aa r X i v : . [ m a t h . QA ] M a r LEVI-CIVITA CONNECTION FOR SU q (2)SUGATO MUKHOPADHYAYIndian Statistical Institute203 B.T. Road, Kolkata, India Abstract.
We prove that the 4 D ± calculi on the quantum group SU q (2) satisfy a metric-independent sufficient condition for the existence of a unique bicovariant Levi-Civita connectioncorresponding to every bi-invariant pseudo-Riemannian metric. Introduction
The quantum group SU q (2) introduced in [7] and the notion of bicovariant differential calculion Hopf algebras was introduced in [8] by Woronowicz. The question of bicovariant Levi-Civitaconnections on bicovariant differential calculi of compact quantum groups have been investigatedby Heckenberger and Schm¨udgen in [5] for the quantum groups SL q ( N ), O q ( N ) and Sp q ( N ). Onthe other hand, Beggs, Majid and their collaborators have studied Levi-Civita connections onquantum groups and homogeneous spaces in various articles, and a comprehensive account can befound in [1].More recently, in [3], bicovariant connections on arbitrary bicovariant differential calculi ofcompact quantum groups and the notion of their metric compatibility with respect to arbitrary bi-invariant pseudo-Riemannian metrics was studied. In that article, the construction of a canonicalbicovariant torsionless connection on a calculus was presented (Theorem 5.3 of [3]), provided thatWoronowicz’s braiding map σ for the calculus satisfies a diagonalisability condition. Also, a metric-independent sufficient condition for the existence of a unique bicovariant Levi-Civita connection(in the sense of Definition 6.3 of [3]) was provided in Theorem 7.9 of [3].In this article, we will investigate the theory of [3] in the context of the 4 D ± calculi of the compactquantum group SU q (2) which were explicitly described in [7] and then [6]. In Section 2, we recallthe notion of covariant Levi-Civita connections on bicovariant differential calculi as formulated in[3]. In Section 3, the 4 D ± calculi on SU q (2) are recalled and we show that Woronowicz’s braidingmap for the 4 D ± calculi satisfy the diagonalisability condition mentioned above. In Section 4,we construct a bicovariant torsionless connection. In Section 5, we will show that the sufficiencycondition of Theorem 7.9 of [3] is satisfied by both calculi, except for at most finitely many valuesof q , and hence we can conclude the existence of a unique bicovariant Levi-Civita connection,corresponding to a bi-invariant pseudo-Riemannian metric.2. Levi-Civita connections on bicovariant differential calculi
In this section, we recall the notion of Levi-Civita connections on bicovariant differential calculias formulated in [3].We say that ( E , ∆ E , E ∆) is a bicovariant bimodule over a Hopf algebra A if E is a bimoduleover A , ( E , ∆ E ) is a left A -comodule, ( E , E ∆) is a right A -comodule, subject to the followingcompatibility conditions:∆ E ( aρ ) = ∆( a )∆ E ( ρ ) , ∆ E ( ρa ) = ∆ E ( ρ )∆( a ) E ∆( aρ ) = ∆( a ) E ∆( ρ ) , E ∆( ρa ) = E ∆( ρ )∆( a ) , where ρ is an arbitrary element of E and a is an arbitrary element of A . If ( E , ∆ E , E ∆) is abicovariant bimodule over A , we say that an element e in E is left (respectively, right) invariant if∆ E ( e ) = 1 ⊗ C e (respectively, E ∆( e ) = e ⊗ C E invariant under the left coaction of A by E , and that of elements invariant under E-mail address : [email protected] . the right coaction of A by E . If E and F are two bicovariant bimodules over A , a C -linear map T : E → F is said to be left covariant if ∆ F ◦ T = (id ⊗ C T ) ◦ ∆ E . T is said to be right covariantif F ∆ ◦ T = ( T ⊗ C id) ◦ E ∆. T is called bicovariant if it is both left-covariant and right-covariant.A (first order) differential calculus ( E , d ) over a Hopf algebra A is called a bicovariant differentialcalculus if the following conditions are satisfied:(i) For any a k , b k in A , k = 1 , . . . , K ,( X k a k db k = 0) implies that ( X k ∆( a k )(id ⊗ C d )∆( b k ) = 0) , (ii) For any a k , b k in A , k = 1 , . . . , K ,( X k a k db k = 0) implies that ( X k ∆( a k )( d ⊗ C id)∆( b k ) = 0) . Woronowicz ([8]) proved that a bicovariant differential calculus is endowed with canonical leftand right-comodule coactions of A , making it into a bicovariant bimodule ( E , ∆ E , E ∆). Moreover,the map d : A → E is a bicovariant map.Next, we state the construction of the associated space of two-forms, Ω ( A ) for a bicovariantdifferential calculus of an arbitrary unital Hopf algebra A as in [8]. To do so, we need to recall thebraiding map σ for bicovariant bimodules. Proposition 2.1. (Proposition 3.1 of [8] ) Given a bicovariant bimodule E on a Hopf algebra A ,there exists a unique bimodule homomorphism σ : E ⊗ A E → E ⊗ A E such that σ ( ω ⊗ A η ) = η ⊗ A ω (1) for any left-invariant element ω and right-invariant element η in E , under the coactions of A . σ isan invertible bicovariant A -bimodule map from E ⊗ A E to itself. Moreover, σ satisfies the followingbraid equation on E ⊗ A E ⊗ A E :(id ⊗ A σ )( σ ⊗ A id)(id ⊗ A σ ) = ( σ ⊗ A id)(id ⊗ A σ )( σ ⊗ A id) . Given a bicovariant first order differential calculus ( E , d ), there exists a unique braiding map σ ,by Proposition 2.1. The space of two-forms is defined to be the bicovariant bimoduleΩ ( A ) := ( E ⊗ A E ) (cid:14) Ker( σ − . The symbol ∧ denotes the quotient map, which is a bicovariant bimodule map, ∧ : E ⊗ A E → Ω ( A ) . The map d : A → E extends to a unique exterior derivative map (to be denoted again by d ), d : E → Ω ( A ) , such that, for all a in A and ρ in E ,(i) d ( aρ ) = da ∧ ρ + ad ( ρ ) , (ii) d ( ρa ) = d ( ρ ) a − ρ ∧ da, (iii) d is bicovariant.Let us, from now on, denote the subspace of left-invariant elements of an arbitrary bicovariantbimodule E by the symbol E . By Proposition 2.5 of [2], the vector space E ⊗ C E can be identifiedwith the space ( E ⊗ A E ) of left-invariant elements of E ⊗ A E . The isomorphism E ⊗ C E → ( E ⊗ A E )is given by ω i ⊗ C ω j ω i ⊗ A ω j (2), where { ω i } i is a vector space basis E .Moreover, by the bicovariance of the map σ : E ⊗ A E → E ⊗ A E , we get the restriction (see Equation18 of [3]): σ := σ | E⊗ C E : E ⊗ C E → E ⊗ C E . (3)From now on, we are going to work under the assumption that σ is a diagonalisable map betweenfinite dimensional vector spaces. In [3], this assumption was crucially used to set up a framework forthe existence of a unique bicovariant Levi-Civita connection on a bicovariant differential calculussatisfying the assumption. Let us introduce some notations and definitions so that we can recall the framework mentionedabove.
Definition 2.2.
Suppose the map σ is diagonalisable. The eigenspace decomposition of E ⊗ C E will be denoted by E ⊗ C E = L λ ∈ Λ V λ , where Λ is the set of distinct eigenvalues of σ and V λ is the eigenspace of σ corresponding to the eigenvalue λ . Thus, for example, V will denote theeigenspace of σ for the eigenvalue λ = 1 . Moreover, we define E ⊗ sym C E to be the eigenspace of σ with eigenvalue , i.e., E ⊗ sym C E := V . We also define F := L λ ∈ Λ \{ } V λ . Finally, we will denote by ( P sym ) the idempotent element inHom ( E ⊗ C E , E ⊗ C E ) with range E ⊗ sym C E and kernel F . Lets us introduce the notion of bi-invariant pseudo-Riemannian metric on a bicovariant A -bimodule E . Definition 2.3. ( [5] , Definition 4.1 of [3] ) Suppose E is a bicovariant A bimodule and σ : E ⊗ A E →E ⊗ A E be the map as in Proposition 2.1. A bi-invariant pseudo-Riemannian metric for the pair ( E , σ ) is a right A -linear map g : E ⊗ A E → A such that the following conditions hold: (i) g ◦ σ = g. (ii) If g ( ρ ⊗ A ν ) = 0 for all ν in E , then ρ = 0 . (iii) The map g is bi-invariant, i.e for all ρ, ν in E , (id ⊗ C ǫg )(∆ ( E⊗ A E ) ( ρ ⊗ A ν )) = g ( ρ ⊗ A ν ) , ( ǫg ⊗ C id)( ( E⊗ A E ) ∆( ρ ⊗ A ν )) = g ( ρ ⊗ A ν ) . Now we can define the torsion of a connection and the compatibility of a left-covariant connectionwith a bi-invariant pseudo-Riemannian metric.
Definition 2.4. ( [5] ) Let ( E , d ) be a bicovariant differential calculus on A . A (right) connectionon E is a C -linear map ∇ : E → E ⊗ A E such that, for all a in A and ρ in E , the following equationholds: ∇ ( ρa ) = ∇ ( ρ ) a + ρ ⊗ A da. The map ∇ is said to a left-covariant, right-covariant or bicovariant connection is it is a left-covariant, right-covariant or bicovariant map, respectively. The torsion of a connection ∇ on E isthe right A -linear map T ∇ := ∧ ◦ ∇ + d : E → Ω ( A ) . The connection ∇ is said to be torsionless if T ∇ = 0 . Our notion of torsion is the same as that of [5], with the only difference being that they workwith left connections.
Definition 2.5. (Definitions 6.1 and 6.3 of [3] ) Let ∇ be a left-covariant connection on a bicovari-ant calculus ( E , d ) such that the map σ is diagonalisable, and g a bi-invariant pseudo-Riemannianmetric. Then we define f Π g ( ∇ ) : E ⊗ C E → E by the following formula : f Π g ( ∇ )( ω i ⊗ C ω j ) = 2(id ⊗ C g )( σ ⊗ C id)( ∇ ⊗ C id) ( P sym )( ω i ⊗ C ω j ) . (4) Next, for all ω , ω in E and a in A , we define f Π g ( ∇ ) : E ⊗ A E → E by f Π g ( ∇ )( ω ⊗ A ω a ) = f Π g ( ∇ )( ω ⊗ C ω ) a + g ( ω ⊗ A ω ) da. Finally, ∇ is said to be compatible with g , if, as maps from E ⊗ A E to E , f Π g ( ∇ ) = dg. This allows us to give the definition of a Levi-Civita connection.
Definition 2.6.
Let ( E , d ) be a bicovariant differential calculus such that the map σ is diagonal-isable and g a pseudo-Riemannian bi-invariant metric on E . A left-covariant connection ∇ on E is called a Levi-Civita connection for the triple ( E , d, g ) if it is torsionless and compatible with g. In [3], it was shown that this suitably generalises the notion of Levi-Civita connections forbicovariant differential calculi on Hopf algebras.Then, we have the following metric-independent sufficient condition for the existence of a uniquebicovariant Levi-Civita connection.
Theorem 2.7. (Theorem 7.9 of [3] ) Suppose ( E , d ) is a bicovariant differential calculus over acosemisimple Hopf algebra A such that the map σ is diagonalisable and g be a bi-invariant pseudo-Riemannian metric. If the map ( ( P sym )) : ( E ⊗ sym C E ) ⊗ C E → E ⊗ C ( E ⊗ sym C E ) is an isomorphism, then the triple ( E , d, g ) admits a unique bicovariant Levi-Civita connection. The D ± calculi on SU q (2) and the braiding map In this section we recall briefly the definition quantum group SU q (2) and the 4 D ± calculi on SU q (2). Then we show that the map σ : E ⊗ C E → E ⊗ C E is actually diagonalisable. Ourmain reference for the details is [6].For q ∈ [ − , \ SU q (2) is the C*-algebra generated by the two elements α and γ , and theiradjoints, satisfying the following relations: α ∗ α + γ ∗ γ = 1 , αα ∗ + q γγ ∗ = 1 ,γ ∗ γ = γγ ∗ , αγ = qγα, αγ ∗ = qγ ∗ α. The comultiplication map ∆ is given by∆( α ) = α ⊗ C α − qγ ∗ ⊗ C γ, ∆( γ ) = γ ⊗ C α + α ∗ ⊗ C γ. This makes SU q (2) into a compact quantum group. We will denote the Hopf ∗ -algebra generatedby the elements α , γ by the symbol A .In [6], it is explicitly proven that there does not exists any three-dimensional bicovariant dif-ferential calculi and exactly two inequivalent four-dimensional calculi for SU q (2). We use thedescription of the two bicovariant calculi, 4D + and 4D − , as given in [6]. We will rephrase someof the notation to fit our formalism. For q ∈ ( − , \{ } , the first order differential calculi E ofboth the 4D + and 4D − calculi are bicovariant A -bimodules such that the space E of one-formsinvariant under the left coaction of A is a 4-dimensional vector space. We will denote a preferredbasis of E by { ω i } i =1 , , , . Here we have replaced the notation in [6] with ω i = Ω i .The following is the explicit description of the exterior derivative d on E for the preferred basis { ω i } i =1 mentioned above. Proposition 3.1. (Equation (5.2) of [6] ) Let d : E → Ω ( A ) be the exterior derivative of the D ± calculus. d ( ω ) = ±√ rω ∧ ω , d ( ω ) = ∓ √ rq ω ∧ ω ,d ( ω ) = ± √ rq ω ∧ ω , d ( ω ) =0 , where the upper sign stand for D + and the lower for D − , and r = 1 + q . Now we show that the map σ for SU q (2) satisfies the diagonalisability condition by givingexplicit bases for eigenspaces of σ . We will use the explicit action of σ on elements ω i ⊗ A ω j , i, j = 1 , , , Proposition 3.2.
For SU q (2) , the map σ is diagonalisable and has the minimal polynomialequation ( σ − σ + q )( σ + q − ) = 0 . Proof.
The proof of this result is by explicit listing of eigenvectors of σ for eigenvalues 1 , q , q − and by a dimension argument. Throughout we make use of the canonical identification ω i ⊗ C ω j ω i ⊗ A ω j as stated in (2). Either by directly applying σ on the following linearly independent two-tensors or from Equation(4.2) of [6], we get that the following are in the eigenspace corresponding to eigenvalue 1: ω ⊗ C ω , ω ⊗ C ω , ω ⊗ C ω + tω ⊗ C ω , ω ⊗ C ω ,ω ⊗ C ω + ω ⊗ C ω , ω ⊗ C ω + q ω ⊗ C ω ,q ω ⊗ C ω + ω ⊗ C ω , t kq √ r ω ⊗ C ω − ω ⊗ C ω − ω ⊗ C ω , t k √ r ω ⊗ C ω + ω ⊗ C ω + ω ⊗ C ω , t kq √ r ω ⊗ C ω + ω ⊗ C ω + ω ⊗ C ω . By explicit computation, the following linearly independent two-tensors are in the eigenspacecorresponding to the eigenvalue q : tqk √ r ω ⊗ C ω − q ω ⊗ C ω − tkq √ r ω ⊗ C ω + ω ⊗ C ω , − tkq √ r ω ⊗ C ω − q ω ⊗ C ω + tqk √ r ω ⊗ C ω + ω ⊗ C ω , − tk √ r ω ⊗ C ω + tk √ r ω ⊗ C ω + t k √ r ω ⊗ C ω − q ω ⊗ C ω + ω ⊗ C ω . By explicit computation, the following linearly independent two-tensors are in the eigenspacecorresponding to the eigenvalue q − : tqk √ r ω ⊗ C ω + ω ⊗ C ω − tkq √ r ω ⊗ C ω − q − ω ⊗ C ω , − tkq √ r ω ⊗ C ω + ω ⊗ C ω + tqk √ r ω ⊗ C ω − q − ω ⊗ C ω , − tk √ r ω ⊗ C ω + tk √ r ω ⊗ C ω + t k √ r ω ⊗ C ω + ω ⊗ C ω − q ω ⊗ C ω . We have thus accounted for 16 linearly independent elements of E ⊗ C E . Since E has dimension4, E ⊗ C E has dimension 16. Hence we have a basis, and in particular bases for the eigenspacedecomposition, of E ⊗ C E . Moreover, σ satisfies the minimal polynomial( σ − σ + q )( σ + q − ) = 0 . (cid:3) A bicovariant torsionless connection
In this section, using the fact that σ is diagonalisable and E ⊗ C E admits an eigenspacedecomposition, we construct a bicovariant torsionless connection on the 4 D ± calculus. Remark 4.1.
Note that since any element ρ in E can be uniquely expressed as ρ = P i ω i a i forsome a i in A (Theorem 2.1 of [8] ), a connection on E is determined by its action on the basis { ω i } i . By Proposition 3.2, we have the eigenspace decomposition E ⊗ C E = Ker( σ − id) ⊕ Ker( σ + q ) ⊕ Ker( σ + q − ) . Since Ker( ∧ ) = Ker( σ − id), we have thatKer( σ + q ) ⊕ Ker( σ + q − ) ∼ = Ω ( A ) , with the isomorphism being given by ∧| Ker( σ + q ) ⊕ Ker( σ + q − ) . Let us denote Ker( σ + q ) ⊕ Ker( σ + q − ) by F from now on. This is consistent with the notation adopted in Definition 2.2. Theorem 4.2.
Let { ω i } i be the preferred basis for the D ± calculus on SU q (2) . For i = 1 , , , ,we define ∇ ( ω i ) = − ( ∧| F ) − ◦ d ( ω i ) ∈ E ⊗ C E . Then, ∇ extends to a bicovariant torsionless connection on E . More explicitly, ∇ ( ω ) = ∓ rt k ( q + 1) (cid:0) tkq √ r ω ⊗ C ω + tqω ⊗ C ω − tqk √ r ω ⊗ C ω + tqω ⊗ C ω (cid:1) ∇ ( ω ) = ± rt k ( q + 1) (cid:0) tqk √ r ω ⊗ C ω − tqω ⊗ C ω − tkq √ r ω ⊗ C ω − tqω ⊗ C ω (cid:1) ∇ ( ω ) = ± qrtk ( q + 1) (cid:0) tk √ r ω ⊗ C ω − tk √ r ω ⊗ C ω − t k √ r ω ⊗ C ω + tqω ⊗ C ω + tqω ⊗ C ω (cid:1) ∇ ( ω ) = 0 Proof.
By the definition of ∇ , ∧ ◦ ∇ ( ω i ) = − ∧ ◦ ( ∧| F ) − ◦ d ( ω i ) = − d ( ω i ) . Therefore, for any element ρ = P i ω i a i in E , ∧ ◦ ∇ ( X i ω i a i ) = ∧ ◦ X i ( ∇ ( ω i ) a i + ω i ⊗ A a i )= − X i (cid:0) ∧ ◦ ( ∧| F ) − ◦ d ( ω i ) a i + ω i ∧ a i (cid:1) = − X i ( d ( ω i ) a i + ω i ∧ a i ) = − X i d ( ω i a i ) . Hence ∇ is a torsionless connection. The construction of ∇ is the same as that in Theorem 5.3of [3]. Hence, by that theorem, our connection ∇ is bicovariant.Now we derive ∇ explicitly on each ω i using the formulas for d ( ω i ) in Proposition 3.1.We have that d ( ω ) = ±√ rω ∧ ω . The decomposition of ω ⊗ C ω as a linear combination of thebasis eigenvectors listed in Proposition 3.2 is given by ω ⊗ C ω = 2 q ( q + 1) (cid:0) q ω ⊗ C ω + ω ⊗ C ω (cid:1) − q √ rk ( q + 1) (cid:0) t k √ r ω ⊗ C ω + ω ⊗ C ω + ω ⊗ C ω (cid:1) − √ rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω − q ω ⊗ C ω + tqk √ r ω ⊗ C ω + ω ⊗ C ω (cid:1) − √ rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω + ω ⊗ C ω + tqk √ r ω ⊗ C ω − q ω ⊗ C ω (cid:1) . Since the first two terms in the above decomposition are elements of Ker( σ − id) = Ker( ∧ ),applying ∧ on both sides, we have ω ∧ ω = ∧ (cid:16) − √ rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω − q ω ⊗ C ω + tqk √ r ω ⊗ C ω + ω ⊗ C ω (cid:1) − √ rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω + ω ⊗ C ω + tqk √ r ω ⊗ C ω − q ω ⊗ C ω (cid:1)(cid:17) , and since the last two terms in the decomposition are from F ,( ∧| F ) − ( ω ∧ ω ) = − √ rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω − q ω ⊗ C ω + tqk √ r ω ⊗ C ω + ω ⊗ C ω (cid:1) − √ rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω + ω ⊗ C ω + tqk √ r ω ⊗ C ω − q ω ⊗ C ω (cid:1) . Thus, by the construction of ∇ , we have ∇ ( ω ) = ∓ (cid:0) − rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω − q ω ⊗ C ω + tqk √ r ω ⊗ C ω + ω ⊗ C ω (cid:1) − rt k ( q + 1) (cid:0) − tkq √ r ω ⊗ C ω + ω ⊗ C ω + tqk √ r ω ⊗ C ω − q ω ⊗ C ω (cid:1)(cid:1) = ∓ rt k ( q + 1) (cid:0) tkq √ r ω ⊗ C ω + tqω ⊗ C ω − tqk √ r ω ⊗ C ω + tqω ⊗ C ω (cid:1) Proposition 3.1 also gives that d ( ω ) = ∓ √ rq ω ∧ ω , d ( ω ) = ± √ rq ω ∧ ω and d ( ω ) = 0. So,similarly, we have ω ⊗ C ω = 2( q + 1) (cid:0) ω ⊗ C ω + q ω ⊗ C ω (cid:1) − q √ rk ( q + 1) (cid:0) t k √ r ω ⊗ C ω − ω ⊗ C ω − ω ⊗ C ω (cid:1) + q √ rt k ( q + 1) (cid:0) tqk √ r ω ⊗ C ω − q ω ⊗ C ω − tkq √ r ω ⊗ C ω + ω ⊗ C ω (cid:1) + q √ rt k ( q + 1) (cid:0) tqk √ r ω ⊗ C ω + ω ⊗ C ω − tkq √ r ω ⊗ C ω − q ω ⊗ C ω (cid:1) , and hence, ∇ ( ω ) = ± rt k ( q + 1) (cid:0) tqk √ r ω ⊗ C ω − tqω ⊗ C ω − tkq √ r ω ⊗ C ω − tqω ⊗ C ω (cid:1) . Moreover, ω ⊗ C ω = 2 tq ( q + 1) (cid:0) ω ⊗ C ω + ω ⊗ C ω (cid:1) + 2 q ( q + 1) (cid:0) ω ⊗ C ω + tω ⊗ C ω (cid:1) − q √ rk ( q + 1) (cid:0) t kq √ r ω ⊗ C ω + ω ⊗ C ω + ω ⊗ C ω (cid:1) − q √ rtk ( q + 1) (cid:0) − tk √ r ω ⊗ C ω + tk √ r ω ⊗ C ω + t k √ r ω ⊗ C ω − q ω ⊗ C ω + ω ⊗ C ω (cid:1) − q √ rtk ( q + 1) (cid:0) − tk √ r ω ⊗ C ω + tk √ r ω ⊗ C ω + t k √ r ω ⊗ C ω + ω ⊗ C ω − q ω ⊗ C ω (cid:1) , and hence, ∇ ( ω ) = ± qrtk ( q + 1) (cid:0) tk √ r ω ⊗ C ω − tk √ r ω ⊗ C ω − t k √ r ω ⊗ C ω + tqω ⊗ C ω + tqω ⊗ C ω (cid:1) Lastly, since d ( ω ) = 0, ∇ ( ω ) = 0Thus, we are done with our proof. (cid:3) Existence of a unique bicovariant Levi-Civita connection
In this section, we prove that except for finitely many q ∈ ( − , \{ } , the 4 D ± calculi admita unique bicovariant Levi-Civita connection for every bi-invariant pseudo-Riemannian metric (asdefined in Definition 2.3) on E . We achieve this by verifying the hypotheses of Theorem 2.7.Recall that for the 4 D ± calculus, we had the decomposition E ⊗ C E = Ker( σ − id) ⊕ Ker( σ + q ) ⊕ Ker( σ + q − ) . We have already fixed the symbol F for Ker( σ + q ) ⊕ Ker( σ + q − ). Let us now denoteKer( σ − id) by E ⊗ sym C E . Moreover, as in Definition 2.2, we define the C -linear map ( P sym ) : E ⊗ C E → E ⊗ C E to be the idempotent with range E ⊗ sym C E and kernel F . Since, ( P sym ) is the idempotent ontothe eigenspace of σ with eigenvalue one, and with kernel the eigenspaces with eigenvalues q and q − , it is of the form (see (22) of [3]) ( P sym ) = σ + q q . σ + q − q − . (5). By Proposition 3.2, the set { ν i } i =1 forms a basis of E ⊗ sym C E , where ν i are given as follows: ν = ω ⊗ C ω , ν = ω ⊗ C ω ,ν = ω ⊗ C ω + tω ⊗ C ω , ν = ω ⊗ C ω ,ν = ω ⊗ C ω + ω ⊗ C ω , ν = ω ⊗ C ω + 1 q ω ⊗ C ω ,ν = ω ⊗ C ω + q ω ⊗ C ω , ν = ω ⊗ C ω + ω ⊗ C ω − t kq √ r ω ⊗ C ω ,ν = ω ⊗ C ω + ω ⊗ C ω + t k √ r ω ⊗ C ω , ν = ω ⊗ C ω + ω ⊗ C ω + t kq √ r ω ⊗ C ω . (6)Thus, an arbitary element of ( E ⊗ sym C E ) ⊗ C E is given by X = P ij A ij ν i ⊗ C ω j , for some complexnumbers A ij . Hence, if we show that ( ( P sym )) ( P ij A ij ν i ⊗ C ω j ) = 0 implies that A ij = 0for all i, j , then ( ( P sym )) is a one-one map from ( E ⊗ sym C E ) ⊗ C E to E ⊗ C ( E ⊗ sym C E ).However, dim(( E ⊗ sym C E ) ⊗ C E ) = dim( E ⊗ C ( E ⊗ sym C E )), so that ( ( P sym )) is a vector spaceisomorphism from ( E ⊗ sym C E ) ⊗ C E to E ⊗ C ( E ⊗ sym C E ). Suppose { A ij } ij are complex numberssuch that ( ( P sym )) ( P ij A ij ν i ⊗ C ω j ) = 0. Then, by (5), we have (cid:0) ( q ( σ ) + 1)(( σ ) + q ) (cid:1) ( X ij A ij ν i ⊗ C ω j ) = 0 . (7)We want to show that except for finitely many values of q , the above equation implies that all the A ij are equal to 0. This involves a long computation, including a series of preparatory lemmas.We will be using the explicit form of σ ( ω i ⊗ C ω j ) as given in Equation (4.1) of [6] as well as (6)to express the left hand side of (7) as a linear combination of basis elements ω i ⊗ C ω j ⊗ C ω k . Thenwe compare coefficients to derive relations among the A ij . We do not provide the details of thecomputation. However, for the purposes of book-keeping, each equation is indexed by a triplet( i, j, k ) meaning that it is obtained by collecting coefficients of the basis element ω i ⊗ C ω j ⊗ C ω k in the expansion of (cid:0) ( q ( σ ) + 1)(( σ ) + q ) (cid:1) ( P mn A mn ν m ⊗ C ω n ). Lemma 5.1.
We have the following equations: A = 0 (1,1,1) A ( q + 2) + ( tA + A + A , t kq √ r )2 q + ( A q + A t k √ r )2 q ( q −
1) = 0 (1,1,2) A ( q + 2 q −
1) + A ( k √ r ( q − q − ))+( A q + A t k √ r )2 q + A ( k √ r q − ( q − A ( − q √ rk ) + A ( q + 1) + ( A q + A t k √ r ) √ rk q + A ( q + 1) = 0 (1,1,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.2.
We have the following equations: A (2 q −
1) + ( tA + A + A , t kq √ r )( q + 1) + ( A q + A t k √ r )( − q ( q − A ( − kq √ r ( q − ) + ( A q + A )( − kq √ r ( q − ) = 0 (1,2,1) tA + A + A , t kq √ r = 0 (1,2,2)( tA + A + A , t kq √ r )( − k √ r ( q − ) + ( tA + A + A , t kq √ r )( − ( q − )+( A q + A t k √ r )2 q + A ( − k √ r ( q − ) = 0 (1,2,3)( tA + A + A , t kq √ r ) q √ rk + ( tA + A + A , t kq √ r )( q + 1)+( A q + A t k √ r )( − q ) + A ( q + 1) = 0 (1,2,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.3.
We have the following equations: A q + A k √ r ( − q ( q − q − ) )+( A q + A t k √ r )( − q + 2 q + 1) + A k √ r ( − ( q − ) = 0 (1,3,1)( tA + A + A , t kq √ r )2 q + ( tA + A + A , t kq √ r ) k √ r ( q − q + q − )+( A q + A t k √ r )( q + 2 q −
1) + A k √ r q − ( q − = 0 (1,3,2)( tA + A + A , t kq √ r )( − q + 2 q ) + A q ( q −
1) + A ( − k √ r q − ( q − )+( A q + A t k √ r )( − q + 6 q −
1) + ( A q + A t k √ r )( − k √ r q − ( q − ) = 0 (1,3,3)( tA + A + A , t kq √ r )( − q + 2 q ) + ( A q + A t k √ r ) √ rk q + A (3( q − + 2 q ) + ( A q + A t k √ r )( q + 1) = 0 (1,3,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7).. (cid:3) Lemma 5.4.
We have the following equations: A ( − q √ rk ) + A ( q + 1) + ( A q + A t k √ r ) q √ rk + A ( q + 1) = 0 (1,4,1)( tA + A + A , t kq √ r ) q √ rk + ( tA + A + A , t kq √ r )( q + 1)+( A q + A t k √ r ) √ rk ( − q ) + A ( q + 1) = 0 (1,4,2) A ( − rk q ) + ( tA + A + A , t kq √ r ) √ rk q + A ( q − A q + A t k √ r ) √ rk q ( q −
1) + ( A q + A t k √ r )( q + 1) = 0 (1,4,3) A = 0 (1,4,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7).. (cid:3) Lemma 5.5.
We have the following equations: A = 0 (2,1,1) A ( q + 2) + A (2 q ) + ( A q − + A t kq √ r )2 q ( q − A ( k √ r q − ( q − ) + ( A q − + A t kq √ r ) k √ r q ( q − q − ) = 0 (2,1,2) A ( q + 2 q −
1) + A k √ r ( q − q − )+( A q − + A t kq √ r )2 q + A k √ r q − ( q −
1) = 0 (2,1,3) A ( − √ rk q ) + A ( q + 1) + ( A q − + A t kq √ r ) √ rk q + A ( q + 1) = 0 (2,1,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.6.
We have the following equations: A (2 q −
1) + A ( q + 1) + ( A q − + A t kq √ r )( − q ( q − A ( kq √ r ( q − ) + ( A q − + A t kq √ r )( − k √ r q ( q − q − )) = 0 (2,2,1) A = 0 (2,2,2) A ( − q + 2 q + 1) + A ( − k √ r ( q − q + 1))+( A q − + A t kq √ r )2 q + A ( − k √ r ( q − ) = 0 (2,2,3) A q √ rk + A ( q + 1) + ( A q − + A t kq √ r ) √ rk ( − q ) + A ( q + 1) = 0 (2,2,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.7.
We have the following equations: A (2 q ) + A ( q + 1) + ( A q − + A t k q √ r )( − q + 2 q + 1)+ A k √ r ( − ( q − ) = 0 (2,3,1) A q + A k √ r ( q − q − ) + ( A q − + A t kq √ r )( q + 2 q − A k √ r q − ( q − = 0 (2,3,2) A q ( q −
1) + A ( − q + 2 q ) + ( A q − + A t kq √ r )( − q + 6 q − A k √ r ( − q − ( q − ) + ( A q − + A t kq √ r ) k √ r ( − q ( q − q − ) ) = 0 (2,3,3) A √ rk q + ( A q − + A t kq √ r ) √ rk q + A (3( q − + 2 q )+( A q − + A t kq √ r )( q + 1) = 0 (2,3,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.8.
We have the following equations: A √ rk ( − q ) + A ( q + 1) + ( A q − + A t kq √ r ) √ rk q + A ( q + 1) = 0 (2,4,1) A √ rk q + A ( q + 1) + ( A q − + A t kq √ r ) √ rk ( − q ) + A ( q + 1) = 0 (2,4,2) A √ rk ( − q ) + A √ rk q + ( A q − + A t kq √ r ) √ rk q ( q − A ( q −
1) + ( A q − + A t kq √ r )( q + 1) = 0 (2,4,3) A = 0 (2,4,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.9.
We have the following equations: A = 0 (3,1,1) A ( q + 2) + A q + A q ( q − A , k √ r q − ( q − + A k √ r q ( q − q − ) = 0 (3,1,2) A ( q + 2 q −
1) + A k √ r ( q − q − ) + A q + A , k √ r q − ( q −
1) = 0 (3,1,3) A √ rk ( − q − ) + A ( q + 1) + A √ rk q + A , ( q + 1) = 0 (3,1,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.10.
We have the following equations: A (2 q −
1) + A ( q + 1) + A q ( − ( q − A , k √ r ( − q − ( q − ) + A k √ r ( − q ( q − q − )) = 0 (3,2,1) A = 0 (3,2,2) A ( − q + 2 q + 1) + A k √ r ( − ( q − q + 1)) + A q + A , k √ r ( − ( q − ) = 0 (3,2,3) A √ rk q + A q +1) + A √ rk ( − q ) + A , ( q + 1) = 0 (3,2,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.11.
We have the following equations: A q + A k √ r ( − ( q − ) + A ( − q + 2 q + 1) + A , k √ r ( − ( q − ) = 0 (3,3,1) A q + A k √ r ( q − q − ) + A ( q + 2 q −
1) + A , k √ r q − ( q − = 0 (3,3,2) A ( − q + 2 q ) + A ( − q + 6 q − A , k √ r ( − q − ( q − ) + A k √ r ( − q ( q − q − ) ) = 0 (3,3,3) A √ rk q + A √ rk q + A , (3( q − + 2 q ) + A ( q + 1) = 0 (3,3,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.12.
We have the following equations: A √ rk ( − q ) + A ( q + 1) + A √ rk q + A , ( q + 1) = 0 (3,4,1) A √ rk q + A ( q + 1) + A √ rk ( − q ) + A , ( q + 1) = 0 (3,4,2) A √ rk ( − q ) + A √ rk q = 0 (3,4,3) A , = 0 (3,4,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.13.
We have the following equations: A = 0 (4,1,1) A ( q + 2) + A q + A , q ( q − A k √ r q − ( q − + A , k √ r q ( q − q − ) = 0 (4,1,2) A ( q + 2 q −
1) + A k √ r ( q − q − ) + A , q + A k √ r q − ( q −
1) = 0 (4,1,3) A √ rk ( − q ) + A ( q + 1) + A , √ rk q + A ( q + 1) = 0 (4,1,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.14.
We have the following equations: A (2 q −
1) + A ( q + 1) + A , q ( q −
1) + A k √ r ( − q − ( q − )+ A , k √ r q ( q − q − ) = 0 (4,2,1) A = 0 (4,2,2) A ( − q + 2 q + 1) + A k √ r ( − q + 2 q −
1) + A , q + A k √ r ( q − = 0 (4,2,3) A √ rk q + A ( q + 1) + A , √ rk ( − q ) + A ( q + 1) = 0 (4,2,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.15.
We have the following equations: A ( q + 2 q −
1) + A k √ r ( q − q − ) + A , ( − q + 2 q + 1)+ A k √ r ( − ( q − ) = 0 (4,3,1) A q + A k √ r ( q − q − ) + A , ( q + 2 q −
1) + A k √ r q − ( q − = 0 (4,3,2) A q ( q −
1) + A ( − q + 2 q ) + A , ( − q + 6 q −
1) + A k √ r ( − q − ( q − )+ A , k √ r ( − q ( q − q − ) ) = 0 (4,3,3) A √ rk q + A , √ rk q + A (3( q − + 2 q ) + A , ( q + 1) = 0 (4,3,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Lemma 5.16.
We have the following equations: A √ rk ( − q ) + A ( q + 1) + A , √ rk q + A ( q + 1) = 0 (4,4,1) A √ rk q + A ( q + 1) + A , √ rk ( − q ) + A ( q + 1) = 0 (4,4,2) A √ rk ( − q ) + A √ rk q + A , √ rk q ( q −
1) + A ( q −
1) + A , ( q + 1) = 0 (4,4,3) A = 0 (4,4,4) Proof.
The above equations are derived by comparing the coeffcients of ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω , ω ⊗ C ω ⊗ C ω and ω ⊗ C ω ⊗ C ω in (7). (cid:3) Theorem 5.17.
For the D ± calculi, the map ( ( P sym )) : ( E ⊗ sym C E ) ⊗ C E → E ⊗ C ( E ⊗ sym C E ) is an isomorphism except for, possibly, finitely many values of q ∈ ( − , \{ } . Hence, for eachbi-invariant pseudo-Riemannian metric g , there exists a unique bicovariant Levi-Civita connectionfor each calculus.Proof. By the discussion preceding the above series of preparatory lemmas, we need to showthat the system of equations given above admit only the trivial solution for A ij , i = 1 , . . . , j = 1 , . . . ,
4. We then proceed to solve these equations for all A ij . Note that the followingvariables are all identically zero in the above over-determined system: A (by (1,1,1)), A (by (1,4,4)), A (by (2,1,1)), A (by (2,2,2)), A (by (2,4,4)), A (by(3,1,1)), A (by (3,2,2)), A , (by (3,4,4)), A (by (4,1,1)), A (by (4,2,2)) and A (by (4,4,4)). This reduces the equations (1,3,1) and (1,4,1) to the following exact system of linear equations inthe variables A and A , with the associated matrix having determinant q ( q + 1) : A q + A k √ r ( − q ( q − q − ) ) = 0 A ( − q √ rk ) + A ( q + 1) = 0Hence the solution for the variables A and A is zero.We repeat this process for the rest of the A ij , identifying a subset of equations which has beenreduced to an exact one due to the previously solved A ij , and then concluding that the set of A ij in the current set are also solved to be 0 except for at most finitely many value of q ∈ ( − , \{ } .(2,2,3) and (2,2,4) reduce to the following system of linear equations in A and A with deter-minant ( q + 1) : A ( − q + 2 q + 1) + A ( − k √ r ( q − q + 1)) = 0 A q √ rk + A ( q + 1) = 0(4,1,3), (4,1,4) and (4,3,1) reduce to the following system of linear equations in A , A , A , with determinant 2 q − q − q + 2: A ( q + 2 q −
1) + A , q + A k √ r q − ( q −
1) = 0 A √ rk ( − q ) + A , √ rk q + A ( q + 1) = 0 A ( q + 2 q −
1) + A , ( − q + 2 q + 1) + A k √ r ( − ( q − ) = 0(4,1,2), (4,2,1), (4,3,3) and (4,4,3) reduce to the following system of linear equations in A , A , A , A , with determinant 4 q + 10 q − q − q + 26 q − q + 4: A ( q + 2) + A q + A , q ( q −
1) + A k √ r q − ( q − = 0 A (2 q −
1) + A ( q + 1) + A , q ( q −
1) + A k √ r ( − q − ( q − ) = 0 A q ( q −
1) + A ( − q + 2 q ) + A , ( − q + 6 q −
1) + A k √ r ( − q − ( q − ) = 0 A √ rk ( − q ) + A √ rk q + A , √ rk q ( q −
1) + A ( q −
1) = 0(3,4,3), (3,1,2), (3,2,1) and (3,3,3) reduce to the following system of linear equations in A , A , A , A with determinant − q ( q − ( q + 1) ( q + 1) : A √ rk ( − q ) + A √ rk q = 0 A ( q + 2) + A q + A (2 q ( q − A k √ r q ( q − q − ) = 0 A (2 q −
1) + A ( q + 1) + A ( − q ( q − A k √ r ( − q ( q − q − )) = 0 A ( − q + 2 q ) + A ( − q + 6 q −
1) + A k √ r ( − q ( q − q − ) ) = 0(2,1,3) and (2,1,4) reduce to the following system of equations in A and A with determinant q ( q + 1) : A ( q + 2 q −
1) + A k √ r ( q − q − ) = 0 A ( − √ rk q ) + A ( q + 1) = 0(1,1,2), (1,2,1), (1,3,3) and (1,3,4) reduce to a system of equations in A , A , A , A withdeterminant a non-zero polynomial in q : A ( q + 2) + tA q + A q q ( q −
1) = 0 A (2 q −
1) + tA ( q + 1) + A q ( − q ( q − tA ( − q + 2 q ) + A q ( q −
1) + A q ( − q + 6 q −
1) + A q ( − k √ r q − ( q − ) = 0 tA ( − q + 2 q ) + A q √ rk q + A q ( q + 1) = 0(2,1,2), (2,2,1), (2,3,3), (2,3,4) and (2,4,3) reduce to a system of equations in A , A , A , A , A with determinant a non-zero polynomial in q : A ( q + 2) + A (2 q ) + ( A q − + A t kq √ r )2 q ( q − A ( k √ r q − ( q − ) + ( A q − + A t kq √ r ) k √ r q ( q − q − ) = 0 A (2 q −
1) + A ( q + 1) + ( A q − + A t kq √ r )( − q ( q − A ( kq √ r ( q − ) + ( A q − + A t kq √ r )( − k √ r q ( q − q − )) = 0 A q ( q −
1) + A ( − q + 2 q ) + ( A q − + A t kq √ r )( − q + 6 q − A k √ r ( − q − ( q − ) + ( A q − + A t kq √ r ) k √ r ( − q ( q − q − ) ) = 0 A √ rk q + ( A q − + A t kq √ r ) √ rk q + A (3( q − + 2 q )+( A q − + A t kq √ r )( q + 1) = 0 A √ rk ( − q ) + A √ rk q + ( A q − + A t kq √ r ) √ rk q ( q − A ( q −
1) + ( A q − + A t kq √ r )( q + 1) = 0(3,3,2) and (3,4,2) reduce to a system of equations in A , A , with determinant q ( q + 1) : A ( q + 2 q −
1) + A , k √ r q − ( q − = 0 A √ rk ( − q ) + A , ( q + 1) = 0Finally, (4,2,3) reduces identically to A = 0.Hence we have shown that all A ij are identically equal to zero except for atmost finitely manyvalues of q ∈ ( − . ( P sym )) | ( E⊗ sym C E ) ⊗ C E is an isomorphism if q does not belongto this finite subset.Since SU q (2) is a cosemisimple Hopf algebra, and we have shown that the map σ is diagonalisable,by Theorem 2.7, for each bi-invariant pseudo-Riemannian metric g , each of the 4 D ± calculi admitsa unique bicovariant Levi-Civita connection for all but finitely many q . (cid:3) The proof of Theorem 2.7, as given in [3], involves explicitly constructing a Levi-Civita connec-tion for each triple ( E , d, g ), subject to the accompanying hypothesis. In Theorem 5.17, we haveshown that the hypothesis holds for the 4 D ± calculi and for any bi-invariant pseudo-Riemannianmetric. In this subsection, we provide the explicit construction of the Levi-Civita connection fora fixed arbitrary bi-invariant pseudo-Riemannian metric g . For this we will need to recall somedefinitions and results from [3]. Definition 5.18.
Let E and g be as above. We define a map V g : E → ( E ) ∗ , V g ( e )( f ) = g ( e ⊗ A f ) . Definition 5.19.
Let g be as above. We define a map g (2) : ( E ⊗ C E ) ⊗ C ( E ⊗ C E ) → C by the formula g (2) (( e ⊗ C e ) ⊗ C ( e ⊗ C e )) = g ( e ⊗ A g ( e ⊗ A e ) ⊗ A e ) for all e , e , e , e in E . We also define a map V g (2) : ( E ⊗ C E ) → ( E ⊗ C E ) ∗ := Hom C ( E ⊗ C E , C ) by the formula V g (2) ( e ⊗ C e )( e ⊗ C e ) = g (2) (( e ⊗ A e ) ⊗ A ( e ⊗ A e )) . Proposition 5.20. (Propositions 4.4 and 4.9 of [3] ) The map V g is one-one and hence a vectorspace isomorphism from E to ( E ) ∗ . Moreover, the map V g (2) is a vector space isomorphism from E ⊗ sym C E onto ( E ⊗ sym C E ) ∗ . Definition 5.21.
Let V and W be finite dimensional complex vector spaces. The canonical vectorspace isomorphism from V ⊗ A W ∗ to Hom C ( W, V ) will be denoted by the symbol ζ V,W . It is definedby the formula: ζ V,W ( X i v i ⊗ A φ i )( w ) = X i v i φ i ( w ) . (8) Lemma 5.22. (Lemma 3.12 of [3] ) The following maps are vector space isomorphisms: ζ E⊗ C E , E : ( E ⊗ sym C E ) ⊗ C ( E ) ∗ → Hom C ( E , E ⊗ sym C E ) ,ζ E , E⊗ C E : E ⊗ C ( E ⊗ sym C E ) ∗ → Hom C ( E ⊗ sym C E , E ) . Definition 5.23.
Given the maps ζ E⊗ sym C E , E , ζ E , E⊗ sym C E , V g , V (2) g and ( P sym )) , the map f Φ g : Hom C ( E , E ⊗ sym C E ) → Hom C ( E ⊗ sym C E , E ) is defined such that the following diagram commutes: Hom C ( E , E ⊗ sym C E ) ( E ⊗ sym C E ) ⊗ C ( E ) ∗ ( E ⊗ sym C E ) ⊗ C E Hom C ( E ⊗ sym C E , E ) E ⊗ C ( E ⊗ sym C E ) ∗ E ⊗ C ( E ⊗ sym C E ) ζ − E⊗ C E , E f Φ g id ⊗ C V − g ( ( P sym )) ζ − E , E⊗ C E id ⊗ C V g (2) Remark 5.24.
In Theorem 5.17, we proved that the map ( ( P sym )) : ( E ⊗ sym C E ) ⊗ C E → E ⊗ C ( E ⊗ sym C E ) is an isomorphism. By Proposition 5.20 and Lemma 5.22, the remaininglegs of the above commutative diagram are isomorphisms. Hence, f Φ g : Hom C ( E , E ⊗ sym C E ) → Hom C ( E ⊗ sym C E , E ) is also an isomorphism. Theorem 5.25.
For a fixed bi-invariant pseudo-Riemannian metric g , the bicovariant Levi-Civitaconnection ∇ is defined on elements of E by ∇ = ∇ + f Φ g − ( dg − f Π g ( ∇ )) , (9) where ∇ is the bicovariant torsionless connection constructed in Theorem 4.2. Here, ∇ and g are considered as restrictions on E and E ⊗ C E respectively.Proof. Let us recall from Remark 4.1, that it is sufficient to define a connection on E to defineit on the whole of E . Next, by (4), f Π g ( ∇ ) is a well-defined map in Hom C ( E ⊗ C E , E ). Themap dg is a well-defined map in Hom C ( E ⊗ C E , E ). (Indeed it is the zero-map, since g maps E ⊗ C E to C , and d maps C to 0. That we write it at all in the formula of ∇ is because of how itappears in the proof of Proposition 7.3 of [3].) We have already remarked that f Φ g is a well-definedisomoprhism from Hom C ( E , E ⊗ sym C E ) to Hom C ( E ⊗ C E , E ). Hence, the right-hand side of (9)is a well-defined map in Hom C ( E , E ⊗ C E ). That it defines the unique bicovariant Levi-Civitaconnection on E follows from the proofs of Proposition 7.3 and Theorem 7.8 of [3], and we leaveout the details. (cid:3) References [1] E.J. Beggs and S. Majid: Quantum Riemannian geometry, Grundlehren der mathematischen Wissenschaften,Springer Verlag, 2019[2] J. Bhowmick and S. Mukhopadhyay: Pseudo-Riemannian metrics on bicovariant bimodules, arxiv:1911.06036v1.[3] J. Bhowmick and S. Mukhopadhyay: Covariant connections on bicovariant differential calculus, arxiv:1912.04689v1.[4] A. Connes: Noncommutative geometry. Academic Press, Inc., San Diego, CA, 1994.[5] I. Heckenberger and K. Schm¨udgen : Levi-Civita Connections on the Quantum Groups SL q ( N ) O q ( N ) andSp q ( N ), Comm. Math. Phys. , 1997 177–196.[6] Piotr Stachura: Bicovariant differential calculi on S µ U(2), Lett. Math. Phys., , 1992, 3, 175–188.[7] S.L. Woronowicz: Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res.Inst. Math. Sci., Kyoto University. Research Institute for Mathematical Sciences. Publications, , 1987, 1,117–181.[8] S.L. Woronowicz: Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math.Phys.,122