On q-deformed Levi-Civita connections
aa r X i v : . [ m a t h . QA ] M a y ON q -DEFORMED LEVI-CIVITA CONNECTIONS JOAKIM ARNLIND, KWALOMBOTA ILWALE AND GIOVANNI LANDI
Abstract.
We explore the possibility of introducing q -deformed connections on the quan-tum 2-sphere and 3-sphere, satisfying a twisted Leibniz’ rule in analogy with q -deformedderivations. We show that such connections always exist on projective modules. Further-more, a condition for metric compatibility is introduced, and an explicit formula is given,parametrizing all metric connections on a free module. For the module of 1-forms on thequantum 3-sphere, a q -deformed torsion freeness condition is introduced and we derive ex-plicit expressions for the Christoffel symbols of a Levi-Civita connection for a general classof metrics satisfying a certain reality condition. Finally, we construct metric connections ona class of projective modules over the quantum 2-sphere. Contents
1. Introduction 12. The quantum 3-sphere 32.1. q -deformed derivations 42.2. A left covariant calculus on S q q -affine connections 53.1. q -affine connections on projective modules 94. A q -affine Levi-Civita connection on Ω ( S q ) 105. The quantum 2-sphere 115.1. A left covariant calculus on S q S q Introduction
In recent years, a lot of progress has been made in understanding the Riemannian aspectsof noncommutative geometry. These are not only mathematically interesting, but also im-portant in physics where noncommutative geometry is expected to play a key role, notablyin a theory of quantum gravity. In Riemannian geometry the Levi-Civita connection andits curvature have a central role, and it turns out that there are several different ways ofapproaching these objects in the noncommutative setting (see e.g. [CFF93, DVMMM96,Maj05, AC10, BM11, Ros13, AW17, BGM19, BGL20]).From an algebraic perspective, the set of vector fields and the set of differential forms are(finitely generated projective) modules over the algebra of functions, a viewpoint which is
Date : 5 May 2020. lso adopted in noncommutative geometry. However, considering vector fields as derivationsdoes not immediately carry over to noncommutative geometry, since the set of derivations ofa (noncommutative) algebra is in general not a module over the algebra but only a moduleover the center of the algebra. Therefore, one is lead naturally to focus on differential formsand define a connection on a general module as taking values in the tensor product of themodule with the module of differential forms. More precisely, let M be a (left) A -module andlet Ω ( A ) denote a module of differential forms together with a differential d : A → Ω ( A ).A connection on M is a linear map ∇ : M → Ω ( A ) ⊗ M satisfying a version of Leibniz rule ∇ ( f m ) = f ∇ m + df ⊗ m (1.1)for f ∈ A and m ∈ M . In differential geometry, for a vector field X one obtains a covariantderivative ∇ X : M → M , by pairing differential forms with X (as differential forms are dualto vector fields). In a noncommutative version of the above, there is in general no canonicalway of obtaining a “covariant derivative” ∇ X : M → M . In a derivation based approach tononcommutative geometry (see e.g. [DV88, DVMMM96]), one puts emphasis on the choiceof a Lie algebra g of derivations of the algebra A . Given a (left) A -module M one definesa connection as a map ∇ : g × M → M , usually writing ∇ ( ∂, m ) = ∇ ∂ m for ∂ ∈ g and m ∈ M , satisfying ∇ ∂ ( f m ) = f ∇ ∂ m + ∂ ( f ) m (1.2)for f ∈ A and m ∈ M , in parallel with (1.1).For quantum groups, it turns out that natural analogues of vector fields are not quitederivations, but rather maps satisfying a twisted Leibniz rule. For instance, as we shall see,for the quantum 3-sphere S q one defines maps X a : S q → S q satisfying X a ( f g ) = f X a ( g ) + X a ( f ) σ a ( g )(1.3)for f, g ∈ S q , where σ a : S q → S q , for a = 1 , ,
3, are algebra morphisms. Thus, in this note,we explore the possibility of introducing a corresponding q -affine connection on a (left) S q -module M . That is, motivated by (1.3) we introduce a covariant derivative ∇ X a : M → M such that ∇ X a ( f m ) = f ∇ X a m + X + ( f )ˆ σ a ( m )(1.4)for f ∈ S q and m ∈ M , where ˆ σ a denotes an extension of σ a to the module M (cf. Section 3).In the following, we make these ideas precise and prove that there exist q -affine connectionson projective modules. Furthermore, we introduce a condition for metric compatibility, andin the particular case of a left covariant calculus over S q , we investigate a derivation baseddefinition of torsion. Then we explicitly construct a Levi-Civita connection, that is a torsionfree and metric compatible connection. Moreover, we construct metric connections on a classof projective modules over the quantum 2-sphere. We note that the Riemannian geometry ofquantum spheres can be studied [BM11] from the point of view of a bimodule connection ondifferential forms satisfying (1.1) as well as a right Leibniz rule twisted by a braiding map. . The quantum 3-sphere
In this section we recall a few basic properties of the quantum 3-sphere [Wor87]. The algebra S q is a unital ∗ -algebra generated by a, a ∗ , c, c ∗ fulfilling ac = qca c ∗ a ∗ = qa ∗ c ∗ ac ∗ = qc ∗ aca ∗ = qa ∗ c cc ∗ = c ∗ c a ∗ a + c ∗ c = aa ∗ + q cc ∗ = for a real parameter q . The identification of S q with the quantum group SEE q (2) is via theHopf algebra structure given by∆( a ) = a ⊗ a − qc ∗ ⊗ c ∆( c ) = c ⊗ a + a ∗ ⊗ c ∆( a ∗ ) = − qc ⊗ c ∗ + a ∗ ⊗ a ∗ ∆( c ∗ ) = a ⊗ c ∗ + c ∗ ⊗ a ∗ with antipode and counit S ( a ) = a ∗ S ( c ) = − qc ǫ ( a ) = 1 ǫ ( c ) = 0 S ( a ∗ ) = a S ( c ∗ ) = − q − c ∗ ǫ ( a ∗ ) = 1 ǫ ( c ∗ ) = 0 . Furthermore, the quantum enveloping algebra U q (su(2)) is the ∗ -algebra with generators E, F, K, K − satisfying K ± E = q ± EK ± K ± F = q ∓ F K ± [ E, F ] = K − K − q − q − . The corresponding Hopf algebra structure is given by the coproduct,∆( E ) = E ⊗ K + K − ⊗ E ∆( F ) = F ⊗ K + K − ⊗ F ∆( K ± ) = K ± ⊗ K ± together with antipode and counit S ( K ) = K − S ( E ) = − qE S ( F ) = − q − Fǫ ( K ) = 1 ǫ ( E ) = 0 ǫ ( F ) = 0 . We recall that there is a unique bilinear pairing between U q (su(2)) and S q given by (cid:10) K ± , a (cid:11) = q ∓ / (cid:10) K ± , a ∗ (cid:11) = q ∓ / h E, c i = 1 h F, c ∗ i = − q − , with the remaining pairings being zero. The pairing induces a U q (su(2))-bimodule structureon S q given by h ⊲ f = f (1) (cid:10) h, f (2) (cid:11) and f ⊳ h = (cid:10) h, f (1) (cid:11) f (2) (2.1)for h ∈ U q (su(2)) and f ∈ S q with Sweedler’s notation ∆( f ) = f (1) ⊗ f (2) (and implicitsum). The ∗ -structure on U q (su(2)), unconventionally denoted here by † (for reasons thatwill become clear momentarily), is given by ( K ± ) † = K ± and E † = F . The action of U q (su(2)) is compatible with the ∗ -algebra structures in the following sense h ⊲ f ∗ = (cid:0) S ( h ) † ⊲ f (cid:1) ∗ f ∗ ⊳ h = (cid:0) f ⊳ S ( h ) † (cid:1) ∗ . et us for convenience list the left and right actions of the generators: K ± ⊲ a n = q ∓ n a n K ± ⊲ c n = q ∓ n c n K ± ⊲ a ∗ n = q ± n ( a ∗ ) n K ± ⊲ c ∗ n = q ± n ( c ∗ ) n E ⊲ a n = − q (3 − n ) / [ n ] a n − c ∗ E ⊲ c n = q (1 − n ) / [ n ] c n − a ∗ E ⊲ ( a ∗ ) n = 0 E ⊲ ( c ∗ ) n = 0 .F ⊲ a n = 0 F ⊲ c n = 0 F ⊲ ( a ∗ ) n = q (1 − n ) / [ n ] c ( a ∗ ) n − F ⊲ ( c ∗ ) n = − q − (1+ n ) / [ n ] a ( c ∗ ) n − and a n ⊳ K ± = q ∓ n a n ( a ∗ ) n ⊳ K ± = q ± n ( a ∗ ) n c n ⊳ K ± = q ± n c n ( c ∗ ) n ⊳ K ± = q ∓ n ( c ∗ ) n a n ⊳ F = q n − [ n ] ca n − ( a ∗ ) n ⊳ F = 0 c n ⊳ F = 0 ( c ∗ ) n ⊳ F = − q n − [ n ] a ∗ ( c ∗ ) n − a n ⊳ E = 0 ( a ∗ ) n ⊳ E = − q n − [ n ] c ∗ ( a ∗ ) n − c n ⊳ E = q n − [ n ] c n − a ( c ∗ ) n ⊳ E = 0where [ n ] = ( q n − q − n ) / ( q − q − ).2.1. q -deformed derivations. The algebra S q comes with a standard set of three q -deformedderivations (which can be used to generate a left covariant differential calculus, see Sec-tion 2.2). Namely, defining X a for a = 1 , , X ≡ X + = √ qEK X ≡ X − = 1 √ q F K X ≡ X z = 1 − K − q − it follows that for f, g ∈ S q (where X a ( f ) denotes either X a ⊲ f or f ⊳ X a ) X + ( f g ) = f X + ( g ) + X + ( f ) σ + ( g ) X − ( f g ) = f X − ( g ) + X + ( f ) σ − ( g ) X z ( f g ) = f X z ( g ) + X + ( f ) σ z ( g ) , with σ + = σ − = K and σ z = K . (2.2)Furthermore, these maps satisfy the following q -deformed commutation relations X − X + − q X + X − = X z (2.3) q X z X − − q − X − X z = (1 + q ) X − (2.4) q X + X z − q − X z X + = (1 + q ) X + . (2.5)For an arbitrary map X : S q → S q one defines X ∗ : S q → S q as X ∗ ( f ) = (cid:0) X ( f ∗ ) (cid:1) ∗ nd it follows that X ∗ + = − K − X − X ∗− = − K − X + X ∗ z = − K − X z , (2.6)satisfying X ∗ + ( f g ) = σ ∗ + ( f ) X ∗ + ( g ) + X ∗ + ( f ) g (2.7) X ∗− ( f g ) = σ ∗− ( f ) X ∗− ( g ) + X ∗− ( f ) g (2.8) X ∗ z ( f g ) = σ ∗ z ( f ) X ∗ z ( g ) + X ∗ z ( f ) g (2.9)with σ ∗ + = σ − = K − , σ ∗− = σ − − = K − , σ ∗ z = σ − z = K − . (2.10)We stress that X ∗ a is different from X † a , as defined above on U q (su(2)).2.2. A left covariant calculus on S q . It is well known that there is a left covariant (firstorder) differential calculus on S q , denoted by Ω ( S q ), generated as a left S q -module by ω = ω + = a dc − qc da ω = ω − = c ∗ da ∗ − qa ∗ dc ∗ ω = ω z = a ∗ da + c ∗ dc, together with the differential d : S q → Ω ( S q ) df = ( X + ⊲ f ) ω + + ( X − ⊲ f ) ω − + ( X z ⊲ f ) ω z (2.11)for f ∈ S q [Wor87]. In fact, Ω ( S q ) is a free left module with a basis given by { ω + , ω − , ω z } .Moreover, Ω ( S q ) is a bimodule with respect to the relations ω z a = q − aω z ω z a ∗ = q a ∗ ω z ω z c = q − cω z ω z c ∗ = q c ∗ ω z ω ± a = q − aω ± ω ± a ∗ = qa ∗ ω ± ω ± c = q − cω ± ω ± c ∗ = qc ∗ ω ± , and, furthermore, Ω ( S q ) is a ∗ -bimodule with ω † + = − ω − ω † z = − ω z satisfying ( f ωg ) † = g ∗ ω † f ∗ for f, g ∈ S q and ω ∈ Ω ( S q ).3. q -affine connections In differential geometry, a connection extends the action of derivatives to vector fields, andfor S q a natural set of ( q -deformed) derivations is given by { X a } a =1 = { X + , X − , X z } . Inthis section, we will introduce a framework extending the action of X a to a connection on S q -modules. Let us first define the set of derivations we shall be interested in. Definition 3.1.
The quantum tangent space of S q is defined as T S q = C (cid:10) X + , X ∗ + , X − , X ∗− , X z , X ∗ z (cid:11) , that is the complex vector space generated by X a and X ∗ a for a = 1 , , T S q to be the analogue of a (complexified) tangent space of S q , we would like tointroduce a covariant derivative ∇ X on a (left) S q -module M , for X ∈ T S q . Since the basiselements of T S q act as q -deformed derivations, the connection should obey an analogous q -deformed Leibniz rule. The motivating example is when M = S q and the action of T S q issimply ∇ X f = X ( f ) for X ∈ T S q and f ∈ S q . In fact, let us be slightly more general and onsider the action on a free module of rank n . Thus, we let M be a free left S q -module withbasis { e i } ni =1 , and write an arbitrary element m ∈ M as m = m i e i for m i ∈ S q , implicitlyassuming a summation over i from 1 to n . Moreover, we assume there exist C -linear mapsˆ σ a , ˆ σ ∗ a : M → M such thatˆ σ a ( f m ) = σ a ( f )ˆ σ a ( m ) and ˆ σ ∗ a ( f m ) = σ ∗ a ( f )ˆ σ ∗ a ( m )for f ∈ S q , m ∈ M and a = 1 , , σ a ( m i e i ) = σ a ( m i ) e i andsimilarly for ˆ σ ∗ a ). Let us define ∇ : T S q × M → M by setting ∇ X a ( m ) = X a ( m i )ˆ σ a ( e i ) and ∇ X ∗ a ( m ) = X ∗ a ( m i ) e i (3.1)for m = m i e i ∈ M (and extending it linearly to all of T S q ). Now, it is easy to check that ∇ X a ( f m ) = f ∇ X a m + X a ( f )ˆ σ a ( m )(3.2) ∇ X ∗ a ( f m ) = σ ∗ a ( f ) ∇ X ∗ a m + X ∗ a ( f ) m (3.3)for f ∈ S q and m ∈ M . Let us generalize these concepts to arbitrary S q -modules supportingan action of σ a . Definition 3.2.
Let M be a left S q -module and let ˆ σ a , ˆ σ ∗ a : M → M be maps such thatˆ σ a ( λ m + λ m ) = λ ˆ σ a ( m ) + λ ˆ σ a ( m ) ˆ σ a ( f m ) = σ a ( f )ˆ σ a ( m )ˆ σ ∗ a ( λ m + λ m ) = λ ˆ σ ∗ a ( m ) + λ ˆ σ ∗ a ( m ) ˆ σ ∗ a ( f m ) = σ ∗ a ( f )ˆ σ ∗ a ( m )for λ , λ ∈ C , f ∈ S q , m , m , m ∈ M and a = 1 , ,
3. Then ( M, ˆ σ a , ˆ σ ∗ a ) is called a σ -module .Moreover, given the σ -modules ( M, ˆ σ a , ˆ σ ∗ a ) and ( ˜ M , ˜ σ a , ˜ σ ∗ a ), a left module homomorphism φ : M → ˜ M is called a σ -module homomorphism if φ (cid:0) ˆ σ a ( m ) (cid:1) = ˜ σ a (cid:0) φ ( m ) (cid:1) z and φ (cid:0) ˆ σ ∗ a ( m ) (cid:1) = ˜ σ ∗ a (cid:0) φ ( m ) (cid:1) for m ∈ M and a = 1 , , σ a , ˆ σ ∗ a are clear from the context, we shall simplywrite M for the σ -module ( M, ˆ σ, ˆ σ ∗ ). Next, motivated by (3.2) and (3.3), we introduceconnections on σ -modules. Definition 3.3.
Let M be a left σ -module. A left q -affine connection on M is a map ∇ : T S q × M → M such that(1) ∇ X ( λ m + λ m ) = λ ∇ X m + λ ∇ X m ,(2) ∇ λ X + λ Y m = λ ∇ X m + λ ∇ Y m ,(3) ∇ X a ( f m ) = f ∇ X a m + X a ( f )ˆ σ a ( m ),(4) ∇ X ∗ a ( f m ) = σ ∗ a ( f ) ∇ X ∗ a m + X ∗ a ( f ) m ,for a = 1 , , m, m , m ∈ M , f ∈ S q , X ∈ T S q and λ , λ ∈ C . Remark . Note that given ∇ X a for a = 1 , ,
3, one can set ∇ X ∗ + = − ˆ σ ∗− ◦ ∇ X − ∇ X ∗− = − ˆ σ ∗ + ◦ ∇ X + ∇ X ∗ z = − ˆ σ ∗ z ◦ ∇ X z satisfying (4) in Definition 3.3, due to (2.6) and (2.10). ext, assume that the module M comes with a hermitian form h : M × M → S q satisfying h ( f m , m ) = f h ( m , m ) h ( m , m ) ∗ = h ( m , m ) h ( m + m , m ) = h ( m , m ) + h ( m , m )for f ∈ S q and m , m , m ∈ M . On a free module with basis { e i } ni =1 , a hermitian form isgiven by h ij = h ∗ ji ∈ S q by setting h ( m , m ) = m i h ij ( m j ) ∗ (3.4)for m = m i e i ∈ ( S q ) n and m = m i e i ∈ ( S q ) n . In the case of the q -affine connection ∇ in(3.1), one finds that X + (cid:0) h ( m , m ) (cid:1) = X + (cid:0) m i h ij ( m j ) ∗ (cid:1) = m i X + (cid:0) h ij ( m j ) ∗ (cid:1) + X + ( m i ) σ + (cid:0) h ij ( m j ) ∗ (cid:1) = m i h ij X + (cid:0) ( m j ) ∗ (cid:1) + m i X + ( h ij ) σ + (cid:0) ( m j ) ∗ (cid:1) + X + ( m i ) σ + (cid:0) h ij ( m j ) ∗ (cid:1) , and assuming that X + ( h ij ) = 0 one obtains X + (cid:0) h ( m , m ) (cid:1) = m i h ij (cid:0) X ∗ + ( m j ) (cid:1) ∗ + σ + (cid:0) ( σ − ◦ X + )( m i ) h ij ( m j ) ∗ (cid:1) = h (cid:0) m , ∇ X ∗ + ( m ) (cid:1) − σ + (cid:0) h ( ∇ X ∗− m , m ) (cid:1) , by using that X ∗− = − σ − ◦ X + . Corresponding formulas are easily worked out for ∇ X − , ∇ X z ,and we shall take this as a motivation for the following definition. Definition 3.5. A q -affine connection ∇ on a left σ -module M is compatible with thehermitian form h : M × M → S q if X + (cid:0) h ( m , m ) (cid:1) = − σ + (cid:0) h ( ∇ X ∗− m , m ) (cid:1) + h (cid:0) m , ∇ X ∗ + m (cid:1) (3.5) X − (cid:0) h ( m , m ) (cid:1) = − σ − (cid:0) h ( ∇ X ∗ + m , m ) (cid:1) + h (cid:0) m , ∇ X ∗− m (cid:1) (3.6) X z (cid:0) h ( m , m ) (cid:1) = − σ z (cid:0) h ( ∇ X ∗ z m , m ) (cid:1) + h (cid:0) m , ∇ X ∗ z m (cid:1) , (3.7)for m , m ∈ M .Note that (3.5) and (3.6) are equivalent since (cid:16) X + (cid:0) h ( m , m ) (cid:1) + σ + (cid:0) h ( ∇ X ∗− m , m ) (cid:1) − h (cid:0) m , ∇ X ∗ + m (cid:1)(cid:17) ∗ = − K − (cid:16) X − (cid:0) h ( m , m ) (cid:1) + σ − (cid:0) h ( ∇ X ∗ + m , m ) (cid:1) − h (cid:0) m , ∇ X ∗− m (cid:1)(cid:17) . In the case of a q -affine connection on a free module, one can derive a convenient parametriza-tion of all connections that are compatible with a given hermitian form. To this end, letus introduce some notation. Let ( S q ) n be a free σ -module with basis { e i } ni =1 . A q -affineconnection ∇ on ( S q ) n can be determined by specifying the Christoffel symbols ∇ X a e i = Γ jai e j , with Γ jai ∈ S q for a = 1 , , i, j = 1 , . . . , n , and setting ∇ X ∗ + e i = − ˆ σ ∗− (cid:0) ∇ X − e i (cid:1) ∇ X ∗− e i = − ˆ σ ∗ + (cid:0) ∇ X + e i (cid:1) ∇ X ∗ z e i = − ˆ σ ∗ z (cid:0) ∇ X z e i (cid:1) ∇ X a ( m i e i ) = m i ∇ X a e i + X a ( m i )ˆ σ a ( e i ) ∇ X ∗ a ( m i e i ) = σ ∗ a ( m i ) ∇ X ∗ a e i + X ∗ a ( m i ) e i . s we shall see, the metric compatibility of ∇ is conveniently formulated in terms of e Γ ai,j = Γ kai σ a (˜ h akj ) ˜ h aij = h (cid:0) ˆ σ ∗ a ( e i ) , e j (cid:1) . (3.8)The hermitian form h is assumed to be invertible (i.e. inducing an isomorphism of themodule and its dual) which implies that there exists h ij such that h ij h jk = h kj h ji = δ ki .In this case, one finds that ˜ h aij (for a = 1 , ,
3) is invertible as well, implying that one mayinvert (3.8) as Γ iaj = e Γ aj,k σ a (cid:0) ˜ h kia (cid:1) . Proposition 3.6.
Let ( S q ) n be a free σ -module with a basis { e i } ni =1 and let ∇ be a q -affineconnection on ( S q ) n given by the Christoffel symbols ∇ a e i = Γ jai e j . Furthermore, assumethat h is an invertible hermitian form on ( S q ) n . Then ∇ is compatible with h if and only ifthere exist hermitian matrices α, β, ρ ∈ Mat n ( S q ) such that e Γ + i,j = X + ( h ij ) + K ( α ij ) + iK ( β ij )(3.9) e Γ − i,j = X − ( h ij ) + K ( α ij ) − iK ( β ij )(3.10) e Γ zi,j = X z ( h ij ) + K ( ρ ij ) , (3.11) with h ij = h ( e i , e j ) .Proof. Starting from ∇ X a e i = Γ jai e j one obtains − σ + (cid:0) h ( ∇ X ∗− e i , e j ) (cid:1) + h ( e i , ∇ X ∗ + e j ) = σ + (cid:0) h (ˆ σ ∗ + (Γ k + i e k ) , e j ) (cid:1) − h (cid:0) e i , ˆ σ ∗− (Γ k − j e k ) (cid:1) = Γ k + i σ + (cid:0) h (ˆ σ ∗ + ( e k ) , e j ) (cid:1) − h ( e i , ˆ σ ∗− ( e k )) σ ∗− (Γ k − j ) ∗ = e Γ + i,j − σ ∗− (cid:0)e Γ − j,i (cid:1) ∗ giving the metric compatibility equation (3.5) as e Γ + i,j = X + (cid:0) h ij (cid:1) + K − (cid:0)e Γ − j,i (cid:1) ∗ . (3.12)Similarly, equation (3.7) may be written as e Γ zi,j = X z (cid:0) h ij (cid:1) + K − (cid:0)e Γ zj,i (cid:1) ∗ . (3.13)To solve (3.12) one may freely choose e Γ − i,j and define e Γ + i,j accordingly. Without loss ofgenerality, let us write e Γ − i,j in the following form e Γ − i,j = X − ( h ij ) + K ( α ij ) − iK ( β ij )for arbitrary α ij = α ∗ ji and β ij = β ∗ ji . Then one obtains e Γ + i,j = X + (cid:0) h ij (cid:1) + K − (cid:0)e Γ − j,i (cid:1) ∗ = X + (cid:0) h ij (cid:1) + K (cid:0) X − ( h ji ) ∗ (cid:1) + K (cid:0) K ( α ji ) ∗ (cid:1) + iK (cid:0) K ( β ji ) ∗ (cid:1) = X + (cid:0) h ij (cid:1) − K (cid:0) K − X + ( h ij ) (cid:1) + K (cid:0) K − ( α ij ) (cid:1) + iK (cid:0) K − ( β ij ) (cid:1) = X + ( h ij ) + K ( α ij ) + iK ( β ij ) ence, every solution of (3.5) may be written in the above form. For the metric equation(3.7) one writes e Γ zi,j = K ( ρ ij + ig ij ), with ρ ij = ρ ∗ ji and g ij = g ∗ ji , and notes that (3.13) isequivalent to K ( ρ ij + ig ij ) = X z ( h ij ) + K ( ρ ij − ig ij ) ⇔ iK ( g ij ) = X z ( h ij ) . This is compatible with the requirement that g ij = g ∗ ji since (cid:0) K − X z ( h ji ) (cid:1) ∗ = K (cid:0) X z ( h ji ) ∗ (cid:1) = − K (cid:0) K − X z ( h ji ) (cid:1) = − K − X z ( h ij ) . Hence, the general solution to (3.13) can be written as e Γ zi,j = X z ( h ij ) + K ( ρ ij )for arbitrary ρ ij = ρ ∗ ji , which concludes the proof. (cid:3) q -affine connections on projective modules. As expected, q -affine connections ex-ist on projective modules. More precisely, one proves the following result. Proposition 3.7.
Let M = (cid:0) ( S q ) n , ˆ σ a , ( σ a ) ∗ (cid:1) be a free σ -module and let ∇ be a q -affineconnection on M . If p : ( S q ) n → ( S q ) n is a projection and ˆ σ a = p ◦ ˆ σ a , ˆ σ ∗ a = p ◦ (ˆ σ a ) ∗ , then ( p ( S q ) n , ˆ σ a , ˆ σ ∗ a ) , is a σ -module and p ◦ ∇ is a q -affine connection on ( p ( S q ) n , ˆ σ a , ˆ σ ∗ a ) .Proof. It follows immediately that ( p ( S q ) n , p ◦ ˆ σ a , p ◦ (ˆ σ a ) ∗ ) satisfy the requirements of Defi-nition 3.2, since (cid:0) ( S q ) n , ˆ σ a , ( σ a ) ∗ (cid:1) is a σ -module. For instance,ˆ σ a ( f m ) = p (cid:0) ˆ σ a ( f m ) (cid:1) = p (cid:0) σ a ( f )ˆ σ a ( m ) (cid:1) = σ a ( f ) p (cid:0) ˆ σ a ( m ) (cid:1) = σ a ( f )ˆ σ a ( m ) . Since ∇ is a q -affine connection, it is immediate that ∇ = p ◦ ∇ satisfies properties (1) and(2) in Definition 3.3. Moreover, for m ∈ p ( S q ) n ∇ X a ( f m ) = p ∇ X a ( f m ) = f p (cid:0) ∇ X a m (cid:1) + X a ( f )( p ◦ ˆ σ a )( m )= f ∇ X a m + X a ( f )ˆ σ a ( m ) ∇ X ∗ a ( f m ) = σ ∗ a ( f ) p (cid:0) ∇ X ∗ a m (cid:1) + X ∗ a ( f ) p ( m )= σ ∗ a ( f ) ∇ X ∗ a m + X ∗ a ( f ) m, from which we conclude that ∇ is a q -affine connection on p ( S q ) n . (cid:3) Since we have shown in the previous section that one can construct q -affine connections onfree modules, Proposition 3.7 shows that q -affine connections exist on projective modules.Moreover, Let ∇ and ˜ ∇ be q -affine connections on a σ -module M and define α ( X, m ) = ∇ X m − ˜ ∇ X m. Then α : T S q × M → M satisfies α ( λX + Y, m ) = λα ( X, m ) + α ( Y, m )(3.14) α ( X, f m + m ) = f α ( X, m ) + α ( X, m )(3.15) or m , m ∈ M , X ∈ T S q , f ∈ S q and λ ∈ C . Conversely, every q -affine connection on aprojective module M can be written as ∇ X m = p ( ∇ X m ) + α ( X, m ) . where ∇ is the connection defined in (3.1) and α : T S q × M → M is an arbitrary mapsatisfying (3.14) and (3.15). Next, let us show that a connection on a projective module iscompatible with the restricted metric if the projection is orthogonal. Proposition 3.8.
Let ∇ be a q -affine connection on the free σ -module ( S q ) n and assumefurthermore that ∇ is compatible with a hermitian form h on ( S q ) n . If p : ( S q ) n → ( S q ) n isan orthogonal projection, i.e. h (cid:0) p ( m ) , m (cid:1) = h (cid:0) m , p ( m ) (cid:1) for all m , m ∈ ( S q ) n , then ˜ ∇ = p ◦ ∇ is compatible with h restricted to p ( S q ) n .Proof. Let us explicitly check one of the conditions in Definition 3.5 for m , m ∈ p ( S q ) n : − σ + (cid:0) h ( ˜ ∇ X ∗− m , m ) (cid:1) + h (cid:0) m , ˜ ∇ X ∗ + m (cid:1) = − σ + (cid:0) h ( p ∇ X ∗− m , m ) (cid:1) + h (cid:0) m , p ∇ X ∗ + m (cid:1) = − σ + (cid:0) h ( ∇ X ∗− m , p ( m )) (cid:1) + h (cid:0) p ( m ) , ∇ X ∗ + m (cid:1) = − σ + (cid:0) h ( ∇ X ∗− m , m ) (cid:1) + h (cid:0) m , ∇ X ∗ + m (cid:1) = X + (cid:0) h ( m , m ) (cid:1) . The remaining conditions are checked in an analogous way. (cid:3) A q -affine Levi-Civita connection on Ω ( S q )In this section we shall construct a q -affine connection on the free left module Ω ( S q ), com-patible with a hermitian form h , satisfying a certain torsion freeness condition. The moduleΩ ( S q ) is a free S q -module of rank 3 with basis ω + , ω − , ω z which implies that the results ofProposition 3.6 may be used. To start with, one has to endow Ω ( S q ) with the structureof a σ -module. Firstly, the actions (2.1) are extended to forms by requiring they commutewith the differential d . Then, one checks directly that K ⊲ ω + = q − ω + , K ⊲ ω − = qω − , K ⊲ ω z = ω z , while K ⊳ ω a = ω a , for a = 1 , ,
3. Let us work with the right action (withoutindicating it explicitly). The σ -module structure is then introduced as K ( ω a ) = ω a ⇒ ˆ σ a ( m b ω b ) = σ a ( m b ) ω b ˆ σ ∗ a ( m b ω b ) = σ ∗ a ( m b ) ω b . Furthermore, we assume that h is an invertible hermitian form on Ω ( S q ), such that K ( h ab ) = h ab , with h ab = h ( ω a , ω b ), for a, b = 1 , ,
3. (Note that for such a metric one has X z ( h ab ) = 0.)With these choices, one finds that e Γ ab,c = Γ pab h pc where the Christoffel symbols are definedas ∇ X a ω b = Γ cab ω c for a, b = 1 , ,
3. In the case of a q -affine connection on Ω ( S q ), there is anatural definition of torsion, motivated by (2.3)–(2.5). Definition 4.1. A q -affine connection ∇ on Ω ( S q ) is torsion free if ∇ − ω + − q ∇ + ω − = ω z (4.1) q ∇ z ω − − q − ∇ − ω z = (1 + q ) ω − (4.2) q ∇ + ω z − q − ∇ z ω + = (1 + q ) ω + . (4.3) n terms of e Γ ab,c , the conditions for a torsion free connection may be reformulated as e Γ − + ,a − q e Γ + − ,a = K (cid:0) ˜ h + za (cid:1) (4.4) q e Γ z − ,a − q − e Γ − z,b σ − (˜ h bc − ) σ z (˜ h zca ) = (1 + q ) K (˜ h z − a )(4.5) q − e Γ z + ,a − q e Γ + z,b σ + (˜ h bc + ) σ z (˜ h zca ) = − (1 + q ) K (˜ h z + a )(4.6)for a = 1 , , σ -module structure on Ω ( S q ). For the particular case whenthe metric is invariant by K , that is K ( h ab ) = h ab , and K ( ω a ) = ω a for a, b = 1 , ,
3, thetorsion free equations become e Γ − + ,a − q e Γ + − ,a = h za (4.7) q e Γ z − ,a − q − e Γ − z,a = (1 + q ) h − a (4.8) q e Γ + z,a − q − e Γ z + ,a = (1 + q ) h + a . (4.9)Out of these, in Section 6, we derive an explicit expression for a torsion free and metricconnection on Ω ( S q ), and show that such a connection exists if the following reality conditionis satisfied (cf. eq. (6.32)): (cid:0) X + ( h − z − q − h z + ) (cid:1) ∗ = X + ( h − z − q − h z + ) . (4.10)The connection is not unique, and the solution depends on 6 parameters. For the particularcase when h ab = hδ ab (obviously satisfying (4.10)), one can set all parameters to be zero: τ = τ = γ + − = f = µ = ρ zz = 0, in the notation of Section 6. It follows that a torsionfree and metric connection is given by ∇ + ω + = X + ( h ) h − ω + ∇ + ω − = − q − ω z ∇ + ω z = ω + + X + ( h ) h − ω z ∇ − ω + = ω z ∇ − ω − = X − ( h ) h − ω − ∇ − ω z = − q − ω − + X − ( h ) h − ω z ∇ z ω + = − q (2 + q ) ω + + q X + ( h ) h − ω z ∇ z ω − = (2 + 2 q − − q − ) ω − + q − X − ( h ) h − ω z ∇ z ω z = − q X − ( h ) h − ω + − q − X + ( h ) h − ω − . The quantum 2-sphere
The noncommutative (standard) Podle´s sphere S q [Pod87] can be considered as a subalgebraof S q by identifying the generators B , B + , B − of S q as B = cc ∗ B + = ca ∗ B − = ac ∗ = B ∗ + , satisfying then the relations B − B = q B B − B + B = q − B B + B − B + = q B (cid:0) − q B (cid:1) B + B − = B (cid:0) − B (cid:1) . hese elements generate the fix-point algebra of the right U (1)-action α z ( a ) = az α z ( a ∗ ) = a ∗ ¯ z α z ( c ) = cz α z ( c ∗ ) = c ∗ ¯ z for z ∈ U (1) and a ∈ S q , related to the U (1)-Hopf-fibration S q ֒ → S q .Now, the left action of the X a does not preserve the algebra S q : one readily computes, X + ⊲ B = qa ∗ c ∗ X − ⊲ B = − q − ca X z ⊲ B = 0 X + ⊲ B + = q ( a ∗ ) X − ⊲ B + = c X z ⊲ B + = 0 X + ⊲ B − = q ( c ∗ ) X − ⊲ B − = − q − ( a ) X z ⊲ B − = 0 . On the other hand, the right action of X a does preserve the algebra S q . Let us denote Y a = X a for the right action. Then, it is easy to check that B ⊳ Y + = q − B − B ⊳ Y − = − q − B + B ⊳ Y z = 0 B + ⊳ Y + = q − q (1 + q ) B B + ⊳ Y − = 0 B + ⊳ Y z = − q (1 + q ) B + B − ⊳ Y + = 0 B − ⊳ Y − = − q − + q − (1 + q ) B B − ⊳ Y z = (1 + q − ) B − . Note that when restricted to S q the Y a are not independent. A long but straightforwardcomputation shows that they are indeed related as (cid:0) ( f ⊳ Y + ) B + q + ( f ⊳ Y − ) B − q − (cid:1) (1 + q ) + ( f ⊳ Y z ) (cid:18) − q q B (cid:19) = ( f ⊳ Y z ) q − (cid:18) − q q (2 q + q + 1) B − (1 − q ) B (cid:19) + ( f ⊳ K ) q − (1 + q ) (cid:0) ( q − B + (1 − q ) B (cid:1) , (5.1)for f ∈ S q . This can be checked on a vector space basis for the algebra S q , a basis whichcan be taken as X ( m )( B ) n for m ∈ Z , n ∈ N with X ( m ) = ( B + ) m for m ≥ X ( m ) = ( B − ) − m for m < A left covariant calculus on S q . Since the element K acts (on the left) as the identityon S q , the differential (2.11) when restricted to f ∈ S q becomes df = ( X − ⊲ f ) ω − + ( X + ⊲ f ) ω + . (5.2)Note that X ± ⊲ f / ∈ S q . In particular one finds dB + = q ( a ∗ ) ω + + c ω − ,dB − = − q ( c ∗ ) ω + − q − a ω − ,dB = c ∗ a ∗ ω + − q − ca ω − which can be inverted to yield ω + = q − a dB + − q c dB − + (1 + q ) ac dB ω − = ( c ∗ ) dB + − q ( a ∗ ) dB − − (1 + q ) c ∗ a ∗ dB , mplying that the differential in (5.2) can be expressed as df = (cid:0) q − ( X + ⊲ f ) a + ( X − ⊲ f ) ( c ∗ ) (cid:1) dB + − (cid:0) q ( X + ⊲ f ) + c q ( X − ⊲ f ) ( a ∗ ) (cid:1) dB − + (1 + q ) (cid:0) ( X + ⊲ f ) ac − ( X − ⊲ f ) c ∗ a ∗ (cid:1) dB . From this expression one finds that the differential d on S q can be written in terms of theright acting operators Y a . Lemma 5.1.
For f ∈ S q , the differential in (5.2) can be written as (5.3) df = ( f ⊳ V + ) dB + + ( f ⊳ V − ) dB − + ( f ⊳ V ) dB where V + = Y + (cid:0) − q − (1 + q ) B (cid:1) q − − Y z B − q − (1 + q )1 + q + Y z B − − q (1 + q )(1 + q ) V − = − Y − (cid:0) − q (1 + q ) B (cid:1) q + Y z B + q − (1 + q )1 + q − Y z B + − q (1 + q )(1 + q ) V = (cid:0) Y + B + q − − Y − B − q (cid:1) (1 + q ) + Y z B (1 − q )(1 + q )1 + q − Y z B − q q . Proof.
By acting on the vector space basis X ( m )( B ) n (as introduced previously), one explic-itly checks the equality of (5.2) and (5.3) via a tedious but straightforward computation. (cid:3) Remark . When q = 1 the derivative (5.3) reduces to df = 2 (cid:0) ( f ⊳ Y + ) B + − ( f ⊳ Y − ) B − (cid:1) dB + (cid:0) ( f ⊳ Y + ) (1 − B ) − ( f ⊳ Y z ) B − (cid:1) dB + + (cid:0) − ( f ⊳ Y − ) (1 − B ) + ( f ⊳ Y z ) B + (cid:1) dB − . (5.4)Classically, the vector field X a are the left invariant vector fields on S = SU (2) with dualleft invariant forms ω a . Thus they do not project to vector fields on the base space S with commuting coordinates ( B + , B − , B ) and relation B + B − = B (1 − B ): X a ⊲ f is nota function on S even when f is. On the other hand, the vector fields Y a are the rightinvariant vector fields on SU (2) and thus they project to vector fields on S , where they arenot independent any longer and are related by2( B + Y + + B − Y − ) + (1 − B ) Y z = 0 , (5.5)which is just the relation to which (5.1) reduces when q = 1.By changing coordinates B = (1 − x ) so that the radius condition for S is written as r = 4 B + B − + x , the exterior derivative operator in (5.4) becomes df = ∂ x f dx + ∂ + f dB + + ∂ − f dB − − (∆ f ) ( x dx + 2 B − dB + + 2 B + dB − )where ∆ = x ∂ x + B + ∂ + + B − ∂ − is the Euler (dilatation) vector field. One then computes dr = 2(1 − r )( x dx +2 B − dB + +2 B + dB − ), which vanishes when restricting to S : r − he form (5.2) of the differential that uses left invariant vector fields and forms can beseen as identifying the cotangent bundle of S with the direct sum of the line bundles of‘charge’ ±
2, that is Ω ( S ) ≃ L − ω − ⊕ L +2 ω + . This identification can be used also for thequantum sphere S q with the line bundles defined as in (5.6) below.5.2. Connections on projective modules over S q . The definitions of σ -modules and q -affine connections apply equally well to the algebra S q . Note that the right action of K preserve S q B ⊳ K = B B + ⊳ K = qB + B − ⊳ K = q − B − . implying that σ a , σ ∗ a leave S q invariant.In this section, we will construct q -affine connections on a class of projective modules over S q (cf. [BM98, HM99, Lan18]). For n ≥ µ = 0 , , . . . , n , let (Ψ n ) µ , (Φ n ) µ ∈ S q begiven as (Φ n ) µ = √ α nµ c n − µ a µ (Ψ n ) µ = p β nµ ( c ∗ ) µ ( a ∗ ) n − µ with α nµ = n − µ − Y k =0 − q n − k ) − q k +1) β nµ = q µ µ − Y k =0 − q − n − k ) − q − k +1) . It is straight-forward to check that n X µ =0 (Φ n ) ∗ µ (Φ n ) µ = n X µ =0 (Ψ n ) ∗ µ (Ψ n ) µ = , implying that ( p n ) νµ = (Ψ n ) µ (Ψ n ) ∗ ν = p β nµ β nν ( c ∗ ) µ ( a ∗ ) n − µ a n − ν c ν ( p − n ) νµ = (Φ n ) µ (Φ n ) ∗ ν = √ α nµ α nν c n − µ a µ ( a ∗ ) ν ( c ∗ ) n − ν satisfy p n = p n and p − n = p − n . Moreover, it is easy to see that ( p n ) νµ , ( p − n ) νµ ∈ S q , whichimplies that M n = ( p n ( S q ) n +1 if n ≥ p −| n | ( S q ) | n | +1 if n < S q -modules for n ∈ Z . Let us recall that these modules areisomorphic to the components in a (vector space) decomposition of S q S q = ⊕ n ∈ Z L n with L n = { f ∈ S q : α z ( f ) = ¯ z n f } , (5.6)and it follows that L = S q , as well as L n L m ⊆ L n + m . For f ∈ S q and f n ∈ L n one has α z ( f f n ) = α z ( f ) α z ( f n ) = ¯ z n f f n , hich implies that L n is a left S q -module. Furthermore, it is easy to see that the rightaction of U q (su(2)) leaves each L n invariant. Let { e µ } nµ =0 be a basis of ( S q ) n +1 and let φ n : ( S q ) n +1 → L n be defined as φ n ( m µ e µ ) = ( m µ (Ψ n ) µ n ≥ m µ (Φ | n | ) µ n < , and we note that φ n ( m ) ∈ L n since (Ψ n ) µ ∈ L n and (Φ n ) µ ∈ L − n (for n ≥ Lemma 5.3.
Let m ∈ ( S q ) n +1 . If p n ( m ) = 0 then φ n ( m ) = 0 .Proof. If p n ( m ) = 0 then (for n ≥ n X µ =0 m µ ( p n ) νµ = 0 ∀ ν ⇒ n X µ,ν =0 m µ ( p n ) νµ (Ψ n ) ν = 0 ⇒ (cid:16) n X µ =0 m µ (Ψ n ) µ (cid:17)(cid:16) n X ν =0 (Ψ n ) ∗ ν (Ψ n ) ν (cid:17) = 0 ⇒ n X µ =0 m µ (Ψ n ) µ = 0which is equivalent to φ n ( m ) = 0. The proof for n < (cid:3) The above result implies that φ n descends to a module homomorphism φ n : M n → L n ,and one can show that φ n is in fact an isomorphism. To simplify the presentation, let usin the following assume that n ≥
0. As generators of M n one can choose ˆ e µ = p n ( e µ ) for µ = 0 , , . . . , n , and one notes that φ n (ˆ e µ ) = ( p n ) νµ (Ψ n ) ν = (Ψ n ) µ , implying that { (Ψ n ) µ } nµ =0 generates L n . Since the S q -module L n is a subset of S q which isalso invariant under the right action of U q (su(2)), it is naturally a σ -module with respect tothe right action of σ a and σ ∗ a ; one finds that(Ψ n ) µ ⊳ K = q
12 ( n − µ ) (Ψ n ) µ (Ψ n ) ∗ µ ⊳ K = q −
12 ( n − µ ) (Ψ n ) ∗ µ giving σ ± (cid:0) (Ψ n ) µ (cid:1) = q n − µ (Ψ n ) µ σ z (cid:0) (Ψ n ) µ (cid:1) = q n − µ ) (Ψ n ) µ σ ∗± (cid:0) (Ψ n ) µ (cid:1) = q − ( n − µ ) (Ψ n ) µ σ ∗ z (cid:0) (Ψ n ) µ (cid:1) = q − n − µ ) (Ψ n ) µ . Correspondingly, we would like to define a σ -module structure on M n such that φ n is amorphism of σ -modules. To this end, we start by introducing a σ -module structure on( S q ) n +1 in analogy with Proposition 3.7; namely, starting fromˆ σ ± ( e µ ) = q n − µ e µ ˆ σ z ( e µ ) = q n − µ ) e µ (ˆ σ ± ) ∗ ( e µ ) = q − ( n − µ ) e µ (ˆ σ z ) ∗ ( e µ ) = q − n − µ ) e µ we set ˆ σ a = p ◦ ˆ σ a : M n → M n and ˆ σ ∗ a = p ◦ (ˆ σ a ) ∗ : M n → M n . Since( p n ) νµ ⊳ K = (cid:0) (Ψ n ) µ (Ψ n ) ν (cid:1) ⊳ K = q − ( µ − ν ) ( p n ) νµ ne finds that ˆ σ ± (ˆ e µ ) = ( p n ◦ ˆ σ ± ) (cid:0) ( p n ) νµ e ν (cid:1) = σ ± (cid:0) ( p n ) νµ (cid:1) ( p n ◦ ˆ σ ± )( e ν )= q − µ − ν ) q n − ν p n ( e ν ) = q n − µ ˆ e µ ˆ σ z (ˆ e µ ) = q n − µ ) ˆ e µ (and similarly for ˆ σ ∗ a ) implying that φ n is a σ -module isomorphism. Finally, let us now turn tothe question of finding q -affine connections on M n that are compatible with a given hermitianform. Thus, assume that h is a hermitian form on ( S q ) n +1 for which p n is an orthogonalprojection. According to Proposition 3.6, every q -affine connection ∇ on ( S q ) n +1 may bewritten in the form ∇ X a e µ = Γ νaµ e ν with e Γ + µ,ν = X + ( h µν ) + K ( α µν ) + iK ( β µν ) e Γ − µ,ν = X − ( h µν ) + K ( α µν ) − iK ( β µν ) e Γ zµ,ν = X z ( h µν ) + K ( ρ µν ) , (5.7)for an arbitrary choice of hermitian α, β, ρ ∈ Mat n +1 ( S q ) and, furthermore, Proposition 3.7shows that ∇ = p n ◦ ∇ is a q -affine connection on M n , compatible with the restriction of h to M n . For the generators { ˆ e µ } nµ =0 one finds that ∇ X + ˆ e µ = ( p n ◦ ∇ ) (cid:0) ( p n ) νµ e ν (cid:1) = p (cid:0) ( p n ) νµ Γ κ + ν e κ + X + (cid:0) ( p n ) νµ (cid:1) ˆ σ + ( e ν ) (cid:1) = (cid:16) ( p n ) νµ Γ κ + ν + q n − µ X + (cid:0) ( p n ) κµ (cid:1)(cid:17) ˆ e κ ∇ X − ˆ e µ = (cid:16) ( p n ) νµ Γ κ − ν + q n − µ X − (cid:0) ( p n ) κµ (cid:1)(cid:17) ˆ e κ ∇ X z ˆ e µ = (cid:16) ( p n ) νµ Γ κzν + q n − µ ) X z (cid:0) ( p n ) κµ (cid:1)(cid:17) ˆ e κ . In the current case, one finds that˜ h ± µν = h (cid:0) (ˆ σ ± ) ∗ ( e µ ) , e ν (cid:1) = q µ − n h µν ⇒ ˜ h µν ± = h µν q n − ν ˜ h zµν = h (cid:0) (ˆ σ z ) ∗ ( e µ ) , e ν (cid:1) = q µ − n ) h µν ⇒ ˜ h µνz = h µν q n − ν ) giving Γ ν ± µ = e Γ ± µ,ρ σ ± (cid:0) ˜ h ρµ ± (cid:1) = q ν − n e Γ ± µ,ρ K ( h ρν )Γ νzµ = e Γ zµ,ρ σ z (˜ h ρνz ) = q ν − m ) e Γ zµ,ρ K ( h ρν )Thus, for the choice α = β = ρ = 0 in (5.7), one obtains the formulas ∇ X ± ˆ e µ = (cid:18) q κ − n ( p n ) νµ X ± ( h νρ ) K ( h ρκ ) + q n − µ X ± (cid:0) ( p n ) κµ (cid:1)(cid:19) ˆ e κ ∇ X z ˆ e µ = (cid:18) q κ − n ) ( p n ) νµ X z ( h νρ ) K ( h ρκ ) + q n − µ ) X z (cid:0) ( p n ) κµ (cid:1)(cid:19) ˆ e κ . . Computation of Christoffel symbols
In this section we work out an explicit expression for a metric and torsion free connectionon Ω ( S q ), as presented in Section 4, for the case when the metric is invariant by K , thatis K ( h ab ) = h ab , and K ( ω a ) = ω a for a, b = 1 , ,
3. As previously noted, for such a metric, e Γ ab,c = Γ pab h pc and the torsion free equations become (cf. (4.7)-(4.9)): e Γ − + ,a − q e Γ + − ,a = h za q e Γ z − ,a − q − e Γ − z,a = (1 + q ) h − a q e Γ + z,a − q − e Γ z + ,a = (1 + q ) h + a . These can be solved as e Γ − + ,a = h za + qK ( γ (1) a )(6.1) e Γ + − ,a = − q − h za + q − K ( γ (1) a )(6.2) e Γ z − ,a = q − (1 + q ) h − a + q − K ( γ (2) a )(6.3) e Γ − z,a = − q (1 + q ) h − a + q K ( γ (2) a )(6.4) e Γ + z,a = q − (1 + q ) h + a + q − K ( γ (3) a )(6.5) e Γ z + ,a = − q (1 + q ) h + a + q K ( γ (3) a )(6.6)for arbitrary γ (1) a , γ (2) a , γ (3) a ∈ S q . On the other hand, Proposition 3.6 gives general expressionsfor e Γ ab,c for a metric connection, and combining the two results yields the following set ofequations to be solved: X − ( h + a ) + K ( γ ∗ a + ) = h za + qK ( γ (1) a )(6.7) X + ( h − a ) + K ( γ − a ) = − q − h za + q − K ( γ (1) a )(6.8) X z ( h − a ) + K ( ρ − a ) = q − (1 + q ) h − a + q − K ( γ (2) a )(6.9) X − ( h za ) + K ( γ ∗ az ) = − q (1 + q ) h − a + q K ( γ (2) a )(6.10) X + ( h za ) + K ( γ za ) = q − (1 + q ) h + a + q − K ( γ (3) a )(6.11) X z ( h + a ) + K ( ρ + a ) = − q (1 + q ) h + a + q K ( γ (3) a )(6.12)where γ ab = α ab + iβ ab . These equations can be rewritten as γ − a = − K − X + ( h − a ) − q − h za + q − γ (1) a (6.13) γ a + = K − X + ( h a + ) + h az + q ( γ (1) a ) ∗ (6.14) ρ − a = − K − X z ( h − a ) + q − (1 + q ) h − a + q − K − ( γ (2) a )(6.15) γ az = K − X + ( h az ) − q (1 + q ) h a − + q ( γ (2) a ) ∗ (6.16) γ za = − K − X + ( h za ) + q − (1 + q ) h + a + q − γ (3) a (6.17) ρ + a = − K − X z ( h + a ) − q (1 + q ) h + a + q K − ( γ (3) a ) . (6.18)When solving for γ ab and ρ ab from the above equations, one finds several ambiguities andconstraints. Namely, there are multiple expressions for γ − + , γ − z , γ z + and γ zz , and we require from Proposition 3.6) that ρ ∗ ++ = ρ ++ , ρ ∗−− = ρ −− and ρ ∗ + − = ρ − + . Let us consider theseconstraints one by one. γ − + : Equations (6.13) and (6.14) give two expressions for γ − + : γ − + = − K − X + ( h − + ) − q − h z + + q − γ (1)+ γ − + = K − X + ( h − + ) + h − z + q ( γ (1) − ) ∗ which coincide if γ (1)+ = qK − X + ( h − + ) + q − h z + + qτ γ (1) − = − q − K − X − ( h + − ) − q − h z − + q − τ ∗ for arbitrary τ ∈ S q , giving γ − + = τ . γ − z : Equations (6.13) and (6.16) give two expressions for γ − z : γ − z = − K − X + ( h − z ) − q − h zz + q − γ (1) z γ − z = K − X + ( h − z ) − q (1 + q ) h −− + q ( γ (2) − ) ∗ which coincide if γ (1) z = qK − X + ( h − z ) + q − h zz + qτ (6.19) γ (2) − = q − K − X − ( h z − ) + (1 + q ) h −− + q − τ ∗ for arbitrary τ ∈ S q , giving γ − z = τ . γ z + : Equations (6.14) and (6.17) give two expressions for γ z + : γ z + = K − X + ( h z + ) + h zz + q ( γ (1) z ) ∗ γ z + = − K − X + ( h z + ) + q − (1 + q ) h ++ + q − γ (3)+ which coincide if γ (1) z = q − K − X − ( h + z ) − q − h zz + q − τ ∗ (6.20) γ (3)+ = q K − X + ( h z + ) − (1 + q ) h ++ + q τ for arbitrary τ ∈ S q , giving γ z + = τ . γ zz : Equations (6.16) and (6.17) give two expressions for γ zz : γ zz = K − X + ( h zz ) − q (1 + q ) h z − + q ( γ (2) z ) ∗ γ zz = − K − X + ( h zz ) + q − (1 + q ) h + z + q − γ (3) z which coincide if γ (2) z = q − K − X − ( h zz ) + (1 + q ) h − z + q − τ ∗ γ (3) z = q K − X + ( h zz ) − (1 + q ) h + z + q τ for arbitrary τ ∈ S q , giving γ zz = τ . he above considerations determine γ (1)+ , γ (1) − , γ (2) − , γ (3)+ , γ (2) z , γ (3) z , γ (1) z , but give two expres-sions for γ (1) z ; equating them gives the following. γ (1) z : Equations (6.19) and (6.20) give two expressions for γ (1) z : γ (1) z = qK − X + ( h − z ) + q − h zz + qτ γ (1) z = q − K − X − ( h + z ) − q − h zz + q − τ ∗ which coincide if τ = − K − X + ( h − z ) − q − h zz + q − µ τ = K − X + ( h z + ) + h zz + qµ ∗ for arbitrary µ ∈ S q , giving γ (1) z = µ and γ (2) − = q − K − X − ( h z − ) + (1 + q ) h −− − q − h zz + q − µ ∗ γ (3)+ = q K − X + ( h z + ) − (1 + q ) h ++ + q h zz + q µ ∗ . Let us summarize what we have obtained so far: γ (1)+ = qK − X + ( h − + ) + q − h z + + qτ (6.21) γ (1) − = − q − K − X − ( h + − ) − q − h z − + q − τ ∗ (6.22) γ (1) z = µ (6.23) γ (2) − = q − K − X − ( h z − ) + (1 + q ) h −− − q − h zz + q − µ ∗ (6.24) γ (3)+ = q K − X + ( h z + ) − (1 + q ) h ++ + q h zz + q µ ∗ (6.25) γ (2) z = q − K − X − ( h zz ) + (1 + q ) h − z + q − τ ∗ (6.26) γ (3) z = q K − X + ( h zz ) − (1 + q ) h + z + q τ , (6.27)for arbitrary µ , τ , τ . It remains to solve the constraints ρ ∗ ++ = ρ ++ , ρ ∗−− = ρ −− and ρ ∗ + − = ρ − + . ρ ++ : From equation (6.18) one obtains ρ ++ = − q (1 + q ) h ++ + q K − ( γ (3)+ )by using that X z ( h ab ) = 0. Requiring ρ ∗ ++ = ρ ++ gives γ (3)+ = K (cid:0) ( γ (3)+ ) ∗ (cid:1) and inserting (6.25) yields K ( µ ) − µ ∗ = q − K − X + ( h z + ) + q − KX − ( h + z )which is solved by µ = q − K − X − ( h + z ) + K − ( f )(6.28)for arbitrary hermitian f ∈ S q . −− : From equation (6.15) one obtains ρ −− = q − (1 + q ) h −− + q − K − ( γ (2) − )by using that X z ( h ab ) = 0. Requiring ρ ∗−− = ρ −− gives γ (2) − = K (cid:0) ( γ (2) − ) ∗ (cid:1) and inserting (6.24) gives K ( µ ) − µ ∗ = qK − X − ( h z − ) + qKX + ( h − z )which is solved by µ = qK − X + ( h − z ) + K − ( f )(6.29)for arbitrary hermitian f ∈ S q .Equations (6.28) and (6.29) give two expressions for µ , which coincide if f − f = qX + ( h − z ) − q − X − ( h + z ) . (6.30)Since f − f is hermitian, a necessary condition for this equation to have a solution is that H = qX + ( h − z ) − q − X − ( h + z )is hermitian. Using K − X ± = q ∓ X ± K − and K − ( h ab ) = h ab , H = H ∗ is equivalent to X + ( h − z − q − h z + ) = − q X − ( h z − − q − h + z ) . (6.31)which can also be written as the reality condition (cid:0) X + ( h − z − q − h z + ) (cid:1) ∗ = X + ( h − z − q − h z + ) . (6.32)If the above condition is fulfilled, then (6.30) is solved by f = qX + ( h − z ) − q − X − ( h + z ) + f f = − qX + ( h − z ) + q − X − ( h + z ) + f for arbitrary hermitian f ∈ S q . This gives µ = qK − X + ( h − z ) + q − K − X − ( h + z ) + K − ( f ) . (6.33) ρ + − : Equations (6.15) and (6.18) give ρ + − = − q (1 + q ) h + − + q K − (cid:0) γ (3) − (cid:1) ρ − + = q − (1 + q ) h − + + q − K − (cid:0) γ (2)+ (cid:1) and requiring ρ ∗ + − = ρ − + yields − q (1 + q ) h − + + q K (cid:0) ( γ (3) − ) ∗ (cid:1) = q − (1 + q ) h − + + q − K − (cid:0) γ (2)+ (cid:1) which is solved by γ (3) − = (1 + q ) h + − + q − K ( µ ∗ ) γ (2)+ = − (1 + q ) h − + + q K ( µ )for arbitrary µ ∈ S q . hus, we have obtained the following equations γ (1)+ = qK − X + ( h − + ) + q − h z + + qτ γ (1) − = − q − K − X − ( h + − ) − q − h z − + q − τ ∗ γ (1) z = qK − X + ( h − z ) + q − K − X − ( h + z ) + K − ( f ) γ (2)+ = − (1 + q ) h − + + q K ( µ ) γ (2) − = q − K − X − ( h z − ) − q − K − X + ( h z + ) + (1 + q ) h −− − q − h zz + q − K ( f ) γ (2) z = q − K − X − ( h zz ) + (1 + q ) h − z + q − τ ∗ γ (3)+ = q K − X + ( h z + ) − q K − X − ( h z − ) − (1 + q ) h ++ + q h zz + q K ( f ) γ (3) − = (1 + q ) h + − + q − K ( µ ∗ ) γ (3) z = q K − X + ( h zz ) − (1 + q ) h + z + q τ giving all 27 Christoffel symbols as e Γ ++ , + = X + ( h ++ ) − q X − ( h + − ) + h + z + q K ( τ ) e Γ + − , − = − q − X − ( h + − ) − q − h z − + q − K ( τ ∗ ) e Γ + z,z = X + ( h zz ) + K ( τ ) e Γ ++ , − = X + ( h + − ) + K ( γ + − ) e Γ + − , + = X + ( h − + ) + K ( τ ) e Γ + z, + = X + ( h z + ) − q X − ( h z − ) + h zz + qK ( f ) e Γ ++ ,z = X + ( h + z ) − q (1 + q ) h + − + q µ ∗ e Γ + z, − = q − (1 + q ) h + − + q − K ( µ ∗ ) e Γ + − ,z = X + ( h − z ) + q − X − ( h + z ) − q − h zz + q − f e Γ − + , + = q X + ( h − + ) + h z + + q K ( τ ∗ ) e Γ −− , − = X − ( h −− ) + q − X + ( h − + ) − q − h − z + q − K ( τ ) e Γ − z,z = X − ( h zz ) + K ( τ ∗ ) e Γ − + , − = X − ( h + − ) + K ( τ ∗ ) e Γ −− , + = X − ( h − + ) + K ( γ ∗ + − ) e Γ − + ,z = X − ( h + z ) + q X + ( h − z ) + h zz + qf e Γ − z, + = − q (1 + q ) h − + + q − K ( µ ) e Γ −− ,z = X − ( h − z ) + q (1 + q ) h − + + q − µ e Γ − z, − = X − ( h z − ) − q − X + ( h z + ) − q − h zz + q − K ( f ) Γ z + , + = q X + ( h z + ) − q X − ( h z − ) − q (1 + q ) h ++ + q h zz + q K ( f ) e Γ z − , − = q − X − ( h z − ) − q − X + ( h z + ) + q − (1 + q ) h −− − q − h zz + q − K ( f ) e Γ zz,z = K ( ρ zz ) e Γ z + , − = K ( µ ∗ ) e Γ z − , + = K ( µ ) e Γ z + ,z = q X + ( h zz ) − q (1 + q ) h + z + q K ( τ ) e Γ zz, + = − q X − ( h zz ) − q (1 + q ) h z + + q K ( τ ∗ ) e Γ z − ,z = q − X − ( h zz ) + q − (1 + q ) h − z + q − K ( τ ∗ ) e Γ zz, − = − q − X + ( h zz ) + q − (1 + q ) h z − + q − K ( τ )depending on the parameters τ , τ , µ , γ + − , ρ zz , f ∈ S q (with f ∗ = f ). Acknowledgements:
The paper is partially supported by INFN-Trieste. GL is supportedby INFN, Iniziativa Specifica GAST, and by INDAM - GNSAGA. JA is supported by grant2017-03710 from the Swedish Research Council. Furthermore, JA would like to thank theDepartment of Mathematics and Geosciences, University of Trieste for hospitality.
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Publ. Res. Inst. Math. Sci. , 23(1):117–181, 1987.(Joakim Arnlind)
Dept. of Math., Link¨oping University, 581 83 Link¨oping, Sweden
E-mail address : [email protected] (Kwalombota Ilwale) Dept. of Math., Link¨oping University, 581 83 Link¨oping, Sweden
E-mail address : [email protected] (Giovanni Landi) Matematica, Universit`a di Trieste,Via A. Valerio, 12/1, 34127 Trieste, ItalyInstitute for Geometry and Physics (IGAP) Trieste, Italy and INFN, Trieste, Italy
E-mail address : [email protected]@units.it