = T − i ◦ K i ◦ T i a direct calculation gives us K = − E + E − E + E K = E + yE − E + E − E K = E − E + E − E K = − E + E + E − E Noticing that K is a sub-case of K and that ( ω , K ) is isomorphic to ( ω , K ). Indeed wehave L ∗ ω = ω and L − i ◦ K ◦ L i = K with L = − . We complete the proof by showing that ( ω , K ) is not isomorphic to ( ω , K ). Indeed, thesymplectomorphism group of ω is generated by L = a , a , a , + a , a , a , a , and L = a , a , a , − a , − a simple calculation gives us f (( L − ◦ K ◦ L − K )( f )) = and f (( L − ◦ K ◦ L − K )( f )) = − L − ◦ K ◦ L , K and L − ◦ K ◦ L , K .10 . Application: Curvature properties of four-dimensional para-K¨ahler Lie algebras Let now ( g , ω, K ) denote a four-dimensional para-K¨ahler Lie algebra. Let ∇ : g × g −→ g be theLevi-Civita product associated to a left-invariant pseudo-Riemannian metric h ( X , Y ) = ω ( KX , Y ).The connection ∇ is also called Hess connection. The curvature tensor is then described in termsof the map R : g × g −→ gl ( g )( X , Y ) R ( X , Y ) = ∇ [ X , Y ] − [ ∇ X , ∇ Y ] . (2)The Ricci tensor is the symmetric tensor ric given by ric ( X , Y ) = tr ( Z R ( X , Z ) Y ) and theRicci operator Ric : g −→ g is given by the relation h Ric ( X ) , Y i = ric ( X , Y ) . The scalar curvatureis defined in the standard way by s = tr ( Ric ).Recall that: ( g , h ) is called flat if R =
0, Ricci flat if
Ric = L X h + ric = λ h , (3)where X = x e + x e + x e + x e is a vector field and λ is a real constant, in that case if X = h is called Einstein metric and if λ is positive, zero, or negative then h is called ashrinking, steady, or expanding Ricci soliton, respectively. We give in the following theoremsome geometrical situations for the left invariant four-dimensional dimensional para-K¨ahler Liegroups. Theorem 3.1.
Let ( g , ω, K ) be a class of para-K¨ahler Lie algebras obtained in Theorem 1.1. Theassociated para-K¨ahler metric and some of his properties are given in the following tablesLie algebra Para-K¨ahler metric R = = λ X rh ∓ (˙ e − ˙ e ) Yes Yes 0 (0 , , x , x )˙ e − ˙ e + ˙ e Yes Yes 0 (0 , , x , x ) rr , ± (˙ e + ˙ e ) Yes Yes 0 (0 , , x , x )˙ e + x ˙ e + ˙ e , x , No No No rr , − ± (˙ e + ˙ e ± ˙ e ) No Yes 0 (0 , , , x )˙ e + ˙ e + ˙ e + ˙ e + x ˙ e Yes Yes 0 ( x , , , x ) r r ± ( − ˙ e − µ ˙ e + e + µ ˙ e + ˙ e ) No Yes , , , µ > − ˙ e − µ ˙ e + ˙ e No Yes , , , e + e − ˙ e Yes Yes − x ( x , x , x , x ) − ˙ e + x ˙ e + ˙ e + x ˙ e − x ˙ e + x ˙ e No No x (0 , , , − ˙ e ± ˙ e Yes Yes − x ( x , , x , r r ∓ ˙ e + ˙ e + x ˙ e , x , No No − x ( x , , , µ = − ˙ e − e − ˙ e Yes Yes − x ( x , , x , e + x ˙ e + ˙ e + y ˙ e , xy , , x , y No No No ˙ e + x ˙ e + ˙ e + x ˙ e , x , No No x (0 , , , e + x ˙ e + ˙ e , x , No No − x (0 , , x , e + ˙ e Yes Yes − x ( x , , x , e + x ˙ e + x ˙ e + ˙ e + x ˙ e , x , No No − x ( x , , x , e + x ˙ e − ˙ e + x ˙ e No No x (0 , , , − ˙ e − ˙ e Yes Yes − x ( x , , , h , xy , No No No r ′ h , y , No No − y (0 , , , (2 + x )˙ e + ( x + e + ˙ e ) − x ˙ e , x , No No No − e + ˙ e + ˙ e Yes Yes − x ( x , , x , e + ˙ e + ˙ e + ˙ e + x ˙ e , x , No No x ( − x , , , e + ˙ e + ˙ e + ˙ e Yes Yes − x ( x , , , r , ± (˙ e ± ˙ e ) No Yes , x , , r , − ˙ e − ˙ e + x ˙ e , x , No No , , , ± (˙ e − ˙ e ) Yes Yes , , , r , − ,β ∓ (˙ e ∓ ˙ e − ˙ e ) , β , No Yes , , , e ∓ ˙ e , β , Yes Yes , , , e − x ˙ e + x ˙ e , β , No No No ∓ (˙ e ∓ ˙ e − ˙ e ) , β = No Yes , , x , e ∓ ˙ e , β = Yes Yes , , x , x )˙ e − x ˙ e + x ˙ e , β = No Yes , , x , r , − , − − x ˙ e − ˙ e − ˙ e − ˙ e x , No Yes , , , − ˙ e − ˙ e − ˙ e Yes Yes , , , ∓ ( ∓ ˙ e − ˙ e − ˙ e ) No Yes , , , − ˙ e ∓ ˙ e Yes Yes , , , e ∓ ˙ e − ˙ e No Yes , , , e + x ˙ e − ˙ e , x , No No , , , e − ˙ e Yes Yes , , , r ,α, − α ∓ (˙ e − ˙ e ∓ ˙ e ) No Yes , , , ∓ ( x ˙ e − ˙ e + ˙ e ) , x , No No No ∓ (˙ e ∓ ˙ e ) Yes Yes , , , − ˙ e ∓ ˙ e − ˙ e No Yes , , , ∓ (˙ e ∓ ˙ e + ˙ e ) Yes Yes x , , , e + x ˙ e + ˙ e ,x , No Yes x (0 , , , e + ˙ e Yes Yes − x ( x , , , x ) ∓ (˙ e − ˙ e ) Yes Yes − x (0 , , , x ) d , − ˙ e + ˙ e − x ˙ e − ˙ e , x , No No No ∓ (˙ e − ˙ e + ˙ e ) No Yes , , , − ˙ e − ˙ e + ˙ e No Yes , , , − ˙ e + x ˙ e + ˙ e , x , No No x (0 , , , ± (˙ e + x ˙ e − ˙ e + ˙ e ) Yes Yes x , , , e ∓ ˙ e + ˙ e No Yes ∓ x , x , , e + ˙ e Yes Yes − x (0 , x , , x )˙ e + x ˙ e − ˙ e , x , No No x (0 , , , e − ˙ e Yes Yes − x (0 , , , x )2 x ˙ e − ˙ e ∓ x ˙ e ∓ ˙ e No No No d , − e ∓ ˙ e + x ˙ e ± ˙ e No No No ˙ e + ˙ e + x ˙ e , No No , , , ∓ (˙ e + ˙ e ) Yes Yes , , , ∓ (˙ e − ˙ e ) Yes Yes , , , − x ˙ e − ˙ e ∓ ˙ e , x , No No , , , ∓ x ˙ e + ˙ e ± ˙ e + x ˙ e , x , No No , , , x ˙ e − x ˙ e − ˙ e − ˙ e + x ˙ e , x , No No , , , d , ˙ e + x ˙ e − ˙ e , x , No No x (0 , , , e ∓ ˙ e Yes Yes − x (0 , , , x ) d ,λ ˙ e ∓ ˙ e + ˙ e No Yes , , , ˙ e − ˙ e + ˙ e No Yes , , , ∓ ˙ e + x ˙ e + ˙ e , x , No No x (0 , , , ∓ ˙ e + x ˙ e − ˙ e , x , No Yes , , , ∓ (˙ e ∓ ˙ e ) Yes Yes − x (0 , , , x ) h ± (˙ e − ˙ e ) No Yes , , , Table 3: Curvature properties of four-dimensional para-K¨ahler Lie algebras h = y (˙ e − ˙ e − ˙ e + ˙ e + ˙ e − ˙ e ) − (2 + x )˙ e + ( x + e + ˙ e ) − x ˙ e h = y (˙ e − ˙ e − ˙ e + ˙ e + ˙ e − ˙ e ) − e + ˙ e + ˙ e Proof.
We report below the details for the case of d , the other cases are treated in the same way.Let { e , e , e , e } denotes the basis used in theorem 1.1 for d , . The non isomorphic para-K¨ahlerstructures in d , are ( ω, K ) and ( ω, K ) with ω = e − e , K = E − E − E + xE + E and K = E − E + E − E .The corresponding compatible metric to ( ω, K i ) is uniquely determined by h i ( X , Y ) = ( K i X , Y ).Hence, para-K¨ahler metrics in d , are of the form h = x
10 0 1 0 x ∈ R and h = −
10 0 − . For h , x ,
0, using the Koszul formula, the Levi-Civita connection is described ∇ e = − x −
10 0 0 00 1 0 00 − x , ∇ e = x
00 0 0 0 − x . ∇ e = − x x x
00 0 − x − x , ∇ e = − − x − . Then we calculate the curvature matrices R ( e i , e j ) (for 1 ≤ i < j ≤
4) and we find R ( e , e ) = − x x x
00 0 − x − x , R ( e , e ) = − x − x x − x R ( e , e ) = − x
00 0 0 0 x , R ( e , e ) = − x
00 0 0 0 x ( e , e ) = x − x − x
00 0 x x and R ( e , e ) = . The Ricci tensor ric and the Ricci operator
Ric are given by ric = x x x x x and Ric = x x x
00 0 0 x The Lie derivative L X h of the metric h with respect to an arbitrary vector field X = x e + x e + x e + x e ∈ g is given by L X h = − x x x x − x − x x − x x x x x − x x xx − x x − x xx − x x . Then, solving equation L X h + ric = λ h , for x , λ = x and X = . Notice that, in this case, the para-K¨ahler metric is a Einstein metric not Ricci flat.For h with x = h ∇ e e = e , ∇ e e = − e , ∇ e e = − e , ∇ e e = e , ∇ e e = e , ∇ e e = − e This para-K¨ahler structure is flat ( R ( e i , e j ) = ≤ i < j ≤ L X h of themetric h , is given by L X h = − x x − x − x − x x − x − x x . Then, solving equation L X h = λ h , for x = h ) we obtain λ = − x and X = x e . . Tables Lie algebra No zero brackets B ,α α < {− , − , } [ e , e ] = − e , [ e , e ] = xe − e , [ e , e ] = − α e B ,α α < {− , − , , } [ e , e ] = − e , [ e , e ] = − x α e , [ e , e ] = − e , [ e , e ] = xe − α e , [ e , e ] = x α e B , − [ e , e ] = − e , [ e , e ] = xe , [ e , e ] = ye − e , [ e , e ] = xe + e , [ e , e ] = − xe B , − [ e , e ] = − e , [ e , e ] = xe , [ e , e ] = ye − xe − e , [ e , e ] = xe + e , [ e , e ] = − xe B , − [ e , e ] = − e , [ e , e ] = − xe , [ e , e ] = ye + xe − e , [ e , e ] = e , [ e , e ] = xe B , − [ e , e ] = − e , [ e , e ] = xe , [ e , e ] = − e , [ e , e ] = xe + e , [ e , e ] = − xe B , [ e , e ] = − e , [ e , e ] = xe , [ e , e ] = − e , [ e , e ] = − xe B , x , e , e ] = − e , [ e , e ] = − y e , [ e , e ] = − xe , [ e , e ] = y x e + y e − e [ e , e ] = ye + xe − e , [ e , e ] = xe − y e B , [ e , e ] = − e , [ e , e ] = xe − e , [ e , e ] = − e B [ e , e ] = − e , [ e , e ] = xe − e − e , [ e , e ] = − e B ,α α , e , e ] = α e , [ e , e ] = [ e , e ] = − e , [ e , e ] = xe + − αα e B ,α α , e , e ] = α e , [ e , e ] = [ e , e ] = x α e − e , [ e , e ] = xe , [ e , e ] = − αα e , [ e , e ] = x ( α − e B , [ e , e ] = e , [ e , e ] = xe − e , [ e , e ] = ye − xe + e , [ e , e ] = − e , [ e , e ] = − xe B , y , e , e ] = e , [ e , e ] = − xe + ye − e , [ e , e ] = ye , [ e , e ] = − x y e + xe + e [ e , e ] = xe + ye − e , [ e , e ] = − ye − xe B , [ e , e ] = e , [ e , e ] = [ e , e ] = xe − e , [ e , e ] = ye − xe + e , [ e , e ] = − xe B , [ e , e ] = e , [ e , e ] = xe + ye − e , [ e , e ] = ye , [ e , e ] = ze , [ e , e ] = ye − e B [ e , e ] = e , [ e , e ] = xe − e , [ e , e ] = ye − e , [ e , e ] = − e B + , [ e , e ] = e , [ e , e ] = xe − e , [ e , e ] = e , [ e , e ] = − xe + e , [ e , e ] = − xe B + , [ e , e ] = e , [ e , e ] = − xe − e , [ e , e ] = xe + e , [ e , e ] = − xe + e , [ e , e ] = xe B + , [ e , e ] = e , [ e , e ] = − xe , [ e , e ] = − xe − xe − e , [ e , e ] = xe + xe + e [ e , e ] = xe + xe + e , [ e , e ] = xe − xe B + , [ e , e ] = e , [ e , e ] = − xe , [ e , e ] = xe − xe − e [ e , e ] = − xe + xe + e , [ e , e ] = xe − xe + e , [ e , e ] = − xe − xe B − , [ e , e ] = e , [ e , e ] = xe + e , [ e , e ] = e , [ e , e ] = − xe + e , [ e , e ] = − xe B − , [ e , e ] = e , [ e , e ] = xe + e , [ e , e ] = xe + e , [ e , e ] = xe + e , [ e , e ] = − xe Table 4: Four dimensional Para-K¨ahler Lie algebras coming from b C , [ e , e ] = e , [ e , e ] = α e , [ e , e ] = − e C , [ e , e ] = e + e , [ e , e ] = e , [ e , e ] = − e C , [ e , e ] = [ e , e ] = e , [ e , e ] = (1 − α ) e , [ e , e ] = α e C , [ e , e ] = [ e , e ] = [ e , e ] = e , [ e , e ] = e C + , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e C − , [ e , e ] = [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e C , [ e , e ] = e C , [ e , e ] = e C , [ e , e ] = [ e , e ] = e , [ e , e ] = e C , [ e , e ] = [ e , e ] = e , [ e , e ] = [ e , e ] = e C , [ e , e ] = [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e C , [ e , e ] = xe , [ e , e ] = ye − e C , [ e , e ] = xe , [ e , e ] = ye , [ e , e ] = − e C , [ e , e ] = xe , [ e , e ] = ye , [ e , e ] = xe − e , [ e , e ] = − xe C , [ e , e ] = xe , [ e , e ] = ye + ze − e , [ e , e ] = ze C , [ e , e ] = [ e , e ] = xe , [ e , e ] = ye + ze − e , [ e , e ] = ( z − x ) e C , [ e , e ] = [ e , e ] = xe − e , [ e , e ] = ye + xe − e C , [ e , e ] = [ e , e ] = xe − e , [ e , e ] = − e , [ e , e ] = xe C + , [ e , e ] = [ e , e ] = xe + ye − e , [ e , e ] = [ e , e ] = ye + xe − e C + , [ e , e ] = [ e , e ] = xe − e , [ e , e ] = [ e , e ] = − e , [ e , e ] = xe C − , [ e , e ] = [ e , e ] = xe + ye − e , [ e , e ] = ye − xe + e , [ e , e ] = − ye + xe − e C − , [ e , e ] = [ e , e ] = xe − e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = xe Table 5: Four dimensional Para-K¨ahler Lie algebras coming from c Source Isomorphism Target B ,α | α | < , α , f = e , f = − x e + e , f = e , f = e r , − , − α B ,α | α | > , α , − f = e , f = e , f = − x e + e , f = − | α | e r , − | α | , | α | B ,α | α | < , α , f = e , f = e , f = − x α e + e , f = − x + αα e + e r , − , − α B ,α | α | > , α , − f = − x | α | e + e , f = e , f = e , f = − x + | α | e + e r , − | α | , | α | , − x , y , f = − ye + e + x e , f = − ye + e , f = ye , f = e + e d , B , − x , , y = f = e + x e , f = e , f = e , f = e + x e r , − , B , − x = f = e , f = − y e + e , f = e , f = e r , − , B , − x , f = e , f = x e + e + x e , f = − xe , f = − x e + ( y + x ) e d , B , − x = f = e , f = − y e + e , f = e , f = e r , − , B , − f = − ye − xe + e , f = e , f = e , f = − e r , − , − B , f = e , f = e , f = − x e + e , f = e rr , − B , f = e , f = e , f = e , f = xe + e rr , − B , x , f = e , f = xe − y e , f = y e + xe − e , f = e r , − , − B , f = e , f = − x e + e , f = e , f = e r , − , − B f = e , f = − e , f = − x e + e , f = e r , − B ,α α > f = e , f = − xe + α − α e , f = − α − α e , f = − e d , α B ,α α < f = − xe + α − α e , f = e , f = α − α e , f = − e d , α − α B , x , f = − xe , f = e , f = xe , f = − e h B , x = f = e , f = e , f = e , f = − e d , B ,α α > f = e + x α e − e , f = e , f = x α e − e , f = − e + α − α e d , α B ,α α < f = − x α e + e + e , f = e , f = − x α e + e , f = α e − e d , α − α B , f = e , f = e , f = − xe + e , f = − e d , B , f = e , f = ye − xe + e , f = − e , f = − e d , B , y , f = e , f = − e − xy e , f = xe − ye + e , f = − y e d , B , x = , y = f = e , f = ze + e , f = − e , f = − e d , B , x , , y = f = zx e − e + x e , f = e , f = − zx e − x e , f = e − x e r r B , x = , y , , z , f = − √ yz y e − e − √ yz e , f = − √ yze − ye + e , f = √ yz y e − e + √ yz e , f = √ yze − ye + e r r B , x = , y , , z = f = e , f = e , f = ye − e , f = − e d , B , xy , , x + yz = f = − ye + e , f = − x y e − x e + e , f = x e + xy e − x e , f = − e d , B , xy , , x + yz > f = z √ x + yz e − x + √ x + yz √ x + yz e + √ x + yz e , f = ( − x + p x + yz ) e − ye + e , f = − z √ x + yz e + x − √ x + yz √ x + yz e − √ x + yz e , f = − x + √ x + yz e − ye + e r r , xy , , x + yz < f = √ − x − yz e − e , f = − x √ − x − yz − z √ − x − yz e − y √ − x − yz + x √ − x − yz e + √ − x − yz e + e , f = √ − x − yz e , f = − x e − ye + e r ′ B x = f = − e , f = ye + e − e , f = e , f = − e d , B x , f = x − yx e − x e , f = − xe + e , f = yx e − e + x e , f = e r r B + , f = − xe − e + e , f = e , f = e , f = e + e d , B + , x , f = e − x e , f = e , f = x e + e , f = x e d , B + , x = f = e + e , f = e , f = e , f = − e + e d , B + , x , f = − e − e , f = e , f = xe + xe + e , f = − x e d , B + , x = f = e , f = e , f = e , f = e d , B + , x , f = e + e , f = e , f = − xe + xe + e , f = − e + e d , B + , x = f = − e + e , f = e , f = e , f = e + e d , B − , f = − xe + e , f = e , f = − e , f = e d , B − , x , f = e + x e , f = e , f = xe + e , f = − x e d , B − , x = f = e + e , f = e , f = − e , f = e + e d , Table 6: Isomorphisms from the Lie algebras obtained in Table 4 onto the Lie algebras in Table 1
Source Isomorphism Target C , − ≤ α < f = e , f = e , f = e , f = − e r , − ,α C , α < − f = e , f = e , f = e , f = − α e r , α , − α C , α > f = e , f = e , f = e , f = − α e r , − α , α C , α = f = e , f = e , f = e , f = e r , − , − C , f = e , f = − e , f = e , f = e r , − C , < α ≤ f = e − e , f = e , f = e , f = α − α e − e d , α C , α < or α > f = e + α e , f = e , f = e , f = e − e d , α − α C , f = e , f = e − e , f = − e , f = − e d , C + , f = e , f = e , f = e , f = e d , C − , f = e , f = − e , f = e , f = e d , C , f = − e , f = e , f = e , f = e rr , C , f = e , f = e , f = e , f = e rh , f = e , f = e , f = e , f = − e d , C , f = e − e , f = e − e , f = − e − e , f = e + e r r C , f = − e , f = e , f = e , f = − e r ′ C , x = f = − e , f = − ye + e , f = e , f = e rr , C , x , f = − x e , f = e , f = − e , f = − xye + xe r r C , x , f = − x e , f = e , f = yx e − e , f = e r r C , x = , y , f = e , f = − ye , f = e , f = − e r , C , x = , y = f = − e , f = e , f = e , f = e rr , C , x , f = yx e − e , f = − xe + e , f = e − x e , f = e r r C , x = , y , f = e , f = − ye , f = e , f = − e r , C , x = , y = f = e − e , f = e , f = e , f = e rr , C , z = , x , f = e , f = − x e , f = − yx e + e , f = − x e r , C , z = , x = f = − e , f = e − e , f = ye − e , f = e rh C , z , , x = f = z e , f = e , f = − ye − ze + e , f = e rr , − C , z , , − ≤ − xz < f = e , f = yzx − z e − ze + e , f = e , f = z e r , − , − xz C , z , , x , , − xz > f = e , f = yzx − z e − ze + e , f = e , f = − x e r , zx , − zx C , z , , x , , − xz < − f = e , f = e , f = yzx − z e − ze + e , f = − x e r , − zx , zx C , z , , − xz = f = − y x e + e + x e , f = e , f = e , f = x e r , − , − C , z = , x = f = − e , f = e − e , f = ye − e , f = e rh C , z = , x , f = e , f = − yx e + e + x e , f = − xe f = − x e d , C , z , , x = f = z e , f = e , f = − ye − ze + e f = e rr , − C , z , , x , , zx = f = − z e , f = − yz e + e , f = e f = − z e h C , , , x , , zx < f = y ( x − z ) z e + e , f = ( x − z ) e + e , f = − x ( x − z ) e , f = − y ( x − z ) x e − x e d , x − zx C , z , , x , , zx > f = ( x − z ) e + e , f = y ( x − z ) z e + e , f = x ( x − z ) e , f = − y ( x − z ) x e − x e d , zx C , f = − ye − xe + e , f = e , f = − xe + e f = − xe − e + e d , C , f = e , f = e , f = − xe + e f = − e d , C + , f = e − e , f = ( − x + y ) e + ( x − y ) e − e + e , f = − ( + x + y ) e − ( + x + y ) e + e + e , f = − ( x + y ) e − ( x + y ) e + e + e r r + , f = e − e , f = − xe − e + e , f = − e − e , f = − xe + e + e r r C − , f = ye − ( x + e + e , f = − ( x + e − ye + e , f = ye − xe + e f = − xe − ye + e r ′ C − , f = − e − e , f = e − xe + e , f = − e f = − xe + e r ′ Table 7: Isomorphisms from the Lie algebras obtained in Table 5 onto the Lie algebras in Table 1
Acknowledgments:
The authors would like to thank sincerely Professor Mohamed Boucetta for his many suggestionswhich were of great help to improve our work.
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