Poisson principal bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson (Lie-Rinehart) connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with Poisson connection, and the base has an inherited Poisson structure and Poisson connection. The Poisson connections are known to be the semiclassical data for a quantum differential calculus and we introduce compatibility conditions to come from a quantum principal bundle. The theory is illustrated by the Poisson level of the $q$-Hopf fibration on the standard $q$-sphere with its standard 3D left covariant quantum differential calculus on $C_q[SU_2]$. We also consider the Poisson level of a quantisation on the bundle and induced covariant derivative on associated bundles, with the Poisson level of the $q$-monopole as an example.


Introduction
It is known since the work of Drinfeld in the 1980's [13,14] that the infinitesimal notion of a Hopf algebra is that of a Lie bialgebra. This is a Lie algebra g equipped with a map δ called a Lie cobracket that forms a Lie 1-cocycle and makes g * into a Lie algebra. The connected and simply connected Lie group G associated to a Lie bialgebra is a Poisson-Lie group, the semiclassical analogue of a quantum group; one can think of standard q-deformed quantum groups C q [G] as the algebraic version of noncommutative deformations of the commutative algebra of functions C ∞ (G) (which is essentially recovered in some algebraic form) if we let or q → 1 or λ → 0, where q = e λ 2 . These are dual to the Drinfeld-Jimbo U q (g) enveloping algebra deformations. The same applies to the bicrossproduct family of quantum groups associated to Lie algebra factorisations [25,23] which tend to coordinate algebras of inhomogeneous groups or dually to enveloping algebras as a parameter λ → 0. Meanwhile, also since the 1980s, the quantum differential geometry of algebras in general and Hopf algebras in particular has been greatly advanced. There are different approaches, notably the one of Connes based on generalising the Dirac operator (a 'spectral triple') using operator algebras and, which is the approach we use, a complementary constructive approach led by the noncommutative geometry of quantum groups, but not limited to them, see for example [3,4,31,24,6]. Another important ingredient here turned out to be the notion of 'bimodule connection' going back to Mourad, Dubois-Violette and Michor [11,12]. In this constructive approach, the first step is to chose a unital possibly noncommutative 'coordinate algebra' A and an A-A-bimodule of first order differential forms Ω 1 . Quantum Riemannian geometry including the notion of metric g ∈ Ω 1 ⊗ A Ω 1 and of quantum Levi-Civita connection can then be developed. Also important in the constructive 'quantum groups' approach is the notion of a quantum principal bundle with a 'total algebra' P (with differential calculus) and a coaction by a quantum group H, due to T. Brzezinski and the first author [7]. The base of the bundle is modelled algebraically as the invariant subalgebra A = P H analogous to functions on X G if classically our model is P = C ∞ (X), H = C ∞ (G) on a total space X and fibre G respectively. Critical is an exact sequence condition which replaces local triviality of the bundle at the algebraic level and which then works globally [7]. The constructive/quantum groups approach is also very different from noncommutative-algebraic geometric approaches such as [37]. Being constructive, it also has been used in models of quantum spacetime such as may be expected to arise from certain quantum gravity effects [25,29,4,31]. Various models of discrete or quantum spacetime have been proposed over the years, such as ideas of Snyder [35], but noncommutative geometry in one of its forms provides a mathematical framework to explore the concept systematically.
Our topic in this paper is to continue a programme [1,30,5,17] to determine how such noncommutative geometry beyond the algebra itself semiclassicalises at the Poisson level. The semiclassical notion of quantum differential calculus on an algebra was studied in [19,1] and amounts to a contavariant or Lie-Rinehart [20,16,9] connection. The work [30] specifically looked at how this specialises in the case of a Poisson-Lie group, showing that translation-invariant associative connected calculi correspond at the semiclassical level to the dual of the Lie algebra, g * , being a pre-Lie or Vinberg algebra. Here we want to extend this to the semiclassical theory of quantum principal bundles, i.e. now with Poisson-Lie group fibre. The most well-known quantum principal bundle is the q-Hopf fibration where P = C q [SU 2 ] with H = C[t, t −1 ], the algebraic circle as fibre, and A = C q [S 2 ] as studied in [7,18,27]; our analysis will include this as a key example both of the bundle and of associated 'Poisson-monopole' bundles. This is a complementary direction to the Poisson-Riemannian geometry explored in [5].
An outline of the paper is as follows. Section 2.1 recalls elements of constructive noncommutative geometry on an algebra, in particular the notion of quantum differential forms Ω 1 on any unital algebra. We also review its semiclassicalisation as a Poisson-compatible Lie-Rinehart connection on a Poisson-manifold. One can think of this as a partially defined covariant derivative along Hamiltonian vector fields given byâ = {a, } for the Poisson structure given by {a, b} = ω(da, db) for a, b ∈ C ∞ (M ) and ω the Poisson bivector. Or, which we prefer here, one can think of this as covariant differentiation 'along forms'. Section 2.2 reviews some basic aspects of Drinfeld's theory of Poisson-Lie groups.
Section 3 starts with a modest reworking of the notion of a Poisson-Lie group; we show that this is equivalent to a group with a Poisson-bracket which is g-covariant in the sense ξ▷{f, g} = {ξ▷f, g} + {f, ξ▷g} + (δ 1 ξ ▷f )(δ 2 ξ ▷g) for all f, g ∈ C ∞ (G), ξ ∈ g acting as left-invariant vector fields and δ ∶ g → g ⊗ g defined from the Poisson bracket near the group identity. This covariance property characterisation then serves as a model for the rest of the paper. It is presumably a known property of Poisson-Lie groups albeit we have not found it elsewhere and include a short proof. We similarly characterise translation-invariant calculi on Hopf algebras previously studied at the semiclassical level in [1,30], now as the covariance property in Proposition 3.3, which is rather more direct than the technical analysis in the previous works. Here∇ is determined by a map Ξ ∶ g → g ⊗ g forming in the flat case a pre-Lie structure on g * as shown in [30]. Section 4 polarises this to the situation of a Poisson-Lie group G right acting on a Poisson manifold X with the action a Poisson map [22] (this is not the same thing as a Poisson action of a group on a Poisson manifold), expressed in the same style as above and extended to G-covariance of a semiclassical differential structure on X defined by a Poisson-compatible Lie-Rinehart connection. We then analyse in Section 5 the semiclassical analogue of a quantum principal bundle. In particular, we find that the canonical map ver (which generates vertical vector fieldsξ) has to be compatible with the Poisson connections for the calculi on the fibre and total space in the form of a novel transversality condition for a Poisson principal bundle with PLG fibre, namely for all ξ ∈ g and τ, η ∈ Ω 1 (X).
The final Section 6 then proceeds to the semiclassical analogue of quantum connections on quantum principal bundles and quantum associated bundles, and includes the example of a charge n q-monopole at the Poisson level. We note that some previous works such as [16] consider contravariant connections on principal bundles which is certainly of interest but a different question from the one we address (our connections on the principal bundle are deformations of usual connections). The paper concludes with some directions for further work.

Preliminaries
2.1. Elements of noncommutative geometry. We briefly discuss the formalism of noncommutative differential geometry in a bimodule approach and its semiclassicalisation. The classical model is A = C ∞ (M ) for a smooth real manifold M with exterior algebra Ω(M ), the semiclassical theory of course equips M with a Poisson structure, while the fully general 'quantum' formulation works over any algebra with a algebraically defined exterior algebra Ω A .
Thus, a 'differential calculus' on A consists of a graded algebra Ω A = ⊕Ω n A with n-forms Ω n A for n ≥ 0, an associative product ∧ ∶ Ω n A ⊗ Ω m A → Ω n+m A and an exterior derivative d ∶ Ω n A → Ω n+1 A satisfying the graded-Leibniz rule.
Here Ω 0 A = A and we assume that Ω is generated by degree 0, 1 and that Ω 1 is spanned by products of A and dA. Both conditions are slightly stronger than a DGA in homological algebra.
Vector bundles are expressed as (typically projective) A-modules E (classically this would be the space of sections). A left connection on this is ∇ E ∶ E → Ω 1 A ⊗ A E obeying the left Leibniz rule ∇ E (ae) = da ⊗ e + a∇ E e. We say that we have a bimodule connection if E is a bimodule and there is a bimodule map If σ E is well-defined then it is uniquely determined, so its existence is a property of a left connection on a bimodule. Bimodule connections extend to tensor products, giving us a monoidal category A E A where objects are pairs (E, ∇ E ); the tensor product connection on E ⊗ F uses σ E to reorder the output of ∇ F .
In the case of a connection on E = Ω 1 A we define the torsion as T ∇ = ∧∇ − d and a metric as g ∈ Ω 1 A ⊗ a Ω 1 A which is invertible in the sense of a bimodule map ( , ) ∶ Ω 1 A ⊗ A Ω 1 A → A inverse to g in the obvious way, which, however, turns out to require g to be central [4]. A quantum Levi-Civita connection (QLC) then makes sense as a bimodule connection on Ω 1 A such that T ∇ = 0 and ∇g = 0. We now assume that all our algebraic structures are built on classical counterparts as vector spaces but have a deformation parameter λ. This can be said formally in terms of power-series but for our limited purposes it is enough to say that all structures should be expandable order by order in λ (we will be interested mainly in the first order only, i.e. the semiclassical level). In this context of a flat deformation it is known that . for some Poisson bracket { , }. We denote the associated Poisson bivector by ω so that {a, b} = ω(da, db). Similarly for a first order calculus, [20,16,9,19] characterised by∇ aη = a∇ η ,∇ η (aζ) = ω(η, da)ζ + a∇ η ζ and the further Poisson-compatibility property [η, τ ] =∇ η τ −∇ τ η with respect to a 'Schouten bracket' [η, τ ] of 1-forms defined by [da, db] = d{a, b} along with [aη, bτ ] = ab[η, τ ] + aω(η, db)τ − bω(τ, da)η. Here∇ da = ∇â = γ(a, ) is the 'Poisson-preconnection' in the notations of [1,5] and the above are the natural extension to general 1-forms of the conditionŝ The preconnection ∇â point of view is that of a usual connection but only partially defined namely only on Hamiltonian vector fieldsâ = {a, }. In the symplectic case the data is equivalent to a symplectic connection with the contravariant∇ its pull back along the map ω # ∶ Ω 1 (M ) → Vect(M) given by ω # (η) = ω(η, ) with ω # (da) =â. Both points of view are useful. We will simply call such data a Poisson connection for brevity.
For deformations, it will be useful to complexify our algebras and then to suppose for our purposes of discussion that there are dense subalgebras where we can work algebraically, for example when discussing Hopf algebras (or quantum groups) and their coactions. The notion of a Hopf algebra works over any field k and there are several texts e.g. [23]. Briefly, this means a unital algebra H which is also a coalgebra with counit ǫ ∶ H → k, coproduct H → H ⊗ H with these being algebra homs, and for which an antipode S ∶ H → H exists. The latter is required to obey (Sh (1) )h (2) = ǫ(h) = h (1) Sh (2) in the 'Sweedler notation' ∆h = h (1) ⊗ h (2) (sum of such terms implicit). An left action of a classical group on a vector space V corresponds in our function algebra point of view to a right coaction ∆ R ∶ V → V ⊗H obeying the arrow-reversal of the usual axioms for a right action. If A is functions on a set then a right group action on the set induces a left action on A and makes A a comodule algebra, i.e. ∆ R is an algebra homomorphism in this case.
In particular, a Hopf algebra coacts on itself as a comodule algebra from both the left and the right via ∆. A calculus Ω 1 H is left (resp. right) covariant if the coaction extends to Ω 1 by making it a Hopf module with coaction commuting with d. A calculus is called bicovariant if both coactions extend and strongly bicovariant if the coactions similarly extend to Ω H making the latter a super-Hopf algebra with super-coproduct on degree 1 given by ∆ L + ∆ R (this is true for the canonical 'Woronowicz' exterior algebra construction [36]), see [32].
Finally, and equally briefly, the data for a quantum principal bundle is P a right H-comodule algebra with 'base' the algebra A = P H = {p ∈ P ∆ R p = p ⊗ 1} of elements fixed under ∆ R p = p( 0) ⊗ p( 1) in a compact notation. We assume that Ω 1 P is H-covariant in that ∆ R extends in a similar way to right-covariance of Ω 1 H above. In this case Ω 1 A = AdA computed in Ω 1 P is the inherited calculus. We also assume that Ω 1 H is bicovariant, which puts it in the form H.Λ 1 H where Λ 1 H is the space of left-invariant 1-forms. Next, we assume that there is a well-defined map ver ∶ Ω 1 P → P ⊗ Λ 1 H (the generator of vertical vector fields) given by (2.1) ver(pdq) = pq( 0) ⊗ (Sq( 1) (1) )dq( 1)((2) , ∀p, q ∈ P.
Finally, as a replacement for local triviality, we require a short exact sequence of left P -modules [7,27]. We will at certain points need to discuss this algebraic theory, which we suppose sits inside a putative full deformation version, however only to the point of extracting Poisson level data which then make sense in the smooth setting.

2.2.
Elements of Poisson-Lie theory. It is well known following the work of Drinfeld [14] that the semiclassical objects underlying Hopf algebras are Poisson-Lie groups (PLGs). This means a Lie group G which also a Poisson manifold such that the multiplication map G × G → G is a Poisson map, where G × G is equipped with the product Poisson structure. In terms of the Poisson tensor ω, this is for all g, h ∈ G. Here R g ∶ G → G and L g ∶ G → G are right and left translation by g and have derivatives R h * ∶ T g G → T gh G and L g * ∶ T h G → T gh G. It is well-known that the corresponding data at the Lie algebra level is a Lie bialgebra, i.e., (g, δ) where g is a Lie algebra and δ ∶ g → g ∧ g is a Lie coalgebra (so its dual is a Lie bracket map g * ⊗ g * → g * on g * ), with δ a 1-cocycle on g relative to the adjoint representation of g on g ∧ g in the sense δ([ξ, η]) = ad ξ (δ(η)) − ad η (δ(ξ)), ∀ξ, η ∈ g.
Next, the right and left actions of the group on itself define (respectively) left and right actions on the algebra of functions on the group C ∞ (G). We define the left-translation invariant formulation as follows. There is canonically a right action (a◁h)(g) = a(hg), ∀g, h ∈ G, a ∈ C ∞ (G) (usually made into a left action via the group inverse, but we refrain from this). Setting g = e tξ and differentiating at t = 0 gives a right action of the Lie algebra g, (a◁ξ)(g) = d dt 0 a(e tξ g).
Using this second formulation, we can view any Poisson structure on a Lie group as the right translation of some map D ∶ G → g ⊗ g, i.e, ω(g) = R g * (D(g)). We define Drinfeld showed that if (g, δ) is a Lie bialgebra then the associated connected and simply connected Lie group G is a Poisson-Lie group by exponentiating δ from a Lie algebra to a Lie group cocycle. Conversely, if (G, D) is a Poisson-Lie group, then its Lie algebra is a Lie bialgebra by differentiating D at the identity. Here δ on a Lie bialgebra is a 1-cocycle hence exponentiates to a 1-cocycle D ∈ Z 1 Ad (G, g ⊗ g). More details are in [23].

Poisson-Lie groups and their semiclassical calculi revisited
We start with a slightly different point of view on Poisson-Lie groups. Instead of the algebra of C ∞ (G) being deformed, one can again following [14] take the view that U (g) gets deformed to a quantum version U λ (g) say, with the same algebra in first approximation and coproduct where λ is a deformation parameter. The coproduct homomorphism property applied to ∆(ξη − ηξ) leads at order λ to the cocycle axiom mentioned in the above section with δ = ∆−τ ○∆, with τ the transposition map. Here δ is the Lie cobracket which can be viewed as a Lie bracket in g * which in turn is the Poisson bracket of linear functions on G near the identity. Here classically U (g) and C ∞ (G) or more precisely an algebraic model C[G] in the case of a Lie group of classical type, are dually paired by ⟨h, as matrix elements of a defining representation ρ ∶ G ⊂ M n , which we also use as a representation of U (g). This extends at the algebraic level to the quantum deformations. Our first observation is a more geometric characterisation of the PLG condition.
Proposition 3.1. Let G be a connected Lie group with Poisson bracket { , } vanishing at the identity and associated map D ∶ G → g ⊗ g with differential at the identity δ(ξ) = δ 1 ξ ⊗ δ 2 ξ (sum of such terms understood). Then Proof. We recall that {a, b}(g) = ⟨ω(g), da ⊗ db⟩, where evaluation here is the standard one between sections of the tangent bundle and cotangent bundle extended to tensor products. Now ξ▷{a, b} =ξ({a, b}) = ⟨Lξω, da ⊗ db⟩ + ⟨ω, dξ(a) ⊗ db + da ⊗ dξb⟩ with the last two terms just {ξ▷a, b} + {a, ξ▷b}. For the first term, if the PLG condition holds then as in the stated covariance condition. Conversely, if this covariance condition holds then the above calculation tells is that the differential at . If we suppose that D(e) = 0 and that G is connected then this requires that D is a 1-cocycle on the group (by the same argument as in the proof of [23, Thm. 8.4.1], namely the cocycle condition as a function in one of the variables is constant and hence zero given the value at e). Hence G is a PLG.
Next we consider the semiclassical data for differential calculus, i.e. a Poisson connection∇. For any (contravariant) connection we define∇ da ∶ C ∞ (G, g * ) → C ∞ (G, g * ) by∇ da s ∶=˜○∇ daŝ . The starting point in [1] is that∇ necessarily has the form for some mapΞ ∶ G × g * × g * → g * and is left translation covariant (in a manner corresponding to a calculus being left covariant in a Hopf algebra sense) if and only ifΞ(g, φ, ψ) is independent of g for all φ, ψ ∈ g * , i.e. given by Ξ ∶ g * ⊗g * → g * . In this case the (contravariant) connection is Poisson-compatible (a Poisson connection in our sense) if and only if This work uses the left-translation invariant formulation associated with right action ◁. We equally well have a right-translation invariant formulation with left action ▷, and in fact we will focus proofs on this case. In this and is right translation covariant if and only ifΞ is again given by a constant Ξ ∶ g * ⊗ g * → g * with Poisson-compatibility if and only if (3.2) holds. The latter and the connection being flat is equivalent to Ξ being a pre-Lie structure for g * [30]. Our next goal is to give a more geometric formulation of this covariance based on the following motivational lemma in which we suppose that C ∞ (G) has a formal deformation C ∞ λ (G). An algebraic version of this would also make the same point.
Suppose that a deformation U λ (g) left acts on C ∞ λ (G) as a module algebra to lowest order. Then which taking commutators implies the PLG covariance condition in Proposition 3.1. Similarly, if U λ (g) acts on a deformed calculus Ω 1 (C ∞ λ (G)) so as to be covariant then giving at lowest order . This extends to general 1-forms as the condition stated in the following Proposition 3.3.
Proof. The first part is immediate on taking commutators of • products. For the last part, given the covariance condition (3.4) on exact 1-forms, we first extend this to general 1-form τ = cdb with b, c ∈ C ∞ (G) (a sum of such terms understood). Then Using condition the PLG covariance condition in Proposition 3.1) to expand the first term and (3.4) on the connection to expand the last, this is This motivates the following result at the Poisson level.
for all τ, η ∈ Ω 1 (G) and ξ ∈ g if and only ifΞ is constant on G.
Proof. We prove the exact forms version (3.4). In terms of s ∈ C ∞ (G, g * ) in the right-translation invariant formulation, we have using that ξ▷ is the action ofξ on functions and commutes with d. Note thatξ(Ξ) isξ acting onΞ as a function on G (its first argument). We see that (3.4) holds if and only ifΞ is constant on its first argument.
If we takeΞ constant as a characterisation of right covariance cf [1,30] then we see that that condition is equivalent to (3.4) or the general form stated. Or we can take this as a definition of covariance as the semiclassical level as shown in the lemma, and find as stated that this is equivalent toΞ constant.
Remark 3.4. In the original left translation-invariant conventions of [1,30], our covariance condition for a PLG in Proposition 3.1 comes out equivalently as This holds if and only if the originalΞ in (3.1) is constant on G.
If one has both left and right covariance then we say that our Lie-Rinehart connection is bicovariant and the compatibility between the two was given in the above context in [30] by (2) . For our purposes we dualise this to the equivalent form for all η, ξ ∈ g, where Ξ * , δ ∶ g → g ⊗ g are written explicitly (with a sum of such terms understood).
Example 3.5. We consider here an example of left covariance by considering the Hopf algebra C q [SL 2 ] with relations ba = qab, ca = qac, db = qbd, dc = qcd, bc = cb, and we refer the reader to any standard literature on quantum groups for the Hopf algebra structure. For our purposes we set q = e λ 2 , for λ deformation parameter and use the conventions of [23]. There is a left translation covariant calculus of left invariant 1-forms due to Woronowicz [36]. It is generated by these as a left module while the right module structure is given by the bimodule relations for homogeneous f of degree f where a = c = 1 and b = d = −1 and exterior derivative The above are quantum algebra formulae recalled for reference. From now on we work at the corresponding semiclassical level albeit focussing on the polynomial subalgebra generators. Indeed, classically, the basis e 0 , e ± of 1-forms is dual to the basis {H,X ± } of left-invariant vector fields generated by the Chevalley basis {H, X ± } of su 2 . In the classical limit these are easily computed, since we have Now, according to Remark 3.4 and our discussion in Section 2.2, for left translation invariant cacluli the covariance condition for a PLG in Proposition 3.1 is achieved via a right action on the function algebra. Thus we require the action of the right invariant vector fieldsξ ∈ su 2 which works out as One can easily check (at the classical level q = 1) that e i ◁ξ = 0 for all i as expected.
With all the machinery in place, now proceed to check the left covariance conditions Similarly for all other generators.
(2) From the bimodule relations, [30] provides the Poisson connection aŝ We can now write condition (3.7) rather trivially as for action by the above-mentioned right invariant vector fields.

PLG actions in Poisson manifolds
As a step towards principal bundles we now 'polarise' the above to a general Poisson manifold X right acted upon by a Lie group G with action β ∶ X × G → X, β(x, g) = x.g, ∀x ∈ X, g ∈ G.
As before, we work in a coordinate algebra language with P = C ∞ (X) with action (g▷f )(x) = f (x.g) for all x ∈ X, g ∈ G and f ∈ P . If we have a Hopf algebraic As in Drinfeld's theory, we now suppose further that G is a PLG and ask that β is a Poisson map where X × G has the direct product Poisson action [22], which we refer to as a PLG action. Note that in classical Poisson geometry where G is just a Lie group it is more normal just to consider each map β( , g) ∶ X → X as a Poisson map, which effectively means the zero Poisson bracket on G, but we are not supposing this. Rather, this is the semiclassical level of the further assumption that there is a quantum deformation C ∞ λ (X) which is module algebra under a quantum deformation U λ (g), or in an algebraic setting a comodule algebra under a quantisation C λ [G]. As before, we do not need these actual Hopf algebras other than as motivation, working at the smooth manifold level.
Adopting a similar notation as in the preceding section, we let ω X and ω G be the respective Poisson tensors and we define and extensions to act on tensor products in the usual way. Then the condition for a PLG action is clearly for all x ∈ X, g ∈ G. Following our previous notation we also denote byξ = ξ▷ the vertical vector field on X associated to ξ ∈ g. This is the left action of g = e tξ on functions differentiated at t at t = 0, i.e.ξ(x) = β x * (ξ) if we view g = T e G. Since these vector fields are defined for each element ξ, we can think of them all together as a single map such that when evaluated against element ξ of the Lie algebra we recoverξ in the form ver ξ = (id ⊗ ξ)ver = iξ ∶ Ω 1 (X) → C ∞ (X).
Proof. This follows the same pattern as the proof of Proposition 3.1.
To put this in context, suppose that a deformation U λ (g) left acts on C ∞ λ (X) as a module algebra to lowest order. Then the action on ξ▷(p•q) has the same form as in Lemma 3.2 with a, b now replaced by p, q ∈ C ∞ λ (X), and taking commutators gives the PLG action condition in Proposition 4.1. Similarly, if this action extends to a deformed calculus Ω 1 (C ∞ λ (X)) then ξ▷(p • dq) has the same form as in Lemma 3.2 again with a, b replaced by p, q ∈ C ∞ (X). Taking commutators of this gives to lowest order the following.
Lemma 4.2. The covariance condition for a deformed calculus on C ∞ λ (X) at semiclassical order is , where ξ ∈ g and p, q ∈ C ∞ λ (X), and generalises to arbitrary 1-forms as for all ξ ∈ g and η, τ ∈ Ω 1 (X).
Proof. The proof exactly follows that of Proposition 3.3, replacing the group product as a right action on itself by β. Details are therefore omitted.
We take either condition in Lemma 4.2 as a definition of what we mean at the Lie action level by a PLG-covariant Lie-Rinehart connection∇ as extracted from the semiclassical part of a putative deformation. To explain the next formula, note that the Lie derivative can be formally extended to one along antisymmetric tensors V by the formula → Ω(X) lowers degree by that of V , cf the ⊥ operation in [28]. In particular, i v∧w = i v i w (by which we mean a sum of such terms) depends antisymmetrically on v, w and so descends to the wedge product, giving a well-defined degree -2 interior product on the exterior algebra. This in turn defines a Lie derivative L v∧w = [i v∧w , d] along antisymmetric bivector fields. Another key ingredient in [28] is the Leibnizator Moreover, if v ∧ w is an antisymmetric product of vector fields on X and τ, η are 1-forms, we have allowing us to write the δ ξ terms as stated in terms of the bivectorδ ξ =δ 1 ξ ∧δ 2 ξ .
This formula reduces to d applied to the ξ▷{a, b} covariance identity if τ, η are exact. It also extends in principle to higher degree forms.
in the construction of the standard q-sphere at a quantum algebraic level. At the semiclassical level this means G = S 1 acting from the right on X = SU 2 (as a diagonal subgroup) with X having the Poisson bracket and Poisson connection ∇ for the 3D quantum calculus as in Example 3.5. G is a PLG with the zero Poisson bracket as S 1 remains unquantied, so δ = 0, but the notation C q 2 [S 1 ] indicates q 2commutation relations in the differential calculus. Writing C[S 1 ] = C[t, t −1 ] at the algebraic level, this has dt.t = (1 + λ)t.dt at lowest order so that (4.3)∇ G dt dt = −tdt is the corresponding Poisson connection on Ω 1 (S 1 ) (it will play more of a role in the next section). The calculus here has left-invariant basic 1-form t −1 dt with dual ξ = t ∂ ∂t as a left-invariant vector field on C[S 1 ]. Here∇ t −1 dt (t −1 dt) = −t −1 dt for the left-invariant 1-form dual toξ. From this we read off for this calculus that Again at the algebraic level, the right action of S 1 on SU 2 corresponds to coaction ∆ R (f ) = f ⊗ t f , (this also works at the quantum level). The Maurer-Cartan form has value t − f dt f = f t −1 dt which we can evaluate by duality asξ(f ) = f f (at the quantum level this will be a q-integer). One can also right this asξ = H▷( ) =H as a vertical vector field on SU 2 using the partial derivative in the e 0 direction as given in Example 3.5. It is then easy to check that { , } in Example 3.5 is right covariant, which since δ = 0 reduces to a usual Poisson action. For example, for a, b ∈ C ∞ (SU 2 ), we have It is sufficient to verify the covariance condition in Lemma 4.2 on the left-invariant forms e i , which then implies it for the general case by the connection property. Again, δ = 0 in the condition. For example, Similarly for other cases, including for e − .
Remark 4.5. We also have right covariance condition for a PLG left action α ∶ G × X → X, namely where ◁ξ on functions is the vertical vector fieldξ. For the action extends to a differential calculus at semiclassical order we need ∇ η τ ◁ξ =∇ η◁ξ τ +∇ η (τ ◁ξ) + (iδ 1 ξ η)(τ ◁δ 2 ξ ). We also have that the Poisson-compatibility condition leads to where( ) refers to the vertical vector field for the left action. The proofs follow those above, swapping left covariance for right covariance.

Semiclassical principal bundles
A principal bundle in a smooth setting is a smooth manifold X with a smooth, free and proper action of a group G and a local triviality condition so that M = X G is a smooth manifold and X a fibre bundle over it with fibre G. We have discussed actions and now we consider the further data we need for a principal bundle at the quantum and hence semiclassical level. From a practical perspective, the key expression of local triviality is transversality: the C(X)-module of horizontal forms Ω 1 hor (the pull back of Ω 1 (M ) along the canonical projection π ∶ X → M ) is precisely the joint kernel of the vertical vector fields. The invariant horizontal forms are then the forms on the base, Ω 1 (M ) ↪ Ω 1 hor (X). In the quantum case, the transversality is exactness of the sequence (2.2), the horizontal forms are P Ω 1 A P with (under reasonable conditions) invariants under the coaction of H recovering the quantum calculus Ω 1 A ⊆ Ω 1 (P ). In the following we suppose for the sake of discussion that deformations C ∞ λ (X), C ∞ λ (G) have dense algebraic versions C λ [X], C λ [G] forming a quantum bundle at the algebraic level with base some C λ [M ].
Lemma 5.1. Let X be a principal bundle with fibre G a PLG and base M = X G and suppose that this data deforms to a quantum principal bundle with P = C λ [X] right coacted on by H = C λ [G]. Let∇ G be bicovariant as defined by Ξ * and∇ X covariant for the differentials of the quantum bundle. Then for all p, q ∈ C ∞ (X), ξ ∈ g. This extends to 1-forms τ, η ∈ Ω 1 (X) as Proof. (1) We begin from the definition of ver(pdq) in (2.1) and note that the counit projection π ǫ ∶ H → H + given by π ǫ (h) = h−ǫ(h) obeys π ǫ (hg) = (π ǫ h)g +ǫ(h)π ǫ (g). Then working in the quantum deformation, we can write where we used that Sp( 1)(1) commutes at lowest order to cancel it at order λ. Hence ver∇ X dp dq ={p, ver(dq ={p, ver(dq) 1 } ⊗ ver(dq) 2 + ver(dp) 1 ver(dq) 1 ⊗∇ G ver(dp) 2 ver(dq) 2 where everything is now classical so that we can bring Sp( 1)(1) into the subscript of ∇ G and use the Leibniz rule to change the order. Finally, we use the contravariant connection property of∇ G to bring the Sq( 1)(1) term to be acted upon by∇ G . We also use S{h, g} = {Sg, Sh} for any functions h, g on a Poisson Lie group (as easily established by considering if we write the quantum product explicitly). We also adopted the notation ver(dq) = ver(dq) 1 ⊗ ver(dq) 2 .
(2) Now consider iξ(∇ X dp (f dq)) for f ∈ C ∞ (X), from the connection property and the condition in (1), we have on τ = f dq that Now letting η = f dp, which then gives the general condition as stated.
This leads to the following key definition; In principle we could need more conditions, but we will find in later sections that the above is enough for a Poisson-level version of spin connections and associated bundles, thereby justifying the definition.
Proof. We compute using Poisson-compatibility, Using that Ξ is a pre-Lie structure for the bracket on g * allows us to recogniseδξ from its cocommutator. We also use antisymmetry of ω X . We then put the result as a bivector interior product as discussed before.
In the exact case, this correctly reduces to the covariance condition on ξ▷{p, q} for the Poisson bracket. We also want to know that our definition is fit for purpose and implies that the base is not only a manifold but a Poisson manifold with an induced Poisson connection.
Proposition 5.4. Let G be a PLG with a free smooth proper right PLG action on a Poisson manifold X. Suppose that G has a bicovariant Poisson connection ∇ G and X a right covariant Poisson connection∇ X as in Lemma 4.2 and obeys the transversality condition in Lemma 5.1 (i.e., is a Poisson-principal bundle). Then M = X G becomes a Poisson manifold with Poisson connection∇ M given by restriction of∇ X .
Proof. Here C ∞ (M ) can be identified with smooth functions on X which are killed by all vertical vector fields. By the invariance of the Poisson bracket it follows that if p, q are killed by allξ then so is {p, q}, so the Poisson bracket restricts. Now let∇ M be the restriction of∇ X to the differentials of such functions. By the covariance condition, it follows that the output of∇ M is invariant under all ξ▷ = Lξ. By the bundle condition it follows that the output of∇ M is killed by iξ. However, if ∑ p i dq i with p i , q i ∈ C ∞ (X) has these properties then by the latter ∑ p iξ (q i ) = 0 and moreover by transversality of the classical bundle we know that the form is horizontal, so we can assume q i ∈ C ∞ (M ). Then by the first condition, ∑ξ(pi)dqi = 0 so that taking the dq i independent, the p i are also in C ∞ (M ), i.e. our form is element of Ω 1 (M ). Thus∇ M is defined, and in this case inherits the connection properties.
Example 5.5. We continue Example 4.4 for the Poisson version of SU 2 → S 2 = SU 2 S 1 and verify that the Poisson brackets and connections there obey the transversality condition in Lemma 5.1, so that we have a Poisson principal bundle. For example, where ξ = H andξ is the associated vertical vector field for the action by righttranslation, soξ =H the left-invariant vector fieldH in Example 3.5. This is dual to e 0 ∈ Ω 1 , soξ(a) = a and iξ(e 0 ) = ⟨ξ, e 0 ⟩ = 1. We also used that Ξ * ξ = −ξ ⊗ ξ from (4.3). For another example, since ⟨ξ, e + ⟩ = 0. The right hand side is similarly zero for this reason given the form of Ξ * ξ . A similar computation can be done for e − . Hence we obtain a Poisson bracket and connection on M = S 2 S 1 by Proposition 5.4, which we now describe. If we use the complexified coordinates z = cd, z * = −ab, x = −bc then the algebra relations are given by From which, the Poisson bracket and Poisson connection can be computed as We also in the present case have a left-translation covariance of the Poisson bracket and 3D calculus in Example 3.5 on SU 2 commuting with the right action of S 1 , so this necessarily descends to an action of su 2 on the sphere as x◁H = 0, z◁H = −2z, z * ◁H = 2z * , x◁X + = −z, z◁X + = 0, z * ◁X + = 2x − 1, with respect to which { , } S 2 and∇ S 2 are covariant in our Poisson sense of Remark 4.5. For example, which is as expected since δ(H) = 0. For another example, On the other hand from which we see that the covariance condition holds.

Connections on Poisson bundles
Once we have a quantum principal bundle P , the next order of business is to to find a 'spin connection' ω ∶ Λ 1 H → Ω 1 P which is equivariant (where H coacts on Λ 1 H by assumption, which we now use, that Ω 1 H is bicovariant) and such that ver(ω(v)) = 1 ⊗ v. This provides an equivariant splitting of Ω 1 P in the form of an associated Π ω defined by Π ω (dp) = p( 0) ω(̟π ǫ p( 1) ), equivariant for the right coaction of H, a left P -module map, idempotent and with kerΠ ω = P Ω 1 A P the horizontal forms [7]. We also require ω to be such that (id − Π ω )dP ⊆ Ω 1 A P , in which case it defines a connection on P itself by

It is known[6] that this is a bimodule connection if and only if
In this case the generalised braiding is We will focus on ∇ P as the key part of the theory. Note that the above products and connections etc. are quantum, but from now we will denote them explicitly as deformations built on the underlying classical objects or algebraic versions of them. Moreover, we now suppose to be concrete that the bimodule structure of Ω 1 P is quantised symmetrically as in [5] so that , for all p, q ∈ C ∞ (X), η ∈ Ω 1 (X). Similarly on the group, which we look at first.
is a quantum group deforming C ∞ (G) with lefttranslation invariant calculus data∇ G given by Ξ * ∶ g → g ⊗ g. Then the quantum Maurer-Cartan form induced a vector space isomorphism at order λ, where e is the group identity and v represents an element of Λ 1 H = H + I.
Proof. Since the action on Ω 1 H is the same as on the vector space Ω 1 G , we expect the classical answer for the invariant subspace. However, it is a calculation to see what the quantum Maurer-Cartan form looks like, namely (3) )dv (4) where v is a function on G vanishing at e viewed in the deformed H + , and the deformed antipode has the form S λ a = Sa − λ 2 {Sa (1) , a (2) }Sa (3) to lowest order as one may check (assuming as we do that the coalgebra is not deformed). Then using dv (2) ̟π ǫ v (2) which computes to the map Θ as stated when we note that and ⟨ξ, ̟(v)⟩ = d dt 0 v(e tξ ) =ξ(v)(e) for any functions v and ξ, η ∈ g (we apply the first observation to the output of Ξ * ∈ g ⊗ g). We identified Λ 1 H with H + I via the quantum Maurer-Cartan form ̟ λ and Λ 1 G with g * via the classical Maurer-Cartan form ̟.
We can therefore proceed with the quantum connection viewed as ω λ ∈ Ω 1 (X, g) as a vector space, which we write as ω λ = ω i λ ⊗e i where {e i } is a classical basis of g but we must allow for Θ for the values on Λ 1 where ω = ω i ⊗e i is now a classical connection on X as a bundle and α i ∈ Ω 1 (X) are the first quantum correction. We let Ω 1 hor ⊆ Ω 1 (X) denote the classical kernel of ver as explained in Section 5. Rather than develop a full theory of ω λ , we ask for the associated ∇ P λ on P λ = C λ [X] to be a quantum covariant derivative. Lemma 6.2. If we have a quantum connection given by ω, α on a deformed bundle of the form above then at order λ: hor as vector spaces. (ii) iξα i = 0 for all ξ ∈ g and all i, for ∇ P λ to be defined.
(iii) The bimodule connection ∇ P λ to order λ then takes the form Proof. (i) Note that by the transversality condition in Definition 5.2 the∇ τ preserves Ω 1 hor for all τ ∈ Ω 1 (X), since iξ∇ X τ η = 0 for all η ∈ Ω 1 hor . The latter is characterised by iξη = 0 for all ξ ∈ g and applied both to the given ξ and to Ξ * 2 . Now if p, q ∈ P and η ∈ Ω 1 (M ) then p • η • q = ηpq + λ 2∇ X qdp−pdq η + O(λ 2 ). The first part already lives in Ω 1 hor and the second part does by the transversality condition as just remarked.
(ii) We already know that ω is a classical connection which in our notation translates amounts to and defines a classical connection ∇ P p = dp−(e i ▷p)ω i . One can check that iξ∇ P p = 0 so the image of ∇ P is in Ω 1 In the quantum case we have as stated. We used p( 0) ⟨ξ, p( 1) ⟩ =ξ(p) (now the vertical vector field generated by ξ ∈ g) and similarly for two derivatives in the similar manner to (6.1) but now for G acting on X. For the output to remain in Ω 1 hor at order λ, we need (e i ▷p)iξ(α i ) + iξ∇ X dei▷p ω i −Ξ * 1 ξ (Ξ * 2 ξ (p)) = 0. But given the Poisson transversality condition and the property ofω i , the second term is as iξ(ω i ) = ξ i are constants (where ξ = ξ i e i ). Hence we need (e i ▷p)iξ(α i ) = 0 for all p, i. However, since the classical action is free, ver is surjective and it follows that we need the condition stated, i.e. that α i ∈ Ω 1 hor . (iii) For completeness, we now verify directly that for α obeying (ii), we indeed have a quantum bimodule connection at order λ. Thus, We used that f is invariant to move it out of most expressions in the order λ term as well as the Leibniz rule d(f (e i ▷p)) in the argument of∇ X . We then use the properties of a Poisson connection. That the result agrees with df ⊗ A p + f • ∇ P λ p then comes down to X dp df making the canonical identifications for the tensor products (here A is the deformed algebra). Finally, for ∇ P λ to be a bimodule connection to our order we which computes as stated using the invariance of f and Poisson compatibility.
We see that α just enters as an order λ addition to the classical ω and the canonical choice is α = 0. It could, however, be useful to retain this freedom if we want to preserve specific further properties of the quantisation, c.f. [5]. There remains one further classical condition which we have not yet discussed, namely equivariance of ω for the adjoint action. As classically, this is needed for ∇ P λ to be equivariant. Proposition 6.3. Let X be Poisson principal bundle over M with connected PLG structure group G and suppose that∇ G is bicovariant in the sense (3.8). If ω = ω i ⊗ e i is a classical connection on X and α i ∈ Ω 1 (X) with for all ξ ∈ g and all i then ∇ P λ defined in Lemma 6.2 to order λ commutes with the action of G.
Proof. The new part here is the equivariance under the action of G. At the Lie algebra level this is as required. We used equivariance of the classical ∇ P and the equivariance of α to transfer ξ▷α i to [ξ, e i ], then covariance of∇ X in our Poisson sense of Lemma 4.2 to expand ξ▷∇ X and equivariance of the classical ω i as in (6.2) to transfer any actions ξ▷ω i and δ 2 ξ ▷ω i to the relevant e i . We finally used the bicovariance condition (3.8) to recognise the answer.
The significance of this equivariance classically is that ∇ P then restricts to any associated bundle with sections E = C ∞ G (X, V ) where G acts on V and sections are viewed as equivariant functions. At the quantum level, this appears as E = (P ⊗V ) H the space of invariants under the coaction of H and the quantum connection ∇ E λ = (∇ P λ ⊗id) restricted toE. If we take V commuting with A then E is an A-bimodule as P is and the connection becomes a bimodule one.
Proof. If s ∈ E, we regard V -values as constants and inherit a bimodule structure at the Poisson level, which we think of as a partially defined or contravariant connection∇ E df s ∶= {f, s} X on E, where f ∈ C ∞ (M ) is also viewed in C ∞ (X). The result is an element of C ∞ G (X, V ) since the Poisson bracket is G-covariant in the sense of Proposition 4.1 and f is invariant. We have∇ E df (gs) = {f, gs} X = {f, g} X + g{f, s} X = {f, g} M s + g∇ E df s as expected and we extend to 1-forms bŷ ∇ E f dg s = f∇ E dg s = f ω X (dg, ds) = ω X (f dg, ds), which shows that this is well-defined. We also view Ω 1 (M ) ⊗ C ∞ (M) C ∞ G (X, V ) as certain horizontal 1-forms on X with values in V so that ∇ E and the application of∇ X df make sense ignoring the V values. The condition stated is just the content of s to order λ on making the parallel identification of the ⊗ A . This follows from ∇ E λ s ∶= ∇ P λ s similarly ignoring the V -values and has result in the right space due the equivariance in Proposition 6.3. There is also an associated inherited σ E λ obeying . Note that the first two terms in the stated property of ∇ E λ are similarly the leading order part of df ⊗ A s and one can check using the transversality condition for a Poisson principal bundle that again all terms are horizontal and hence can be viewed in The simplest case in the quantum bundle theory is that of a line bundle, where H = C q 2 [S 1 ] and V = C with coaction v ↦ v ⊗t n , in which case E = E n is the degree −n component of P . Thus E n ⊆ P is an A-sub-bimodule of P and equivariance in this context amounts to ∇ P (E n ) ⊆ Ω 1 A ⊗ A E n for the quantum connection, which restriction is ∇ En as a bimodule connection. At the Poisson level we have different grade components where H▷p = p p for p of homogeneous grade and the content of equivariance is that ∇ P λ preserves this so restricts to each E n as associated bundles.
Example 6.5. We continue Example 5.5 for the Poisson level of the bundle on the q-sphere given by the right coaction of H = C q 2 [S 1 ] on P = C q [SU 2 ], now with its q-monopole spin connection ω. Recall that the coaction was given by ∆f = f ⊗ t f for any f ∈ C q [SU 2 ]. The differential of this, expressed via the map ver, is easily computed to be ver(e ± ) = 0, ver(e 0 ) = 1 ⊗ t −1 dt, for the left invariant 3D calculus. Since, by definition, we require ver ○ ω(η) = 1 ⊗ η for η ∈ Ω 1 P , we have ω(t − 1) = ω(t −1 dt) = e 0 and more generally, ω(t n − 1) = [n] q 2 e 0 . It follows after some computation that the quantum connection generators in E ±1 are For example, These are previously known quantum bundle calculations cf. [7,18,27] and bullet products should be understood up to this point. We will henceforth indicate them and label the quantum differential forms explicitly. Then the quantum 1-form )e + to lowest order. As a result, we find that the covariant derivative is undeformed to lowest order ∇ P λ = ∇ P + O(λ 2 ) at least on E ±1 . For example, We now check how this goes at the Poisson level. Since ω λ ∶ Λ 1 H ≅ g * → Ω 1 P (using Lemma 6.1) and g is one dimensional, we deduce ω λ = e 0 ⊗ H, ω λ (t n − 1) = e 0 λ n(1 + λ 2 (n − 1)) = e 0 n(1 + λ 2 where H is our basis element of g and Θ is ⟨H, Θ(t n − 1)⟩ = n(1 + λ 2 n) corresponding to relating the quantum and classical Maurer-Cartan forms to order λ. We see that the quantum connection looks the same as the classical one, the deformation appearing in Θ. This means that α = 0 and (ii) in Lemma 6.2 holds. Finally, the quantum connection has correction on degree ±1 and one can check that this similarly applies to p ∈ C ∞ (X) of any degree. So in this example ∇ P λ p = ∇ P p+O(λ 2 ) for all p. Moreover, Ξ in the present example trivially obeys the bicovariance condition (3.8) so the quantum connection preserves each degree component E n ⊆ P , which is in any case clear as the classical connection does. So restriction gives us a Poisson-level q-monopole connection ∇ E λ on each E n .
Another natural condition is for Π ω to be a P -bimodule map, which at the quantum algebra level is analysed in [6] as ω(v) • p = p( 0) • ω(v◁p (1) ), where products are quantum ones. In the semiclassical theory above this comes down by similar methods to (6.3)∇ X dp ω i ⊗ e i = Ξ * 1 ei (p) ⊗ Ξ * 2 ei for all p ∈ C ∞ (X), which is a rather strong condition at it determines∇ X from { , } X and∇ G and implies in particular that∇ df ω i = 0 for all f ∈ C ∞ (M ) and all i. One can check that it does, however, hold for the Poisson level q-sphere example above.

Concluding remarks
We have shown that Drinfeld's theory of Poisson-Lie groups [13,14] extends in a natural way to principal bundles X which are Poisson manifolds and have Poisson-Lie structure group G. We formulated the PLG condition itself as a covariance of the Poisson bracket on the group, { , } G and extended this to the notion of PLG action on { , } X . We then extended these covariance notions to Poisson-level quantum differential structures controlled by Poisson-compatible contravariant or Lie-Rinehart connections∇ G and∇ X respectively in the sense of [19,1,20,16,30]. In Section 5, we supplemented this by a further transversality condition expressing that X → M = X G is a quantum bundle at the Poisson level. This also implied that M is not only a Poisson manifold but inherits a Poisson-level quantum differential structure∇ M . The theory is rounded off in Section 6 with a theory of quantum 'spin' connections on the bundle at the Poisson level which we show is sufficient to induce Poisson-level 'quantum covariant derivatives' on all associated bundles, which classically is the key application of principal bundles. Every step of the theory was illustrated on the semiclassical limit of the known q-Hopf fibration deforming S 3 → S 2 with U (1) fibre, the latter being a PLG with zero { , } G but nonzero∇ G .
A direction for further work would be a general Poisson version of the theory of quantum homogeneous bundles, of which the q-Hopf fibration above is the simplest example. This theory with semiclassical differentials structures would be rather different from previous discussions of Poisson homogeneous spaces, for example in [15]. Whether our approach can extend to a geometric picture of the full 2parameter Podleś spheres [34], for example, is less clear since a direct principal bundle approach would seem to require coalgebra bundles [8] for which the notion of differential calculus on the fibre is not at all clear. Another issue, even for ordinary quantum principal bundles, is that associative bicovariant calculi on q-deformation quantum groups are often not possible with classical dimension [1]. This, however, is not a problem in our approach and just means at the Poisson level that∇ G is not flat. It should be possible, for example, to construct a nonassociative deformation of the next Hopf fibration S 7 → S 4 with SU 2 fibre, combining the Hopf-quasigroup methods in [21] and the nonassociative bicovariant calculi by twisting in [1,2]. A third interesting direction for further work would be to consider quantum group frame bundles as in [26] at the Poisson level whereby Ω 1 (M ) is an associated bundle to a Poisson principal bundle. This would then extend the above theory to a PLGcovariant version of Riemannian geometry on M as a somewhat different approach to the one in [5]. Indeed, one expects the quantum metrics at semiclassical order to be different even for the Poisson level of the metric for the q-sphere studied above.