Atiyah sequences of braided Lie algebras and their splittings

Associated with an equivariant noncommutative principal bundle we give an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. For the $SU(2)$-principal bundle over the sphere $S^{4}_\theta$ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra $so_\theta(5,1)$ yields a five parameter family of splittings. On the principal $SO_\theta(2n,\mathbb{R})$-bundle of orthonormal frames over the sphere $S^{2n}_\theta$, the splitting of the sequence leads to the Levi-Civita connection for the `round' metric on the $S^{2n}_\theta$. The corresponding Riemannian geometry of $S^{2n}_\theta$ is worked out.


Introduction
The Atiyah sequence of a principal bundle over a manifold M is an important tool in the study of the geometry of Yang-Mills theories [7].It is given as a sequence of vector bundles over M with Lie algebra structures on the corresponding modules of sections, resulting in a short exact sequence of (infinite dimensional) Lie algebras, 0 → Γad(P ) → X G (P ) → X(M) → 0 (1.1) with principal G-bundle P → M.Here X(M) are the vector fields over M (the section of the tangent bundle T M), while X G (P ) consists of G-invariant vector fields on P (the G-invariant sections of the tangent bundle T P along the fibres of P ), and Γad(P ) its Lie subalgebra made of vertical ones (with ad(P ) the bundle associated to P via the adjoint representation of G on its Lie algebra).
A splitting of the sequence corresponds to a G-invariant direct sum decomposition of the tangent space T P in horizontal and vertical parts, that is, to a connection on P → M. The space Γad(P ) is the (infinite dimensional) Lie algebra of the gauge group of the principal bundle P .Its elements are infinitesimal gauge transformations, [7, §3].
Exact sequences of vector bundles (named after Atiyah) and their splittings were introduced in [6] in the complex analytic context, motivated by the study of connections and obstructions to their existence.Later versions are in [20], [17], with a braided case coming from a Z 2 -grading in [14].The sequence can also be seen as a Lie algebroid [18].
In §3 of the present paper, we study a quantum version 0 → g → P → T → 0 of the Lie algebroid (1.1) in the context of braided Lie algebras (of derivations) associated with a triangular Hopf algebra (K, R). 1 Each term in the sequence is a braided Lie-Rinehart pair for a braided commutative algebra B. Given any such an exact sequence, we define a connection to be a splitting of the underlying short exact sequence of vector spaces.The splitting does not need to be a braided Lie algebra morphism: the curvature is defined as the g-valued braided two-form on T that measures the extent to which the connection fails to be such.In §3.4 elements of g are explicitly seen as infinitesimal gauge transformations which act on the set of connections (splittings) of the sequence.Having a connection one can define the covariant derivative of g-valued braided forms on P .The connection and the curvature satisfy a structure equation and a Bianchi identity (Proposition 3.3).In this braided context, the covariant derivative of the curvature does not vanish in general (in agreement with the result in [12]) and [11]), while it does for a K-equivariant connection.
In §4 we consider K-equivariant noncommutative principal bundles, that is K-equivariant Hopf-Galois extensions B ⊂ A, where (K, R) is a triangular Hopf algebra.We have then a corresponding sequence of braided Lie algebras of derivations as in (4.2), → 0 which, when exact, is a noncommutative version of the Atiyah sequence (1.1).The braided Lie algebra of infinitesimal gauge transformations aut R B (A) of a Hopf-Galois extension consists of vertical braided derivations of the algebra A; it was introduced and studied in [4,5].The theory is exemplified with the construction in §4.1 of the Atiyah sequence of braided Lie algebras for the instanton bundle on the noncommutative sphere S 4  θ and a five parameter family of splittings (connections) of it.In a C * -algebra context an Atiyah sequence for noncommutative principal bundles was given in [19].
The last part of the paper in §5 is dedicated to the example of the noncommutative principal SO θ (2n, R)-bundle SO θ (2n + 1, R) → S 2n θ on the noncommutative sphere S 2n θ .This is the noncommutative orthogonal frame bundle through an identification of braided derivations of O(S 2n θ ) as sections of the bundle associated to the principal bundle via the fundamental corepresentation of O(SO θ (2n, R)) on the algebra O(R 2n θ ).The corresponding Atiyah sequence is constructed and an equivariant splitting determined.This leads to a novel and very direct construction of the Levi-Civita connection on S 2n θ ; the connection is explicitly presented and globally defined using global coordinate functions on S 2n θ .Indeed the principal equivariant connection induces a covariant derivative on the associated tangent bundle which is torsion free and compatible with the 'round' metric.We then work out the corresponding Riemannian geometry of S 2n θ .The latter is an Einstein space (the Ricci tensor being proportional to the metric, see (5.27)) and a space form (the scalar curvature being constant, see (5.28)).
The study of Levi-Civita connections in noncommutative geometry is a very active field of research.In the braided context uniqueness and existence of Levi-Civita connections has been actively pursued, especially via Koszul formulae, see [1] and the different contributions there referred to.Our new contribution to this subject -in this paper on connections as splitting of Atiyah sequences-is the explicit and globally defined expression of the Levi-Civita connection on S 2n θ given in (5.23).

Algebraic preliminaries
We work in the category of k-modules with k a commutative field.All algebras are unital and associative and morphisms of algebras preserve the unit.All coalgebras satisfy the corresponding dual conditions.We use standard terminologies and notations in Hopf algebra theory.For H a bialgebra, we call H-equivariant a map of H-modules or Hcomodules.
Recall that a bialgebra (or Hopf algebra) K is quasitriangular if there exists an invertible element R ∈ K ⊗ K (the universal R-matrix of K) with respect to which the coproduct ∆ of K is quasi-cocommutative for each k ∈ K, with ∆ cop := τ • ∆, τ the flip map, and R ∈ K ⊗ K the inverse of R, RR = RR = 1 ⊗ 1. Moreover R is required to satisfy, We write R := R α ⊗R α with an implicit sum.Then R 12 = R α ⊗R α ⊗1, and similarly for R 23 and R 13 .From conditions (2.1) and (2.2) it follows that R satisfies the quantum Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 .The R-matrix of a quasitriangular bialgebra (K, R) is unital: (ε ⊗ id)R = 1 = (id ⊗ ε)R, with ε the counit of K.When K is a Hopf algebra, the quasitriangularity implies that its antipode S is invertible and satisfies The Hopf algebra K is said to be triangular when R = R 21 , with R 21 = τ (R) = R α ⊗ R α .
2.1.Braided Lie algebras of derivations.For the purpose of the present paper we only need to consider braided Lie algebras associated with a triangular Hopf algebra (K, R).A K-braided Lie algebra is a K-module g, with action ⊲: K ⊗ g → g, which is endowed with a bilinear map (a bracket) the coproduct of K, and satisfies the conditions for all u, v, w ∈ g, k ∈ K.A morphism of braided Lie algebras is a morphism of Kmodules that in addition commutes with the brackets.Any K-module algebra A is a K-braided Lie algebra for the braided commutator Also, for A a K-module algebra, the K-module algebra (Hom(A, A), ⊲ Hom(A,A) ) of linear maps from A to A with action is a braided Lie algebra with the braided commutator; here S is the antipode of K.An important example of a braided Lie algebra associated to a K-module algebra is that of braided derivations, [4, §5.2].A braided derivation is any Y ∈ Hom(A, A) which satisfies for any a, a ′ in A. We denote Der R (A) the k-module of braided derivations of A (to lighten notation we often drop the label R).It is a K-submodule of Hom(A, A), with action given by the restriction of ⊲ Hom(A,A) and a braided Lie subalgebra of Hom(A, A) with braided commutator Remark 2.1.When K is not finite dimensional over k the matrix R is in general not an element of K ⊗ K but rather belongs to a suitable topological completion of the tensor product algebra.In the examples of the present paper this fact does not constitute a problem since R acts diagonally in the fundamental representation and we consider representations that are algebraic direct sums of this one.Representation-wise we hence consider the braided monoidal category of K-modules that are algebraic direct sums of finite dimensional representations of K-modules.

2.2.
Brackets of Lie algebra valued maps.Let N be a K-module and g a braided Lie algebra.The space of g-valued multilinear maps η : N ⊗ . . .⊗ N → g which are braided antisymmetric in their arguments (g-valued forms on N), can be given the structure of a (super) braided Lie algebra.In the present paper we just need the bracket between one-forms and between a one-form and a two-form.
Given any two linear maps η, φ : N → g, their bracket is defined by for Y, Y ′ ∈ N. The action on maps is the adjoint one in (2.4).By definition, the bracket is braided antisymmetric in the arguments, η, φ (Y, In particular, when φ = η the formula (2.8) can be written as For a one-form η and a two-form Φ one defines (2.11) with b.c.p. standing for braided cyclic permutations of elements (X, Y, Z) in N. The braided antisymmetry of the two-form Φ implies the same property for both expressions above.A short computation shows that From this one also computes that for X, Y, Z ∈ N, and in parallel with (2.9).
The bracket satisfies a braided Jacobi identity.For one-forms this reads as in the following proposition.Proposition 2.2.For one-forms ω, η, ϕ, we have the identity (2.13) Proof.We first compute separately the three terms in (2.13): from (2.10) and (2.8), Using the braided cyclic permutation we substitute the second term in (2.15) with the following one: which in turn, using (2.2), can be rewritten as Next, using the Yang-Baxter equation on the indices ν, γ, α this reduces to Finally, in the expression one finds that this term and the two positive ones in (2.14) and (2.16) sum up to zero due to the Jacobi identity for derivations.A similar computation shows that the other three terms sum up to zero as well, thus establishing (2.13).

2.3.
Brackets of Lie algebra valued forms on bimodules.For a subalgebra B ⊆ A, the space Hom(A, A) is a left B-module via the left multiplication by elements in B.
In general this is not the case for Der(A).A sufficient condition for that is the quasi- We write Y • b to distinguish the right B-module structure from the evaluation of a derivation on an element in B. From the definition of the bimodule structure it follows compatibility with the K-action: Let N and N ′ be K-modules and B-bimodules as above.The space of right B-linear maps Hom B (N, N ′ ) is a K-module with action )) and a B-bimodule with actions (bη With g a braided Lie algebra over K, the definition of the bracket in §2.2 can be repeated for g-valued forms on N, i.e. g-valued right B-linear maps η : N ⊗ B . ..⊗B N → g which are braided antisymmetric in their arguments.As we shall see this results into a (super) braided Lie-Rinehart algebra.

The Atiyah sequence of braided Lie algebras
The classical Atiyah sequence [6,7] is a sequence of vector bundles over a base space M with Lie algebra structures on the corresponding modules of sections, this resulting in a short exact sequence of (infinite dimensional) Lie algebras.In this paper we generalise the construction to a sequence of braided Lie algebras with braiding implemented by the triangular structure of a symmetry Hopf algebra K.In §4 we shall describe the basic example of this setting, the sequence of braided Lie algebras of a K-equivariant Hopf-Galois extension B = A coH ⊂ A, with structure Hopf algebra H.
3.1.The Atiyah sequence and the Lie-Rinehart structures.Let (K, R) be a triangular Hopf algebra and consider an exact sequence of K-braided Lie algebras, with ı and π braided Lie algebra morphisms.We identify g with its image ı(g) in P , while we write π(Y ) = Y π for Y ∈ P , as the image in T via the projection π.By construction g is an ideal in P for the braided commutator: [Y, V ] ∈ g, when Y ∈ P , V ∈ g.We further take g, P and T to be B-bimodules with structures as in (2.18) and (2.19) for B a quasi-commutative algebra (that is (2.17) holds for A = B) and ι, π to be right B-module morphisms (and hence left ones).
Finally, we take (B, T ) to be a braided Lie-Rinehart pair: the algebra B is a T -module with elements of T acting as braided derivations of B, and there is compatibility with the B-module structure of T , (which also fixes [bX, X ′ • b ′ ], cf.(2.20)).The pair (B, P ) is as well taken to be a braided Lie-Rinehart pair (and so is (B, g) for the trivial action): elements of P act as braided derivations of B via the map π, Y (b) = Y π (b), for Y ∈ P and b ∈ B. The subalgebra g acts trivially on B: elements of g are 'vertical'.For simplicity we take T = Der R (B).
The sequence (3.1) is a braided Lie algebroid with anchor π.

Splittings, connections and curvatures.
A connection on the sequence (3.1) is a splitting of it, that is a right B-module map which is a section of π, ρ :

and recalling the adjoint action property
From the B-bimodule structure of T one also has Remark 3.1.In accordance with the K-adjoint action, k ⊲ (ρ(X)) = (k (1) ⊲ ρ)(k (2) ⊲ X), the connection ρ "acts from the left".Therefore it is natural to take it to be right B-linear.
The corresponding vertical projection, the retract of ı, is the right B-module map Clearly ω(ρ(X)) = 0, for each X ∈ T .We denote the horizontal projection.By definition ω + h = id.
In general, ρ (and so ω) is not a braided Lie algebra morphism.The extent to which it fails to be such is measured by the (basic) curvature for X, X ′ ∈ T .From the K-equivariance of π and π • ρ = id T it follows that π • Ω = 0, and so Ω is g-valued.Also, from ω • ρ = 0 it follows that A braided antisymmetric right B-linear map from T ⊗ B T to g is called a g-valued braided two-form on T .Proposition 3.2.The curvature Ω is a g-valued braided two-form on T .
Proof.We first show that Ω is well-defined on T ⊗ B T , that is Ω(X, b X ′ ) = Ω(X • b, X ′ ), for b ∈ B and X, X ′ ∈ T .Indeed, using (3.2) and (2.19), we compute Then, the right B-linearity of ρ yields Next, using the expression in (2.9) for the bracket: using (3.4) in the last but one equality.While, using using (3.5), in the last but one equality.Now, a direct computation shows that With this, comparing (3.10) and (3.11), we obtain which, when compared with (3.9), amounts to (3.12) On the other hand, recalling that the K-action defined in (2.21) closes on right B-linear maps, A comparison with (3.12) yields Ω(X, Finally, both terms in (3.8) are braided antisymmetric and hence so is the curvature: The curvature can be given as g-valued braided two-form on P (the spatial curvature): This turns out to be basic, that is ) one also computes: This expression can be read as a structure equation: Here the g-valued two-form ω, ω on P is defined as in (2.8).Also, given a g-valued one-form η on P , its exterior derivative dη is the g-valued two-form on P defined by for Y, Y ′ ∈ P .The above is well-defined since g, as mentioned, is an ideal in P for the braided commutator and braided antisymmetric by construction.
One could construct an exterior algebra of g-valued forms on P by extending the definition of the exterior derivative d to a form of any degree.For the sake of the present paper we just need it on one-and two-forms.For Φ a g-valued two-form on P , its exterior derivative is given as for X, Y, Z ∈ P , and b.c.p. standing for braided cyclic permutations of (X, Y, Z).The result is a g-valued three-form on P .Using this definition and (3.15), a lengthy and intricate computation that uses Jacobi identity and Yang-Baxter equation shows d 2 = 0.
3.3.The covariant derivative.Having a connection one can define the covariant derivative of (in particular) g-valued braided forms on P .This uses the horizontal projection h in (3.7).For a one-form η we define while for a two-form Φ its covariant derivative is defined as using (2.12) for the second equality.These definitions can be generalised to forms of any degree, obtaining braided versions of the classical formulas [8, eq.(I.1)], [13, eq. ( 19)].From (3.13) and (3.8) the curvature is written in terms of the horizontal projection as Then one shows the following proposition.
Proposition 3.3.For the connection and the curvature there is a structure equation and a Bianchi identity
Proof.From the structure equation (3.14) and (3.15), and from the definition (3.17) of the covariant derivative, we have Then, for X, Y, Z ∈ P , from definitions (3.18) and (3.19) one computes vanishes by Jacobi identity while the first and fourth terms cancel each other.The remaining one, the second term, then gives The second equality follows from the Jacobi identity (2.13).When the connection is equivariant, k ⊲ h = ε(k)h and the right-hand side of (3.20) vanishes by Jacobi identity, Then the relation in (3.20) is the usual Bianchi identity.
An easy computation also gives that the covariant derivative of the curvature as in (3.18) can be written as (3.20) compares with a similar one in [11, § 5.4].In the (dual) notation of [12] it says that our connection is regular but not necessarily multiplicative.
3.4.The space of connections and the gauge transformations.The space C(T, g) of connections on the sequence (3.1) is an affine space modelled on the linear space of right B-module maps η : T → g (the g-valued one-forms on B).Indeed, given a connection ρ : T → P and such a map η : T → g, one has In the examples of the present paper we shall use this decomposition with ρ a K-equivariant connection.
The space of connections C(T, g) is a subset of the K-module Hom B (T, P ).The latter carries an action of the braided Lie algebra P (cf.[4, §5.3]).
Proposition 3.5.The map δ : P ⊗ Hom B (T, P ) → Hom B (T, P ) given by is an action of the K-braided Lie algebra P , that is Proof.Formula (3.22) follows from the quasi-cocommutativity (2.1) and the explicit expression (2.21) for the K-action on morphisms.Then, using formula (3.22) we compute This also yields These two expressions have two terms in common so their difference is given by using Yang-Baxter equation for the second summand.Finally Jacobi identity gives and coincides with the right hand side of (3.23).
The braided Lie algebra action of P on Hom B (T, P ) gives rise to a map Indeed, from π • ρ = id T , it follows that π • (δ Y ρ) = 0 and so δ Y ρ : T → g.Proposition (3.5) can be generalised.Let M be a right B-module with a compatible P -action ⊲ P .Then there is a P -action δ : P ⊗ Hom B (M, P ) → Hom B (M, P ), (3.24) Since the P -action on T ⊗ T is by braided derivations, when Φ is a P -valued two-form on T , the above becomes Proposition 3.6.The variation δ Y Ω of the curvature of the connection ρ for the action of an element Y ∈ P , as defined in (3.25), explicitly reads Proof.This follows from linearity of the action δ Y and from its braided derivation rule.
In analogy with the classical case, an element V ∈ g acts on a connection and on the corresponding curvature as an infinitesimal gauge transformation: Corollary 3.7.The variation (3.21) of a connection ρ for the action of a vertical element V ∈ g is given by while the variation (3.25) of the curvature is Then in view of (3.23), g is the braided Lie algebra of infinitesimal gauge transformations.The universal enveloping algebra U(g) is the braided Hopf algebra of such transformations [4, §5.3].Remark 3.8.Let Ω ′ be the curvature of the transformed connection ρ ′ = ρ + δ Y ρ, for Y ∈ P .Since the action δ Y is a braided derivation, the variation δ Y Ω differs from the first order term in the difference Ω ′ − Ω (which by construction is a derivation): When the splitting ρ is K-equivariant, k ⊲ ρ = ε(k)ρ, the variations of the connection and of the curvature reduce to Moreover the extra term in the right-hand side of (3.26) vanishes.

Atiyah sequences for Hopf-Galois extensions
Recall that an algebra A is a right H-comodule algebra for a Hopf algebra H if it carries a right coaction δ : A → A ⊗ H which is a morphism of algebras.We write δ(a) = a (0) ⊗ a (1) in Sweedler notation with an implicit sum.The subspace of coinvariants The algebra extension B ⊆ A is called an H-Galois extension if this map is bijective.
In the present paper we deal with H-Galois extensions which are K-equivariant.That is A also carries a left K-action ⊲ : K ⊗ A → A, for K a Hopf algebra, that commutes with the right H-coaction, δ Given a K-equivariant Hopf-Galois extension B = A coH ⊆ A, with triangular Hopf algebra (K, R), the Lie algebra Der(A) of K-braided derivations of A has two distinguished Lie subalgebras.Firstly, the Lie subalgebra of braided derivations that are H-equivariant (that is H-comodule maps), and then its Lie subalgebra of vertical derivations

Each derivation in Der R
M H (A), being H-equivariant, restricts to a derivation on the subalgebra of coinvariant elements B = A coH .Thus, associated to B = A coH ⊆ A, there is the sequence of braided Lie algebras aut R is a version of the Atiyah sequence of a (commutative) principal fibre bundle.When the K-module algebra B is quasi-central in A, see (2.17), the braided Lie algebras in the sequence are also B-bimodules with module structures as in (2.18) and (2.19).The above is then a sequence of braided Lie-Rinehart pairs as in §3.1.
As studied in the previous section, a connection on the bundle can be given as an Hequivariant splitting of the sequence, associating to a derivation X in Der R (B) a unique horizontal derivation in Der R M H (A) projecting onto X.The curvature of the connection measures the extend to which this map fails to be a braided Lie algebra morphism.
An example of this construction with the study of moduli spaces of connections is given in the next two subsections.In §5 we present a sequence for the orthogonal frame bundle over the sphere S 2n θ with a splitting which leads to the Levi-Civita connection.
4.1.The sequence for the SU(2)-bundle over the sphere S 4 θ .This section is devoted to the Atiyah sequence of braided Lie algebras associated with the O(SU(2)) Hopf-Galois extension O(S 4 θ ) ⊂ O(S 7 θ ) constructed in [15] and related connections.
Let θ ∈ R. The * -algebra O(S 7 θ ) has generators z r , z * r , r = 1, 2, 3, 4 with commutation relations determined by the action of a 2-torus.For K the Hopf algebra generated by two commuting elements H 1 , H 2 , the generators z r are eigenfunctions for the action of ), for r = 1, 2, 3, 4, and so are their * -conjugated z * r with eigenvalues −µ r .Then, the commutation relations among the z r read for r, s = ±1, ±2, ±3, ±4 and z −r := z * r .In addition, the generators satisfy a sphere relation z 1 2))-comodule algebra with right coaction which is defined on the algebra generators as Here .
⊗ denotes the composition of the tensor product ⊗ with matrix multiplication.As usual the coaction is extended to the whole O(S 7 θ ) as a * -algebra morphism.The subalgebra B = O(S 7 θ ) coO(SU (2)) of coinvariant elements for the coaction δ is generated by the entries of the matrix p := u• θ u † and is identified with the algebra O(S 4 θ ) of coordinate functions on the 4-sphere S 4 θ in [10].The K-action on O(S 7 θ ) commutes with the O(SU(2))-coaction (4.4) and the sphere O(S 4 θ ) carries the induced action of K. We denote the generators of O(S 4 θ ) by b µ , with eigenvalues µ = (µ 1 , µ 2 ) = (0, 0), (±1, 0), (0, ±1) of the action of H 1 , H 2 .Explicitly, and satisfy the sphere relation From the above, one works out [5, Eq. (4.27)] the following mixed relations: with braiding implemented by the triangular structure of K, which, as in Remark These are extended to the whole algebra O(S 4 θ ) as braided derivations: It is easy to verify that T µ ( showing that the T µ 's are well-defined as derivations of O(S 4 θ ).Moreover, using the sphere relation, one also sees that they are not independent but rather satisfy the relation for the left module structure as in (2.18).The action of H 1 , H 2 , the generators of K, on the generators b µ lifts to the adjoint action (2.6) on the derivations T µ : The derivations T µ are the simplest combinations of the five basis derivations of O(R 5 θ ) that preserve the sphere.In the classical limit θ = 0, the derivations T µ reduce to Proof.We compute the braided commutator of two generators for µ, ν, τ = (0, 0), (±1, 0), (0, ±1).For the last equality we have used that ν * ∧ ν = 0 implies T µ (b ν ) = λ 2µ∧ν T ν (b µ ) for each µ, ν.Then, identity (4.10) is verified.
We conclude this subsection with two Lemmas that we use later on.We first observe that formula (4.10) allows us to write the derivations T µ in terms of their commutators: Lemma 4.2.The generators T µ can be expressed in terms of their commutators as Proof.From the expression (4.10) of the commutators, we compute where we used the sphere relation for the second equality and (4.8) for the last one.
For the braided antisymmetric commutators we introduce the notation The action (4.9) of H 1 and H 2 on the derivations T µ implies From (4.10), on the generators of O(S 4 θ ) these derivations are given by Proposition 4.3.The derivations L π µ,ν give a faithful representation of the braided Lie algebra so θ (5), that is: Proof.From (4.12) a direct computation yields: 13) and the one with indices exchanged The braided commutator is just given by the terms in (4.13) minus the terms in (4.14) with an extra factor coming from the braiding, that is λ 2(µ+ν)∧(τ +σ) .The result is obtained by pairing the terms with the same δ's.For instance, the one coming from µ,σ (b α ) using that τ = −ν to simplify the exponent of λ.The other terms behave similarly.
In the classical limit, the derivations L π µ,ν give the representation [10] of so(5), thus the name so θ (5) for the braided Lie algebra generated by the deformed generators L π µ,ν .
In the limit θ = 0, the sequence of Lie algebras in (4.15) is the Atiyah sequence of the SU(2)-principal Hopf bundle S 7 → S 4 , with aut R O(S 4 ) (O(S 7 )) the (infinite dimensional classical) Lie algebra of infinitesimal gauge transformations.
Remark 4.4.The braided Lie algebras in the sequence (4.15) were obtained in [5] from a twist deformation quantization of the corresponding Lie algebras of the SU(2)-classical fibration S 7 → S 4 .The braided Lie algebra Der R M H (O(S 7 θ )) was generated by derivations H j , j = 1, 2, and E r for r the roots of the Lie algebra so (5).These are related to the generators L µ,ν by with the previous lexicographic order on the µ's.Moreover, for their restrictions, where ϕ 00 := 1, Moreover, each of the non vanishing terms in parenthesis in (4.19) is just one of the derivations K's and W 's in [5,Prop. 4.14].Then, the ten vertical derivations Y µ,ν are those associated to the quantization Y (11) ∝ L (0,1)+(1,0) of the operator Y 11 , the highest weight vector of the representation [10].Proof.From (4.20), it is immediate to see that π • ρ = id.
From Remark 3.1, this connection is left O(S 4 θ )-linear as well since it is equivariant: Proposition 4.6.The connection ρ is invariant under the action (by braided commutators) of the braided Lie algebra so θ (5) of O(SO θ (4))-equivariant derivations on O(SO θ (5)): On the other hand, using (4.12) and the braided commutator in (4.18), we compute As mentioned, the splitting gives an equivariant direct sum decomposition of in horizontal and vertical parts.For the corresponding vertical projection, we have the following.
The last sum is actually limited to τ = µ, ν since for these two choices for the index τ the term in parenthesis vanishes.Then, the ten vertical derivations ω(L µ,ν ) coincide with the generators Y µ,ν in (4.19) of the braided Lie algebra aut R The connection ρ assigns to every derivation X on O(S 4 θ ) a unique horizontal derivation on O(S 7 θ ) that projects on X via π.As mentioned, this needs not be a braided Lie algebra morphism in general: the commutators of horizontal vector fields need not be horizontal.A measure of this failure is the curvature (3.8).
where the Y µ,ν are the derivations defined in (4.19).
Proof.We show that ρ(T µ ), ρ(T ν ) = L µ,ν .(4.24) Using the module structure, we compute θ ), and thus from (4.12) the first two terms in the above expression are equal to As for the third term, using the braided commutator in (4.18), one gets Putting the three terms together one then arrives at For the last term, the O(S 4 θ )-linearity of the horizontal lift and the identity (4.8) give θ ) 4 ) of the vector bundle associated to the fundamental representation of SU(2) [15].When translated to the principal bundle this corresponds to an su(2)-valued one-form ω : Der R M H (O(S 7 θ )) → su (2).Explicitly The computations for the other horizontal derivations are similar.
Table 1.Horizontal vector fields on O(S 7 θ ) 4.2.The conformal algebra and its action on connections.The 'basic' connection ρ we have described in the previous sections is an instanton in the sense that the curvature of the connection one-form in (4.25) is anti-selfdual [10,9].An infinitesimal action of twisted conformal transformations yields a five parameter family of instantons [16].We obtain here a five parameter family of splittings of the Atiyah sequence (4.15) associated with the braided conformal Lie algebra so θ (5, 1).
We know from Proposition 4.6 that the basic connection in (4.21) is invariant under the action of the braided Lie algebra generated by the elements L µ,ν .The remaining five generators T µ will give new, not gauge equivalent connections ρ µ = ρ + δ µ ρ, with While the connection ρ is K-equivariant the connections ρ µ are not: due to (4.26) one finds H j ⊲ ρ µ = µ j δ µ ρ.On the generators T ν one computes, using (4.21) and (4.24), The last equality follows from ω(T ν ) = T µ − ρ(T π µ ) = T µ − ρ(T µ ) and τ b * τ T τ = 0. Lemma 4.9.On generators of Der R (O(S 4 θ )) the variation of the curvature is given by Proof.From the general theory, see (3.27), the variation of the curvature is For the last two summands, using formula (4.10) for the commutator of the T µ one gets: and similarly For the first summand, from Ω(T µ , T ν ) = −ω(L µ,ν ) = −Y µ,ν and their explicit expression in (4.19), we compute where we used formulas for the third equality, and (4.21) for the fourth equality.Finally, recalling that γ b * γ T γ = 0, we have Finally, using (4.23), we obtain hence concluding the proof.

5.
The Riemannian geometry of the noncommutative sphere θ .In analogy with the classical case this is thought of as the bundle of orthonormal frames on S 2n θ via the identification of derivations Der R (O(S 2n θ )) with sections of the associated bundle for the fundamental corepresentation of the Hopf algebra O(SO θ (2n, R)) on the algebra O(R 2n θ ).An equivariant splitting of the Atiyah sequence of the frame bundle then leads to an explicit and globally defined expression for the Levi-Civita connection of the 'round' metric on O(S 2n θ ).The corresponding curvature and Ricci tensors and the scalar curvature are then computed.
The Hopf algebra O(SO θ (2n + 1, R)) is the noncommutative algebra generated by the entries of a matrix N = (n IJ ), I, J = 1, . . .2n + 1, modulo the orthogonality conditions and det(N) = 1 for the quantum determinant.There the matrix Q is given as In slight more generality, we may assume the matrix Q to be symmetric and have a single entry equal to 1 in each row.For fixed index J, we set J ′ to be the unique index such that Q JJ ′ = 1.Clearly (J ′ ) ′ = J.We denote by 0 the unique index J ∈ {1, . . ., 2n + 1} such that Q JJ = 1 and by I the subset consisting of the n indices J such that J < J ′ .For the choice of Q in (5.2) one has J ′ = n + J for J ≤ n, 0 = 2n + 1 and I = {1, . . ., n}.
Consider the n functionals t j : O(SO θ (2n + 1, R)) → C, j ∈ I, defined by t j (n KL ) = δ jK δ jL − δ jK ′ δ jL ′ and t j (aa ′ ) = t j (a)ε(a ′ ) + ε(a)t j (a ′ ), for any a, a ′ ∈ O(SO θ (2n + 1, R)).They commute under the convolution product and give the cotriangular structure of O(SO θ (2n + 1, R)) as R = e 2πiθ jk t j ⊗t k (sum over j, k ∈ I is understood).They generate the Hopf algebra U(t n ), the universal enveloping algebra of the Lie algebra t n of the (commutative) n-torus.This Hopf algebra has triangular structure R = e 2πiθ jk t j ⊗t k and therefore the Hopf algebra The functionals t j are lifted to the derivations H j = (t j ⊗ id) • ∆ and H j = (id ⊗ t j ) • ∆ of O(SO θ (2n + 1, R)), which are right and left O(SO θ (2n + 1, R))-invariant, respectively.They define an action of the Hopf algebra given explicitly by It is not difficult to see that cotriangularity of O(SO θ (2n + 1, R)) is then equivalent to quasi-commutativity of the (K, R)-module algebra O(SO θ (2n + 1, R)) (cf.also [4,Ex. 5.10] and are extended to the whole algebra as braided derivations.They are braided antisymmetric, L IJ = −λ IJ L JI , and in particular L jj ′ = H j .The K-action on the derivations L IJ is the lift of the action on O(SO θ (2n + 1, R)) given in (5.6).Since these derivations are right O(SO θ (2n + 1, R))-invariant they commute with the left invariant ones H j and the action of the latter is trivial.As for the action of H j one finds It follows that for these generators the braided commutator in (2.7) explicitly reads Proposition 5.1.The derivations L IJ close the braided Lie algebra so θ (2n + 1): (5.8) Proof.From (5.7) one computes and similarly Then, using (5.5) to simplify products of λ's, one gets and equation (5.8) is verified.
It is clear that any derivation in (5.7) restricts to a derivation of the subalgebra of O(SO θ (2n + 1)) generated by the entries of any column of N, thus in particular to a derivation of O(S 2n θ ).We denote π : Der R M H (O(SO θ (2n + 1, R))) → Der R (O(S 2n θ )) the map which associates to X ∈ Der R M H (O(SO θ (2n + 1, R))) its restriction X π to O(S 2n θ ), π(X) := X π .For the L IJ in (5.7), one easily computes (5.9) The restrictions L π IJ close the braided Lie algebra so θ (2n + 1), as in (5.8), too.Proof.We establish the lemma by showing that Since both sides are braided derivations it is enough to show the equality on the generators of O(S 2n θ ).From (5.9), and using the relation Then, A comparision with (5.9) shows that this coincides with the evaluation L π IJ (u K ).
Notice that the sphere relation implies the generators are constrained as J u J ′ T J = 0. Proof.From (5.11) we compute In the limit θ IJ = 0, the derivations L π IJ give a representation of the Lie algebra so(2n + 1) as vector fields on the sphere S 2n .5.1.The sequence and the equivariant connection.From the definition of the generators where we used (5.7) and the orthogonality condition (5.3) for the third equality.
Proposition 5.4.The connection ρ is invariant under the action of the braided Lie algebra so θ (2n + 1): for every The proof is analogous to that of Proposition 4.6 and we omit it.Due to this proposition the connection ρ is left O(S 2n θ )-linear as well.Using (5.7), the map ρ satisfies ρ( J u J ′ T J ) = I,J u J ′ • θ u I ′ L IJ = 0 as it should be due to J u J ′ T J = 0.
The kernel of the projection π is generated, as an O(S 2n θ )-module, by the derivations where θ ) (O(SO θ (5, R))) was studied in [5] in the context of twist deformation quantization.
The braided Lie algebras above give rise to a short exact sequence, (5.15)The sequence is split by the map ρ in (5.12), the horizontal lift.The corresponding vertical projection is then (5.14): As for the curvature one has the following.Proposition 5.5.On the generators T ν ∈ Der R (O(S 2n θ )), the curvature is given by Ω(T I , T Proof.We need to show that [ρ(T I ), ρ(T J )] = L IJ .For this we use the expression for ρ(T J )(n ST ) in (5.13) and that ρ(T Next for the braided commutator: where the last equality follows from (5.7).
The so θ (2n)-valued connection 1-form on the bundle O(S 2n θ ) ⊆ O(SO θ (2n + 1, R)), corresponding to the splitting of the Atiyah sequence (5.15), is the projection ω | so θ (2n) to so θ (2n) of the Maurer-Cartan form The differential calculus on the algebras O(SO θ (2n + 1, R)) was constructed in [3,9].The commutation relations among degree-zero and degree-one generators of the differential algebra Ω(SO θ (2n + 1, R)) are given by With E JK the elementary matrices (with component 1 in position JK and zero otherwise) one computes where in the last equality we have defined the d(d − 1)/2 matrices K JL with d = 2n + 1.
More in general, for any even or odd d we have the following.
Lemma 5.6.The matrices Proof.The first equality is clear, the second follows from a direct computation.For the last statement we compute and renaming the indices, proving (5.16).
The module generators above are not independent: J u J ′ φ (J) = 0 for u J = n J0 the generators of the sphere O(S 2n θ ) and module structure written as (bφ (J) )(e α ) = b• θ φ (J) (e α ) , for b ∈ O(S 2n θ ).Indeed, from (5.3), The O(S 2n θ )-module of equivariant maps E T can be realised as the image of the free module via a suitable projection with entries in O(S 2n θ ).Define 2n vectors |ϕ α , α = α ′ , with components |ϕ α J := n Jα ′ , J = 1, . . ., 2n + 1.From the orthogonality condition of the matrix N in (5.3), these are orthonormal ϕ α , ϕ β = J n J ′ α n Jβ ′ = δ αβ and we get a matrix projection p := α =α ′ |ϕ α ϕ α | .The entries of p are in O(S 2n θ ): using again the orthogonality condition one computes (5.20) This projector has rank 2n, its trace is • 1 , with 1 the constant function.We can then identify E T = (O(S 2n θ ) 2n+1 )p.In this way the rows of p are a set of generators for the module E T .
The module E T gives the 'tangent module', that is, the derivations Der R (O(S 2n θ )).This can be seen in two different ways.On the one hand, from the expression (5.20), the rows of p, that is, the generators of E T , are the components of the generators T J in (5.11).
On the other hand, let |∆ be the unit vector with components (u J , J = 1, . . ., 2n + 1), and let p N = |∆ ∆| be the 'normal' projection with corresponding 'normal' bundle E N . (5.21) The scalar curvature defined by r = J R(g −1 (θ J ′ ) , T J ) is computed to be r = 2n(2n − 1). (5.28) Proof.For the Ricci tensor, from (5.25) and (5.26), R(T J , T K ) := The first term is given by I (δ I ′ I ′ − u I ′ • θ u I ) T J (u K ) = 2n T J (u K ) while the second one is − I λ IJ (δ I ′ J ′ − u I ′ • θ u J ) T I (u K ) = −T J (u K ) + u J • θ I u I ′ (T I (u K )) = −T J (u K ), being I u I ′ T I = 0.When added they give (5.27) using that T J (u K ) = g(T J , T K ).
For the scalar curvature one has then r = (2n − 1) as stated.
) for all b ∈ B, a ∈ A. When the K-module algebra B is quasi-central in A, the braided Lie algebra Der(A) inherits the left B-module structure (bY )(a) := b Y (a) , (2.18) for Y ∈ Der(A), b ∈ B and a ∈ A. Then a right B-module structure is given by b ∈ B} .The linear spaces Der R M H (A) and aut R B (A) are K-braided Lie subalgebras of Der(A), [4, Prop.7.2].Elements of aut R B (A) are regarded as infinitesimal gauge transformations of the K-equivariant Hopf-Galois extension B = A coH ⊆ A, [4, Def.7.1].

. 20 )
This shows the surjectivity of π : Der R M H (O(S 7 θ )) → Der R (O(S 4 θ )), and thus the exactness of the sequence in (4.15) (since the elements of aut R O(S 4 θ ) (O(S 7 θ )) are vertical).We know from the general theory in §3.4 that the braided Lie algebra aut R O(S 4 θ ) (O(S 7 θ

Proposition 4 . 8 .
On the generators T µ of the braided Lie algebra Der R (O(S 4 θ )) the curvature of the connection in (4.21) is given explicitly by

Proposition 5 . 3 .
The braided Lie algebra structure of Der R (O(S 2n θ )) is given by [T I , T J ] = u I T J − λ IJ u J T I .
From p ⊕ p N = id we read the direct sum decomposition O(S 2n θ ) 2n+1 = E T ⊕ E N .We have as a consequence a module isomorphism between the O(S 2n θ )-module Der R (O(S 2n θ )) of derivations of O(S 2n θ ) and the O(S 2n θ )-module E T .We then identify the generators T J of Der R (O(S 2n θ )) in (5.10) with the generators φ (J) of the module E T in (5.19) via the module mapΓ : Der R (O(S 2n θ )) → E T , T J → φ (J) J = 1, . . ., 2n + 1.The matrix Q defining the orthogonality condition (5.1) is used for a metric on E T ,g : Der R (O(S 2n θ )) × Der R (O(S 2n θ )) → O(S 2n θ ).This is the restriction of the standard metric on the free module O(S 2n θ ) 2n+1 to the moduleE T = (O(S 2n θ ) 2n+1 )p ≃ Der R (O(S 2n θ )).Thinking of the generators T J as the rows of the projection (5.20), on these the metric is defined byg(T J , T K ) := L,I (T J ) L • θ Q LI • θ (T K ) Iand computed to be g(T J , T K ) = L,I