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-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant co...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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Інститут математики НАН України
2021
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| Цитувати: | Poisson Principal Bundles. Shahn Majid and Liam Williams. SIGMA 17 (2021), 006, 23 pages |
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| citation_txt | Poisson Principal Bundles. Shahn Majid and Liam Williams. SIGMA 17 (2021), 006, 23 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the 𝑞-Hopf fibration on the standard 𝑞-sphere. We also construct the Poisson level of the spin connection on a principal bundle.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 006, 23 pages
Poisson Principal Bundles
Shahn MAJID and Liam WILLIAMS
School of Mathematical Sciences, Queen Mary University of London,
Mile End Rd, London E1 4NS, UK
E-mail: s.majid@qmul.ac.uk, l.williams@qmul.ac.uk
Received June 11, 2020, in final form January 05, 2021; Published online January 13, 2021
https://doi.org/10.3842/SIGMA.2021.006
Abstract. 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-compatible
contravariant connection, the fibre is a Poisson–Lie group in the sense of Drinfeld with bico-
variant Poisson-compatible contravariant connection, and the base has an inherited Poisson
structure and Poisson-compatible contravariant connection. The latter are known to be the
semiclassical data for a quantum differential calculus. The theory is illustrated by the Pois-
son level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson
level of the spin connection on a principal bundle.
Key words: noncommutative geometry; quantum group; gauge theory; symplectic geometry;
poisson geometry; Lie bialgebra; homogenous space; q-monopole
2020 Mathematics Subject Classification: 58B32; 53D17; 17B37; 17B62
1 Introduction
It is known since the work of Drinfeld in the 1980’s [13, 14] that the infinitesimal notion of
a quantum group is that of a Lie bialgebra. This is a Lie algebra g equipped with a ‘Lie
cobracket’ map δ that forms a Lie 1-cocycle and makes g∗ into a Lie algebra. The associated
connected and simply connected Lie group G is a Poisson–Lie group, the semiclassical analogue
of a quantum group; one can think of standard q-deformed quantum groups Cq[G] as in [33], as
the algebraic version of noncommutative deformations of C∞(G) (which is essentially recovered
in some algebraic form) if we let λ → 0, where q = e
λ
2 . These are dual to the Drinfeld–
Jimbo Uq(g) enveloping algebra deformations. The same applies to the bicrossproduct family of
quantum groups associated to Lie algebra factorisations [25] which tend to coordinate algebras
of inhomogeneous groups or dually to their enveloping algebras and have a role in quantum
spacetime models.
Meanwhile, quantum groups also led in the 1990s to the emergence of a constructive ap-
proach to quantum differential geometry including but not limited to their quantum geometry.
By now, there is a significant body of work and we refer to the text [4] and references therein.
This approach is somewhat different in character from Connes’ [10] well-known approach to
noncommutative geometry based on generalising the Dirac operator (a ‘spectral triple’) using
operator algebras, as well as from noncommutative-algebraic geometric approaches such as [34].
The constructive approach starts with the notion of 1-forms Ω1 over a possibly noncommu-
tative algebra A. Sections of vector bundles appear in the nicest case as A-A-bimodules E
and connections on them as bimodule connections ∇E : E → Ω1 ⊗A E in the sense of [16, 31].
There is also a notion of quantum principal bundles with quantum group fibre, ‘spin connec-
tions’ on them and associated bundles [7]. The most well-known example is the q-Hopf fibra-
tion [4, 7, 19, 28], with total space algebra P = Cq[SU2], quantum group fibre H = Cq2
[
S1
]
This paper is a contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in
honour of Giovanni Landi. The full collection is available at https://www.emis.de/journals/SIGMA/Landi.html
mailto:s.majid@qmul.ac.uk
mailto:l.williams@qmul.ac.uk
https://doi.org/10.3842/SIGMA.2021.006
https://www.emis.de/journals/SIGMA/Landi.html
2 S. Majid and L. Williams
and base algebra A = Cq
[
S2
]
. We introduce a Poisson-level theory relevant to the first order
data for deformation-quantisation of such quantum principal bundles and connections on them.
This fits within a programme [1, 3, 18, 30] to semiclassicalise various aspects of noncommutative
differential geometry, including quantum Riemannian geometry.
Our results effectively extend Drinfeld’s work to a Poisson–Lie group G as the structure
group of a classical principal bundle, the total space X of which is a Poisson manifold; the
action of G is a Poisson action and, which is the critical new part, both G and X are equipped
with the leading order data for quantum differential forms Ω1 in a compatible way. In this
regard, [9, 17, 21] previously looked at deformations of vector bundles and found their Poisson-
level data as a contravariant or Lie–Rinehart connection ∇̂ (we will use a hat to remind us that
these are not usual connections). The specific case of the deformation of the bimodule of 1-
forms Ω1 was studied in [1, 2, 20] and requires ∇̂ to be Poisson-compatible and flat for a strictly
associative calculus at the next order of deformation. A main result [30, Corollary 4.2] in the
case of a Poisson–Lie group G is that flat ∇̂ for the semiclassical version of left translation-
invariant
(
Ω1,d
)
correspond to g∗, the dual of the Lie algebra of G, being a pre-Lie or Vinberg
algebra. Meanwhile, one of the powerful lessons of quantum geometry model building is that
very often no quantum differential calculus of the expected classical dimension exists. There are
broadly two remedies to this ‘quantum anomaly for differentiation’. One is to allow an extra
dimension in the cotangent bundle to absorb the anomaly [27, 29], which exits a deformation
theory setting. The other is to allow a breakdown of associativity at order λ2 in the deformation
parameter λ, as in [1, 2]. This corresponds at the Poisson level to allowing ∇̂ to have curvature,
which we do throughout.
Because this work stands at a triple point of Poisson geometry, noncommutative differential
geometry and deformation theory, there are necessarily significant preliminaries, collected in
Section 2. This will entail compromises in notation (such as both left invariant and right
invariant vector fields, instead of keeping everything to the left). We also include key Lemmas 3.1
and 4.1 known to experts on Poisson–Lie groups but which should be helpful to those coming
from noncommutative geometry. The first genuinely new result is Proposition 3.2 which better
characterises one-sided translation invariance of ∇̂ on a Poisson–Lie group as
ξ.∇̂ητ = ∇̂ξ.ητ + ∇̂η(ξ.τ) + i
δ̃1
ξ
(η)
(
δ2
ξ.τ
)
for all ξ ∈ g and 1-forms η, τ , with notation δ(ξ) = δ1
ξ ⊗ δ2
ξ (sum of terms understood). Here i is
interior product and ξ. is Lie derivative along the left-invariant vector field ξ̃ for the infinitesimal
action of G on itself. Theorem 3.5 gives a Lie-bialgebra characterisation of 2-sided translation
invariant (bicovariant) ∇̂ in terms of data Ξ: g∗ ⊗ g∗ → g∗. Section 4 polarises these concepts
to the case of a Poisson Lie group G Poisson-acting on a Poisson manifold X with a G-covariant
Poisson-compatible contravariant connection ∇̂ on X.
This then feeds into our novel formulation of a Poisson principal bundle in Section 5. The
key new idea here is a transversality condition
iξ̃∇̂ητ = πX(η,diξ̃τ) + i
Ξ̃∗1ξ
(η)i
Ξ̃∗2ξ
(τ)
for all ξ ∈ g and η, τ ∈ Ω1(X), see Definition 5.1. Here πX is the Poisson bivector and Ξ∗(ξ) =
Ξ∗1ξ ⊗ Ξ∗2ξ (sum of terms understood). An important consequence is that the base M of the
bundle inherits a Poisson structure with Poisson-compatible contravariant connection ∇̂M . This
fits with our point of view of such data being the right notion of a ‘semiclassical manifold’, as
infinitesimal data for the quantisation of both the algebra and its differential structure. Along
the way, Corollaries 4.4 and 5.2 provide some consequences on the Schouten bracket [η, τ ] of
1-forms in this context.
Poisson Principal Bundles 3
The final Section 6 formulates the semiclassical notion of a spin connection on a quantum
principal bundle and quantum associated bundles. We first formulate a quantum spin connection
as an equivariant bimodule connection ∇P : P → Ω1
hor. Then we show that the semiclassical
data for the quantisation of a classical such ∇P is a map Γ: C∞(X)→ Ω1
hor such that
Γ(ap) = aΓ(p) + ∇̂Xda∇P p− ∇̂Xdpda−∇P {a, p}X
for all a ∈ C∞(M), p ∈ C∞(X). The main result, Theorem 6.2, is that every classical spin
connection has a canonical such Γ for the semiclassical level. The proof makes critical use of the
bundle transversality condition as well as bicovariance of ∇̂G on the fibre, bringing together our
results.
The paper concludes with some directions for further work. Note that the seminal work [17]
considered a notion of contravariant connections on ordinary principal bundles over a Poisson
manifold that induce contravariant connections on associated bundles. This is not what we do
(we consider the deformation of usual and not contravariant connections on Poisson principal
bundles), but is certainly of interest. Meanwhile, [6] showed existence and uniqueness of the
deformation-quantisation of principal bundles in sense of C∞(X)[[λ]] a right module over the
∗-product C∞(M)[[λ]] equivariantly under the structure group G, which is not quantised. The
latter could be seen as a special case of a PLG with zero Poisson bracket and even here, the
paper found obstructions for the bimodule case. These are very different considerations from the
ones in the present paper but they support our view that we may need nonassociative geometry
at higher deformation order.
2 Preliminaries
2.1 Elements of noncommutative differential geometry
We outline a formalism of noncommutative differential geometry in a bimodule approach, includ-
ing Hopf algebras with differential structure. The classical model is A = C∞(M) for a smooth
real manifold M with exterior algebra Ω(M), while the general formulation works over any
unital algebra with an algebraically defined exterior algebra ΩA.
Thus, a ‘differential calculus’ on a unital algebra A consists of a graded algebra ΩA = ⊕Ωn
A
with n-forms Ωn
A for n ≥ 0, an associative product ∧ : Ωn
A ⊗ Ωm
A → Ωn+m
A and an exterior
derivative d: Ωn
A → Ωn+1
A satisfying the graded-Leibniz rule. Here Ω0
A = A and we assume
that ΩA is generated by degree 0, 1 and that Ω1
A 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 → Ω1
A ⊗A E obeying the left Leibniz
rule ∇E(ae) = da⊗ e+ a∇Ee for all a ∈ A, e ∈ E. We have a bimodule connection [16, 31] if E
is a bimodule and there is a bimodule map with
σE : E ⊗A Ω1
A → Ω1
A ⊗A E, ∇E(ea) = (∇Ee)a+ σE(e⊗A da).
If σE 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, where the tensor product connection on E ⊗A F uses σE to reorder the output of ∇F .
More details and results are in [4].
The notion of a Hopf algebra is covered in several texts, e.g., [25]. Briefly, this means
a unital algebra H which is also a coalgebra with counit ε : H → k (for k the ground field) and
coproduct ∆: H → H ⊗H, with these maps being algebra homomorphisms, and for which an
antipode S : H → H exists. The latter is required to obey (Sh(1))h(2) = ε(h) = h(1)Sh(2) in the
4 S. Majid and L. Williams
‘Sweedler notation’ ∆h = h(1) ⊗ h(2) (sum of such terms implicit). A 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 V = A then
we say that A is a comodule algebra if ∆R is an algebra homomorphism. If we have a suitable
dually paired Hopf algebra (the algebraic dual typically being to big) then A becomes a module
algebra for the left action . given by evaluation against the left factor of the output of ∆A,
meaning φ.(ab) = (φ(1).a)(φ(2).b) and φ.1 = ε(φ)1 for φ in the dually paired Hopf algebra.
If H coacts from the left then we similarly have a right action / by evaluation against the left
factor of the output of ∆L.
In particular, a Hopf algebra coacts on itself as a comodule algebra from both the left and
the right via ∆. A calculus Ω1
H is left (resp. right) translation covariant if the coaction extends
to Ω1 by
∆L(hdg) = h(1)g(1) ⊗ h(2)dg(2), ∆R(hdg) = h(1)dg(1) ⊗ h(2)g(2)
(making it a ‘Hopf module’) with the coaction commuting with d. A calculus is called bicovariant
if both coactions extend, and strongly bicovariant if ∆L + ∆R makes the exterior algebra ΩH
into a super-Hopf algebra. We will not need higher forms, but this is true for the canonical ΩH
in the bicovariant case in [36].
Finally, and equally briefly, the data for a quantum principal bundle is P a right H-comodule
algebra with ‘base’ the algebra A = PH = {p ∈ P |∆Rp = p ⊗ 1} of elements fixed under
∆Rp = p ¯(0)⊗p ¯(1) in a compact notation. We assume that Ω1
P is H-covariant in that ∆R extends
in a similar way to right-covariance of Ω1
H above. In this case Ω1
A = AdA computed in Ω1
P
is the inherited calculus. We also assume that Ω1
H is bicovariant and let Λ1
H be the space of
left-invariant 1-forms. Next, we assume that there is a well-defined map ver : Ω1
P → P ⊗ Λ1
H
(the generator of vector fields for the infinitesimal action) given by
ver(pdq) = pq ¯(0) ⊗
(
Sq ¯(1)(1)
)
dq ¯(1)((2) = pq ¯(0) ⊗$πε
(
q ¯(1)
)
, ∀ p, q ∈ P, (2.1)
where πε : H → H+ defined by πε(h) = h− 1ε(h) projects onto the kernel of the counit, and the
Maurer–Cartan form $ : H+ → Ω1 is defined by $(h) = Sh(1)dh(2) and affords an identification
of Λ1
H as a quotient of H+. Finally, as a replacement for local triviality, we require a short exact
sequence of left P -modules
0 −→ PΩ1
AP −→ Ω1
P
ver−−→ P ⊗ Λ1
H −→ 0. (2.2)
More details are in [7, 28] and [4, Chapter 5]. We will at certain points need to discuss this
algebraic theory, adapted to a putative formal deformation version, but only to the point of
extracting Poisson level data which then make sense by themselves in the smooth setting.
2.2 Elements of deformation theory
In deformation theory, it is usual to adapt the algebraic context above by now working over the
ring C[[λ]] for a formal deformation parameter λ and with an associative C[[λ]]-linear ∗-product
of the form a ∗ b =
∑∞
i=0Ci(a, b)λ
i on A = C∞(M)[[λ]], see, e.g., [9]. In this context, it is well
known that
a ∗ b− b ∗ a = λ{a, b}+O
(
λ2
)
.
defines a Poisson bracket {a, b} = C1(a, b) − C1(b, a) for a, b ∈ C∞(M). We denote the as-
sociated Poisson bivector by π so that {a, b} = π(da,db). Similarly, if we have a vector
Poisson Principal Bundles 5
bundle over M with space of sections Γ as a projective module over C∞(M), and this is de-
formed to an A-bimodule Γ[[λ]] with left and right bullet products a • η =
∑∞
i=0 Li(a, η)λi and
η • a =
∑∞
i=0Ri(η, a)λi then
a • η − η • a = λ∇̂daη +O
(
λ2
)
for all a ∈ C∞(M), η ∈ Γ, where ∇̂daη := L1(a, η) − R1(η, a) is a contravariant connection or
‘Lie–Rinehart’ covariant derivative [9, 17, 21] characterised by
∇̂aζ = a∇̂ζ , ∇̂ζ(aη) = π(ζ,da)η + a∇̂ζη, (2.3)
for all ζ ∈ Ω1(M). In the strictly associative case, this will be flat (as a special case of a more
general statement in [9]). We will be interested in the case Γ = Ω1(M), the classical space of
1-forms, and for Ω1[[λ]] to obey the Leibniz rule with an undeformed exterior derivative d, which
requires the further Poisson-compatibility
[η, τ ] = ∇̂ητ − ∇̂τη (2.4)
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 ] + aπ(η,db)τ − bπ(τ,da)η. More usually in the literature (see, e.g., [17]), con-
dition (2.4) is called ‘torsionless’ and Poisson-compatible refers to something stronger, whereas
we reserve torsion for its usual meaning for linear connections. Equations (2.3)–(2.4) are the
geometric version on general 1-forms of the conditions [1, 3, 20]
∇̂d(ab)η = a∇̂dbη + (∇̂daη)b, ∇̂da(bη) = b∇̂da(η) + {a, b}η,
d{a, b} = ∇̂dadb− ∇̂dbda
coming out of the order λ analysis. The point of view in [1, 3] was to think if this equivalently
as a partially defined connection or preconnection ∇ by ∇̂da = ∇â, where â = {a, } is the
Hamitonian vector field associated to a ∈ C∞(M) and ∇ is a usual connection but only defined
along such vector fields. Both points of view are useful. In the symplectic case, the data is
equivalent to a linear connection which preserves the Poisson tensor up to torsion, and the
contravariant ∇̂ is its pull back along the map π# : Ω1(M)→ Vect(M) given by π#(η) = π(η, )
with π#(da) = â.
The strictly associative setting is, however, often not compatible with deformation theory in
that for examples related to quantum groups, one may not have Ω1, d with desired covariance
properties and with expected classical dimension over the noncommutative algebra (i.e., even in
the free module case where there is a dimension). This was noted at the Poisson level in [1, 20]
and leads into a somewhat different point of view [3] where we consider formal products of the
form
a ∗ b = ab+
λ
2
{a, b}+O
(
λ2
)
, a • η = aη +
λ
2
∇̂daη +O
(
λ2
)
,
η • a = aη − λ
2
∇̂daη +O
(
λ2
)
(2.5)
but demand associativity and the bimodule property only O
(
λ2
)
. The minimal setting is to
assume only that { , } is a biderivation and ∇̂ a possibly curved contravariant connection
compatible with it. This leads into fully nonassociative geometry, while the most important
special case appears to be the intermediate one where ∗ is associative (deformation-quantising
a Poisson manifold) but • is not necessarily a bimodule structure O
(
λ2
)
, so that { , } is a
Poisson bracket but ∇̂ can be curved. This leads to a self-contained setting, which we adopt, of
a Poisson manifold equipped with possibly curved Poisson-compatible contravariant connection.
6 S. Majid and L. Williams
2.3 Elements of Poisson–Lie theory
Following the work of Drinfeld [14], the semiclassical object underlying a deformation Hopf
algebra is a Poisson–Lie group (PLG). This means a Lie group G which is 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 π, this is
π(gh) = Lg∗(π(h)) +Rh∗(π(g))
for all g, h ∈ G. Here Rg : G → G and Lg : G → G are right and left translation by g and
have derivatives Rh∗ : TgG → TghG and Lg∗ : ThG → TghG. 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]) = adξ(δ(η))− adη(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 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 = etξ
and differentiating at t = 0 gives a right action of the Lie algebra g,
(a/ξ)(g) =
d
dt
∣∣∣
0
a
(
etξg
)
= ξ̀(a)(g),
defining an associated right-invariant vector field ξ̀. This can be defined also as ξ̀(g) = Rg∗(ξ).
The actions extends to 1-forms by
(τ/h)(g) = L∗h(τ(hg))
and /ξ = Lξ̀ = iξ̀d + diξ̀ on forms, where i denotes interior product and L the Lie derivative.
We similarly define the left-invariant vector field ξ̃ associated to ξ ∈ g as ξ̃(g) = Lg∗ξ for
ξ ∈ g equating to ξ. defined below by right translation. We also recall that there is a one to one
correspondence between 1-forms τ ∈ Ω1(G) and τ̃ ∈ C∞(G, g∗) via τ̃(g) = L∗g(τ(g)) or conversely
s ∈ C∞(G, g∗) defines a 1-form which we will denote š(g) = L∗g−1(s(g)). In these terms, the
action is (s/h)(g) = s(hg) as for a function. In particular, v̌(g) = L∗g−1v is the left-translation
invariant 1-form (invariant under /) associated to v ∈ g∗ and 〈ξ̃, v̌〉 = 〈ξ, v〉 is constant. We
also have
〈
ξ,
(
d̃a
)
(g)
〉
= ξ̃(a)(g). Finally, in the right translation-invariant formulation, we start
with the left action
(h.a)(g) = a(gh), (ξ.a)(g) =
d
dt
∣∣∣
t=0
a
(
getξ
)
= ξ̃(a)(g)
is the left-invariant vector field ξ̃ for the infinitesimal action of ξ ∈ g. The group action extends
to differential forms τ ∈ Ω1(G) by
(h.τ)(g) = R∗h(τ(gh))
with Lie algebra action . given by Lie derivative along ξ̃. In this case, the relevant one to one
correspondence between 1-forms τ ∈ Ω1(G) and, say, τ̀ ∈ C∞(G, g∗) is via τ̀(g) = R∗g(τ(g))
Poisson Principal Bundles 7
with inverse s ∈ C∞(G, g∗) defining a 1-form s̆(g) = R∗g−1(s(g)). In these terms, the action is
(h.s)(g) = s(gh) as for a function. In particular, v̆(g) = R∗g−1v is a right-translation invariant 1-
form (invariant under .) associated to v ∈ g∗. We have 〈ξ̀, v̆〉 = 〈ξ, v〉 and 〈ξ, (d̀a)(g)〉 = ξ̀(a)(g)
where ξ̀(g) = Rg∗ξ is the right-invariant vector field associated to ξ ∈ 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., π(g) = Rg∗(D(g)). We define
δξ :=
d
dt
∣∣∣
t=0
D
(
etξ
)
∈ g⊗ g, ∀ ξ ∈ g.
Drinfeld showed that if (g, δ) is a Lie bialgebra then the associated connected and simply con-
nected 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 ∈ Z1
Ad(G, g⊗ g). More details are in [25].
Drinfeld [14] also introduced a deformation theory point of view where quantum groups
Uq(g) are deformations of U(g) as a Hopf algebra with q = e
λ
2 . Suffice it to say that the
coproduct homomorphism property applied to ∆(ξη− ηξ) leads at order λ to the cocycle axiom
for δ := ∆ − τ ◦ ∆, with τ the transposition map. Meanwhile, if given δ, we can suppose a
specific form of deformation,
∆ξ = ξ ⊗ 1 + 1⊗ ξ +
λ
2
δ(ξ) +O
(
λ2
)
(2.6)
for ξ ∈ g as the analogue on the coalgebra side of the specific form in (2.5) on the algebra
side. Here δ can be viewed as a Lie bracket on g∗ which in turn is the Poisson bracket of linear
functions on G near the identity. These quantum groups are dual to corresponding Cq[G] which
again have formal deformation versions. A complication here is that C∞(G) is only a Hopf
algebra with respect to a topological tensor product, but this can be handled in the compact
connected case [5]. In practice, one can work with a dense algebraic model C[G], generated in the
case of classical Lie type by the matrix entry coordinate functions tij according to ρ : G ⊂Mn.
Here ρ extends to U(g) and provides a dual pairing U(g) by 〈h, tij〉 = ρ(h)ij of Hopf algebras.
With such complications understood, we will similarly suppose for the sake of discussion some
formal deformation C∞(G)[[λ]] for the specific leading form of product (2.5) and a dual U(g)[[λ]]
with the leading form (2.6) of coproduct.
3 Poisson–Lie groups and their semiclassical calculi revisited
We start with a more geometric characterisation of the PLG condition.
Lemma 3.1 (cf. [12, 23, 35]). 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) = δ1
ξ ⊗ δ2
ξ (sum of such terms understood). Then
ξ . {a, b} = {ξ . a, b}+ {a, ξ . b}+
(
δ1
ξ . a
)(
δ2
ξ . b
)
, (3.1)
for all ξ ∈ g, a, b ∈ C∞(G) if and only if G is a PLG via { , }.
Proof. The proof in one direction, where the PLG condition holds, is clear from [23] but we
include it for completeness. This can also be found in [12] as well as easily derived from [35,
Proposition 10.7]. We recall that {a, b}(g) = 〈π(g),da ⊗ db〉, where evaluation here is the
8 S. Majid and L. Williams
standard one between sections of the tangent bundle and cotangent bundle extended to tensor
products. Then
ξ.{a, b} = ξ̃({a, b}) = 〈Lξ̃π,da⊗ db〉+
〈
π,dξ̃(a)⊗ db+ da⊗ dξ̃(b)
〉
with the last two terms {ξ.a, b}+ {a, ξ.b}. For the first term, if the PLG condition holds then
(Lξ̃π)(g) = (ξ.π)(g) =
d
dt
∣∣∣
0
Re−tξ∗π
(
getξ
)
=
d
dt
∣∣∣
0
(
Re−tξ∗Lg∗π
(
etξ
)
+ π(g)
)
=
d
dt
∣∣∣
0
Lg∗D
(
etξ
)
= Lg∗δξ,
which evaluates on da ⊗ db to δ̃1
ξ (a)δ̃2
ξ (b), cf. [23, p. 54]. Conversely, if (3.1) holds then we
deduce that the differential at t = 0 of Re−tξ∗π
(
getξ
)
− Re−tξ∗Lg∗π
(
etξ
)
vanishes, i.e., that
Adg∗δξ = d
dt
∣∣
0
D
(
getξ
)
= ξ̃(D)(g). If we suppose that D(e) = 0 and that G is connected, then D
is a 1-cocycle on the group (by the same argument as in the proof of [25, Theorem 8.4.1]).
Hence G is a PLG. �
Next, we consider the semiclassical data for differential calculus, i.e., a Poisson-compatible
contravariant connection ∇̂. For any contravariant connection, we define
∇̃da : C∞(G, g∗)→ C∞(G, g∗), ∇̃das := ˜◦ ∇̂daš,
where š is the 1-form corresponding to s ∈ C∞(G, g∗) and ˜ in this context turns a 1-form back
into an element of C∞(G, g∗), as explained in Section 2.3. The starting point in [1] is that ∇̃
necessarily has the form
(∇̃das)(g) = {a, s}(g) + Ξ̃(g, d̃a(g), s(g)), (3.2)
for some map Ξ̃ : 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 if Ξ̃(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 if and only if
Ξ(φ, ψ)− Ξ(ψ, φ) = [φ, ψ]g∗ , ∀φ, ψ ∈ g∗. (3.3)
This 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 case, ∇̀da : C∞(G, g∗)→ C∞(G, g∗) defined by ∇̀das := `◦ ∇̂das̆ has
the form
(∇̀das)(g) = {a, s}(g) + Ξ̀(g, d̀a(g), s(g))
and is right translation covariant if and only if Ξ̀ is again given by a constant Ξ: g∗ ⊗ g∗ → g∗,
with Poisson-compatibility if and only if (3.3) holds. The latter and the contravariant connection
being flat is equivalent to Ξ being a pre-Lie structure for g∗ [30] (but we do not limit ourselves
to this case.) Our next goal is a more geometric reformulation of these results as following from
Lemma 3.1.
Proposition 3.2. Let G be a connected PLG and ∇̂ a contravariant connection on Ω1(G).
Then ∇̂ is right-translation covariant in the sense
ξ.∇̂ητ = ∇̂ξ.ητ + ∇̂η(ξ.τ) + i
δ̃1
ξ
(η)
(
δ2
ξ.τ
)
(3.4)
for all τ, η ∈ Ω1(G) and ξ ∈ g if and only if Ξ̀ is constant on G.
Poisson Principal Bundles 9
Proof. In terms of s ∈ C∞(G, g∗) in the right-translation invariant formulation,(
ξ.∇̀das
)
(g) = ξ̃({a, s}) + ξ̃(Ξ̀)
(
g, `(da)(g), s(g)
)
+ Ξ̀
(
g, ξ̃(d̀a)(g), s(g)
)
+ Ξ̀
(
g, `(da)(g), ξ̃(s)(g)
)
= {ξ.a, s}+ {a, ξ.s}+
(
δ1
ξ.a
)(
δ2
ξ.s
)
+ ξ̃(Ξ̀)
(
g, `(da)(g), s(g)
)
+ Ξ̀
(
g, ξ̃(d̀a)(g), s(g)
)
+ Ξ̀
(
g, `(da)(g), ξ̃(s)(g)
)
= ∇̀dξ.as+ ∇̀da(ξ.s) +
(
δ1
ξ.a
)(
δ2
ξ.s
)
+ ξ̃(Ξ̀)
(
g, `(da)(g), s(g)
)
using that ξ. is the action of ξ̃ on functions and commutes with d, being the Lie derivative
along ξ̃ on 1-forms. Note that ξ̃(Ξ̀) is ξ̃ acting on Ξ̀ as a function on G (its first argument). We
see that
ξ.∇̂dadb = ∇̂d(ξ.a)db+ ∇̂dad(ξ.b) +
(
δ1
ξ.a
)
d
(
δ2
ξ.b
)
(3.5)
holds for all ξ ∈ g, a, b ∈ C∞(G) if and only if Ξ̀ is constant in its first argument. This extends
to general 1-forms as (3.4) in a straightforward manner, the details of which are omitted. �
Remark 3.3. Let G be connected and simply connected with Lie algebra g. Let U(g)[[λ]] be
a deformation of U(g) with coproduct of the form (2.6), dual to C∞(G)[[λ]] as a deformation
of C∞(G) with product ∗ as in (2.5). That the latter is a left module algebra appears to leading
order as (3.1). That a deformation Ω1(G)[[λ]] is a left and right Hopf module for the • products
as in (2.5) appears as (3.5). In fact, we only need these deformations to O
(
λ2
)
so that the
required notion of covariance applies even when ∇̂G has curvature, as explained in Section 2.2.
Remark 3.4. In the original left translation-invariant conventions of [1, 30], our covariance
condition for a PLG in Lemma 3.1 comes out equivalently as
{a, b}/ξ = {a/ξ, b}+ {a, b/ξ}+
(
a/δ1
ξ
)(
b/δ2
ξ
)
. (3.6)
Similarly, for a left-translation covariant differential structure, we need
∇̂dadb/ξ = ∇̂d(a/ξ)db+ ∇̂dad(b/ξ) +
(
a/δ1
ξ
)
d
(
b/δ2
ξ
)
or on general 1-forms
(∇̂ητ)/ξ = ∇̂η/ξτ + ∇̂η(τ/ξ) + i
δ̃1
ξ
(η)
(
τ/δ2
ξ
)
, (3.7)
and this holds if and only if the original Ξ̃ in (3.2) is constant on G.
If one has both left and right covariance then we say that our contravariant connection is
bicovariant.
Theorem 3.5. A right-covariant contravariant connection given by ∇̃das = {a, s} + Ξ
(
d̃a, s
)
as in (3.2) by Ξ: g∗ ⊗ g∗ → g∗ is bicovariant if and only if
δg∗Ξ(φ, ψ)− Ξ(φ(1), ψ)⊗ φ(2) − Ξ(φ, ψ(1))⊗ ψ(2) = ψ(1) ⊗ [φ, ψ(2)]g∗ ,
where δg∗(φ) = φ(1) ⊗ φ(2), or equivalently in dual form,
Ξ∗[η,ξ] =
[
Ξ∗1η , ξ
]
⊗ Ξ∗2η + Ξ∗1η ⊗
[
Ξ∗2η , ξ
]
+ δ1
ξ ⊗
[
η, δ2
ξ
]
for all η, ξ ∈ g, where Ξ∗, δ : g → g ⊗ g are written explicitly (with a sum of such terms
understood).
10 S. Majid and L. Williams
Proof. Under the first identification Ω1(G) with C∞(G, g∗) in Section 2.3, the left action on
1-forms appears as h.s = L∗gR
∗
hL
∗
h−1g−1(s(gh)) which differentiates d
dt
∣∣
0
for h = etξ to (ξ.s)(g) =
ξ̃(s)(g) + ad∗ξ(s(g)) where adξ(φ) = φ(1)〈φ(2), ξ〉 in terms of the Lie cobracket on g∗. Also note
that ξ.da = dξ̃(a) as the action commutes with d. Then the left-covariance in the form used in
Proposition 3.2 appears in these terms as the condition
ξ̃({a, s}) + ad∗ξ({a, s}) + ξ̃
(
Ξ(d̃a, s)
)
+ ad∗ξΞ
(
d̃a, s
)
= {ξ̃(a), s}+ Ξ
(
d̃
(
ξ̃(a)
)
, s
)
+
{
a, ξ̃(s) + ad∗ξ(s)
}
+ Ξ
(
d̃a, ξ̃(s) + ad∗ξ(s)
)
+ δ̃1
ξ (a)
(
δ̃2
ξ (s) + ad∗δ2
ξ
(s)
)
.
We now expand ξ̃{a, s} using the PLG covariance in Lemma 3.1 (noting that s is a g∗-valued
function under our trivialisation of the bundle) and we expand ξ̃
(
Ξ(d̃a, s)
)
= Ξ
(
ξ̃
(
d̃a
)
, s
)
+
Ξ
(
d̃a, ξ̃(s)
)
since Ξ itself is constant on G, and cancel terms. Then our condition becomes
ad∗ξ({a, s}) + ad∗ξΞ
(
d̃a, s
)
+ Ξ
(
ξ̃
(
d̃a
)
, s
)
= Ξ
(
d̃
(
ξ̃(a)
)
, s
)
+ {a, ad∗ξ(s)}+ Ξ
(
d̃a, ad∗ξ(s)
)
+ δ̃1
ξ (a)ad∗δ2
ξ
(s).
Next, we note that ad∗ξ({a, s}) = {a, ad∗ξ(s)} as ad∗ acts only on the g∗ values and that d̃
(
ξ̃(a)
)
−
ξ̃
(
d̃a
)
= ad∗ξ
(
d̃a
)
deduced from formulae in the preliminaries (or by expanding da = ∂i(a)f i
where ∂i = ẽi in a basis of the Lie algebra with dual basis {f i}). Using these identities, our
condition for bicovariance reduces to
ad∗ξΞ
(
d̃a, s
)
= Ξ
(
ad∗ξ
(
d̃a
)
, s
)
+ Ξ
(
d̃a, ad∗ξ(s)
)
+ δ̃1
ξ (a)ad∗δ2
ξ
(s),
which being true for all s and all a reduces to
ad∗ξΞ(φ, ψ) = Ξ(ad∗ξφ, ψ) + Ξ(φ, ad∗ξψ) +
〈
δ1
ξ , φ
〉
ad∗δ2
ξ
ψ
for all φ, ψ ∈ g∗, and for all ξ ∈ g. This is the first stated form of the condition in terms of the
Lie cobracket (when evaluated against ξ in the second tensor factor and converting the bracket
on g∗ to a cobracket on ξ). �
The stated characterisation of bicovariant Ξ was obtained in the pre-Lie algebra flat connec-
tion case in [30], but now we see that the same holds in general using our new methods.
Example 3.6. At the algebraic level, we consider the standard Hopf algebra Cq[SL2] with
generators a, b, c, d and relations
ba = qab, ca = qac, db = qbd, dc = qcd, bc = cb,
ad− da =
(
q−1 − q
)
bc, ad− q−1bc = 1,
and a standard matrix form of coproduct and antipode [25]. There is a left translation covariant
Ω1(Cq[SU2]) with [36]
e0 = dda− qbdc, e+ = q−1adc− q−2cda, e− = ddb− qbdd
a left basis of left-invariant 1-forms and the right module structure given by the bimodule
relations
e0f = q2|f |fe0, e±f = q|f |e±,
Poisson Principal Bundles 11
for homogeneous f of degree |f |, where |a| = |c| = 1 and |b| = |d| = −1. The exterior derivative
is
da = ae0 + qbe+, db = ae− − q−2be0, dc = ce0 + qde+, dd = ce− − q−2de0.
We now set q = e
λ
2 , for λ a deformation parameter and work at the corresponding semiclassical
level albeit focussing on the polynomial subalgebra generators. Classically, the basis e0, e± of
1-forms is dual to the basis
{
H̃, X̃±
}
of left-invariant vector fields generated by the Chevalley
basis {H,X±} of su2. The latter can be read off from df = H̃(f)e0 + X̃±(f)e± (sum over ±) as
H̃
(
a b
c d
)
=
(
a −b
c −d
)
, X̃+
(
a b
c d
)
=
(
0 a
0 c
)
, X̃−
(
a b
c d
)
=
(
b 0
d 0
)
.
The left coaction covariance of the calculus corresponds in Remark 3.4 to covariance under
a right action of the Lie algebra, with right-invariant vector fields
H̀
(
a b
c d
)
=
(
a b
−c −d
)
, X̀+
(
a b
c d
)
=
(
c d
0 0
)
, X̀−
(
a b
c d
)
=
(
0 0
a b
)
.
(1) From the algebra relations, the Poisson bracket can be read off on the generators as
{a, b} = −1
2ab, {a, c} = −1
2ac, {a, d} = −bc, {b, d} = −1
2bd, {c, d} = −1
2cd
and {b, c} = 0, from which one can easily verify (3.6) as a check on calculations, with δ(X±) =
1
2(X± ⊗H −H ⊗X±) and δ(H) = 0.
(2) From the bimodule relations, [30] provides the Poisson-compatible contravariant connec-
tion as
∇̂
d
(
a b
c d
)ei =
1
2
ti
(
a −b
c −d
)
ei, t0 = −2, t± = −1,
where i ∈ {0,±}, from which one can easily verify (3.7) as a check.
4 PLG actions on 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 version H = C[G] of C∞(G)
then correspondingly P would be a right H-comodule algebra with coaction ∆Rp = p ¯(0)⊗p ¯(1) ∈
P ⊗H.
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 structure [24], 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
to consider each map β( , g) : X → X as a Poisson map, which effectively would mean the zero
Poisson bracket on G.
Adopting a similar notation as in the preceding section, we let πX and πG be the respective
Poisson tensors and we define
βg : X → X, βx : G→ X, βg(x) = βx(g) = x.g
12 S. Majid and L. Williams
with differentials
βg∗ : TxX → Tx.gX, βx∗ : TgG→ Tx.gX,
and extensions to act on tensor products in the usual way. Then the condition for a PLG action
is clearly
πX(x · g) = βg∗πX(x) + βx∗πG(g),
for all x ∈ X, g ∈ G. Following our previous notation, we also denote by ξ̃ = ξ. the vector
field associated to ξ ∈ g for the infinitesimal action on X. This is the left action of g = etξ on
functions differentiated in t at t = 0, i.e., ξ̃(x) = βx∗(ξ) if we view g = TeG. Since these vector
fields are defined for each element ξ, we can think of them all together as a single map
ver : Ω1(X)→ C∞(X)⊗ g∗
such that when evaluated against element ξ of the Lie algebra we recover ξ̃ in the form verξ =
(id⊗ξ)ver = iξ̃ : Ω1(X)→ C∞(X). Our first observation is a known result but cast in a language
closer to a quantum group module algebra.
Lemma 4.1 (cf. [23, p. 54]). Let G be a connected Poisson–Lie group, X a Poisson manifold
and β a smooth right action of the group G on X. This is a PLG action if and only if
ξ . {p, q} = {ξ . p, q}+ {p, ξ . q}+
(
δ1
ξ . p
)(
δ2
ξ . q
)
(4.1)
holds for all p, q ∈ C∞(X), ξ ∈ g.
Proof. This follows the same pattern as in the proof of Lemma 3.1 and is, moreover, equivalent
to [23, p. 54]. Hence we omit any details. �
There is clearly an analogous analysis to [1] as to the natural notion of a Poisson-compatible
contravariant connection ∇̂X on X being compatible, polarising the case of G acting on itself
with contravariant connection ∇̂G. Without giving such an analysis here, we take the polarised
version of (3.4) as a definition, i.e.,
Definition 4.2. Let G be a connected PLG on a PLG-covariant Possion manifold X. A con-
travariant connection ∇̂ on Ω1(X) is similarly G-covariant if and only if
ξ.∇̂ητ = ∇̂ξ.ητ + ∇̂η(ξ.τ) + (i
δ̃1
ξ
η)
(
δ2
ξ.τ
)
, (4.2)
for all ξ ∈ g and η, τ ∈ Ω1(X).
Remark 4.3. One can check that this is compatible with Lemma 4.1 and the axioms of a
contravariant connection in the sense that ξ.∇̂dp(qτ) = ξ.
(
{p, q}τ+q∇̂dpτ
)
expanded using (4.2)
on the left hand side agrees with (4.1) and ξ.∇̂dpτ expanded separately. It similarly respects
Poisson-compatibility, e.g., on exact forms we have
ξ . ∇̂dpdq = ∇̂d(ξ.p)dq + ∇̂dpd(ξ . q) +
(
δ1
ξ . p
)
d
(
δ2
ξ . q
)
,
for all ξ ∈ g and p, q ∈ C∞(X). Subtracting the same with p↔ q gives d{p, q} on using (4.1), an-
tisymmetry of the output of δξ and the Poisson-compatibility applied to d{ξ.p, q} and d{p, ξ.q}
separately. The parallel of Remark 3.3 also applies. If U(g)[[λ]] acts on C∞(X)[[λ]] in a de-
formation context then ξ.(p ∗ q − q ∗ p) computed as a module algebra gives the PLG action
condition (4.1) from the order λ terms. Similarly, if this action extends to a deformed calculus
Ω1(X)[[λ]] then ξ.(p • dq − dq • p) computed as a module algebra requires (4.2). This then
generalises to arbitrary 1-forms as (4.2). In fact, we only need these deformations to O
(
λ2
)
as
our contravariant connections could be curved.
Poisson Principal Bundles 13
To explain the next corollary, note that the Lie derivative can be formally extended to one
along antisymmetric tensors V by the formula LV = iV d− (−1)|V |diV where iV : Ω(X)→ Ω(X)
lowers degree by that of V . In particular, iv∧w = iviw (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 Lv∧w = [iv∧w,d] along antisymmetric bivector fields. We will also need, as in [29], the
Leibnizator LB(τ, η) = B(τ ∧ η)− (Bτ)∧ η− (−1)|B||τ |τ ∧Bη for B an operator on the exterior
algebra of degree |B|.
Corollary 4.4. Let ∇̂ be a PLG-covariant contravariant connection on Ω1(X) in the sense of
Definition 4.2. If this is Poisson-compatible then
ξ.[τ, η] = [ξ.τ, η] + [τ, ξ.η] + 1
2LLδ̃ξ(τ, η)
also holds for all η, τ ∈ Ω1(X) and ξ ∈ g.
Proof. Recall that Poisson-compatibility of the connection is [τ, η] = ∇̂τη−∇̂ητ . Hence in the
covariant case,
ξ.[τ, η] = ξ.(∇̂τη − ∇̂ητ)
= ∇̂ξ.τη + ∇̂τ (ξ.η) +
(
iδ̃1
ξ
τ
)(
δ2
ξ.η
)
− ∇̂ξ.ητ − ∇̂η(ξ.τ)−
(
iδ̃1
ξ
η
)(
δ2
ξ.τ
)
= [ξ.τ, η] + [τ, ξ.η] +
(
iδ̃1
ξ
τ
)(
δ2
ξ.η
)
−
(
iδ̃1
ξ
η
)(
δ2
ξ.τ
)
.
Moreover, if v ∧ w is an antisymmetric product of vector fields on X and τ , η are 1-forms, we
have
Lv∧w(τ ∧ η) = iviw(dτ ∧ η − τ ∧ dη)− d(iviw(τ ∧ η))
= iv∧w(dτ)η − iv(η)iw(dτ) + iv(dτ)iw(η)− iv(dη)iw(τ)
+ iv(τ)iw(dη)− iv∧w(dη)τ − d(iv(η)iw(τ)− iv(τ)iw(η))
= iv∧w(dτ)η − iv∧w(dη)τ + 2iv(τ)(iw(dη) + diw(η)) + 2iw(η)(iv(dτ) + div(τ))
= iv∧w(dτ)η − iv∧w(dη)τ + 2iv(τ)Lwη + 2iw(η)Lvτ
so that
iv(τ)Lwη + iw(η)Lvτ = 1
2
(
Lv∧w(τ ∧ η)− (iv∧wdτ)η + τ(iv∧wdη)
)
= 1
2LLv∧w(τ, η)
allowing us to write the δξ terms as stated in terms of the bivector δ̃ξ = δ̃1
ξ ∧ δ̃2
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.
Remark 4.5. We also have a right-covariance condition for a PLG left action α : G×X → X,
namely
{p, q}/ξ = {p/ξ, q}+ {p, q/ξ}+
(
p/δ1
ξ
)(
q/δ2
ξ
)
,
where /ξ on functions is the vector field ξ̀ for the infinitesimal action. The covariance of the
calculus and the consequence of Poisson-compatibility now appear as
(∇̂ητ)/ξ = ∇̂η/ξτ + ∇̂η(τ/ξ) + (i
δ̀1
ξ
η)
(
τ/δ2
ξ
)
,
[τ, η]/ξ = [τ/ξ, η] + [τ, η/ξ] + 1
2LLδ̀ξ(τ, η).
14 S. Majid and L. Williams
Example 4.6. Here we consider the right coaction ∆R(f) = f⊗t|f | of H = Cq2
[
S1
]
= C
[
t, t−1
]
with dt.t = q2tdt on f ∈ P = Cq[SU2] with its calculus and notations as in Example 3.6. At
the semiclassical level, G = S1 acting from the right on X = SU2 (as a diagonal subgroup) as
a PLG with the zero Poisson bracket, so δ = 0. From the calculus, dt.t = (1 + λ)t.dt + O
(
λ2
)
gives the Poisson-compatible contravariant connection
∇̂Gdtdt = −tdt
on Ω1
(
S1
)
. This has left-invariant basic 1-form t−1dt with dual ξ̃ = t ∂∂t as a left-invariant vector
field. Then ∇̂t−1dt
(
t−1dt
)
= −t−1dt, which translates to
Ξ∗ξ = −ξ ⊗ ξ. (4.3)
It is also clear from the form of the coaction that ξ̃(f) = |f |f so that ξ̃ = H.( ) = H̃ as a vector
field for the infinitesimal action on SU2 using the partial derivative in the e0 direction as given
in Example 3.6. The Poisson bracket on SU2 in Example 3.6 is then easily seen to be right
covariant for the action of S1. This is well-known and details are omitted. For the covariance
condition in Definition 4.2, we compute H.e0 = 0, H.e+ = 2e+, H.e− = −2e−. Then, for
example
H.∇̂
d
(
a b
c d
)e+ = H.
(
1
2
t+
(
a −b
c −d
)
e+
)
= −1
2
(
a b
c d
)
e+ −
(
a −b
c −d
)
e+
= ∇̂
d
(
H.
(
a b
c d
))e+ + ∇̂
d
(
a b
c d
)(H.e+)
as required since δ = 0.
5 Formulation of Poisson principal bundles
A principal bundle in a smooth setting is a smooth manifold X with a smooth, free and proper
action of a Lie 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 Ω1
hor (the pull back of Ω1(M) along the canonical projection X → M) is
precisely the joint kernel of the vector fields for the infinitesimal action. The invariant horizontal
forms are then the forms on the base, Ω1(M) ↪→ Ω1
hor(X). In the quantum case, the transver-
sality is exactness of the sequence (2.2), the horizontal forms are PΩ1
AP with (under reasonable
conditions) invariants under the coaction of H recovering the quantum calculus Ω1
A ⊆ Ω1
P .
Therefore the main missing ingredient we need is the semiclassical version of this transversality
condition. Motivated by the assumption of ∗-product quantisations, see Remark 5.5, we are led
to the following.
Definition 5.1. A Poisson principal bundle means a classical principal bundle X → X/G with
1. X a Poisson manifold and G a Poisson–Lie group such that the action on X is a Poisson–Lie
group action of G in the sense of Lemma 4.1.
2. A bicovariant Poisson-compatible connection on G given by Ξ: g∗ ⊗ g∗ → g∗.
3. A Poisson-compatible connection on X which is left G-covariant in the sense of Defini-
tion 4.2.
Poisson Principal Bundles 15
4. A Poisson transversality condition for all η, τ ∈ Ω1(X) and ξ ∈ g,
iξ̃
(
∇̂Xη τ
)
= i
Ξ̃∗1ξ
(η)i
Ξ̃∗2ξ
(τ) + πX(η,diξ̃(τ)).
In principle, we could need more conditions, but we will see that the above is sufficient for
our purposes.
Corollary 5.2. Let ∇̂ be a left-covariant contravariant connection Ω1(X) obeying the transver-
sality condition (4) in Definition 5.1. If this is Poisson-compatible then
iξ̃[η, τ ] = πX(η,diξ̃(τ))− πX(diξ̃(η), τ)− 1
2 iδ̃ξ(η ∧ τ)
for all η, τ ∈ Ω1(X).
Proof. We compute using Poisson-compatibility,
iξ̃[η, τ ] = iξ̃∇̂
X
η τ − iξ̃∇̂
X
τ η
= i
Ξ̃∗1ξ
(η)i
Ξ̃∗2ξ
(τ) + πX(η,diξ̃(τ))− i
Ξ̃∗1ξ
(τ)i
Ξ̃∗2ξ
(η)− πX(τ,diξ̃(η))
= πX(η,diξ̃(τ))− πX(diξ̃(η), τ) + i
( ˜δξ)1(η)i
( ˜δξ)2(τ).
Here, (3.3) in dual form allows us to recognise δ̃ξ from the cocommutator. We also use antisym-
metry of πX . We then put the result as a bivector interior product as discussed before. �
On exact differentials, this 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-compatible contravariant
connection.
Proposition 5.3. Let G be a PLG with a free smooth proper right PLG action on a Poisson
manifold X, equipped with Poisson-compatible contravariant connections ∇̂G and ∇̂X so as to
form a Poisson-principal bundle as in Definition 5.1. Then M = X/G becomes a Poisson
manifold with Poisson-compatible contravariant connection ∇̂M the restriction of ∇̂X .
Proof. Here C∞(M) can be identified with smooth functions on X which are killed by all
vector fields for the infinitesimal action. By the invariance of the Poisson bracket, it follows that
if p, q are killed by all ξ̃ 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. = Lξ̃. By the bundle condition, it follows that
the output of ∇̂M is killed by iξ̃. However, if
∑
pidqi with pi, qi ∈ C∞(X) has these properties
then
∑
piξ̃(qi) = 0 and moreover, by transversality of the classical bundle, we know that the
form is horizontal, so we can assume qi ∈ C∞(M). Then by the first property,
∑
ξ̃(pi)dqi = 0 so
that taking the dqi independent, the pi are also in C∞(M), i.e., our form is an element of Ω1(M).
Thus ∇̂M is defined, and in this case inherits the connection properties. �
Example 5.4. Following on from Example 4.6 for the Poisson version of SU2 → S2 = SU2/S
1,
one can verify that the Poisson brackets and connections there obey the transversality condi-
tion (4) in Definition 5.1. For example,
iξ̃
(
∇̂X
d
(
a b
c d
)e0
)
= iξ̃
((
−a b
−c d
)
e0
)
=
(
−a b
−c d
)
= −ξ̃
((
a b
c d
))
iξ̃
(
e0
)
+
{(
a b
c d
)
, iξ̃e
0
}
X
= i
Ξ̃∗1ξ
(
d
(
a b
c d
))
i
Ξ̃∗2ξ
(
e0
)
+ πX
(
d
(
a b
c d
)
, d
(
iξ̃e
0
))
,
16 S. Majid and L. Williams
where ξ = H and ξ̃ is the vector field for its infinitesimal action by right-translation as in
Example 3.6. This is dual to e0 ∈ Ω1, so ξ̃(a) = a and iξ̃
(
e0
)
=
〈
ξ̃, e0
〉
= 1. We also used
Ξ∗ξ = −ξ ⊗ ξ from (4.3).
Hence we obtain a Poisson bracket and connection on M = S2/S1 by Proposition 5.3, which
we now describe. If we use the complexified coordinates z = cd, z∗ = −ab, x = −bc then the
algebra relations are given by
zx = (1 + λ)xz, z∗x = (1− λ)xz∗, zz∗ = (1 + 2λ)z∗z − λx, z∗z = x(1− x).
From which, the Poisson bracket and Poisson-compatible contravariant connection can be com-
puted as
{z, x}S2 = xz, {z∗, x}S2 = −xz∗, {z, z∗}S2 = 2z∗zx,
∇̂S2
dzdz = (2x− 1)zdz − 2z2dx, ∇̂S2
dzdz∗ = −(2x− 1)zdz∗ − 2x2dx,
∇̂S2
dz∗dz = (2x− 1)z∗dz + 2x2dx, ∇̂S2
dz∗dz
∗ = −(2x− 1)z∗dz∗ + 2z∗2dx,
∇̂S2
dzdx = −(2x− 1)zdx+ (2x− 1)xdz, ∇̂S2
dz∗dx = (2x− 1)z∗dx− (2x− 1)xdz∗,
∇̂S2
dxdx = (2x− 1)xdx+ 2xzdz∗.
In the present case, we also have left-translation covariance of the Poisson bracket and a 3D
calculus in Example 3.6 on SU2 commuting with the right action of S1. This necessarily descends
to an action of su2 on the sphere as
x/H = 0, z/H = −2z, z∗/H = 2z∗, x/X+ = −z, z/X+ = 0,
z∗/X+ = 2x− 1, x/X− = z∗, z/X− = 1− 2x, z∗/X− = 0
with respect to which { , }S2 and ∇̂S2
are covariant in our Poisson sense of Remark 4.5. For
example,(
∇̂S2
dzdz∗
)
/X+ =
(
−(2x− 1)zdz∗ − 2x2dx
)
/X+ = 2z2dz∗ + 2zdx+ 2x2dz
= 4z(1− x)dx+ 2x(2x− 1)dz
since dx = − 1
2x−1(z∗dz + zdz∗) classically. On the other hand,
∇̂S2
d(z/X+)dz
∗ + ∇̂dzd(z∗/X+) = 2∇̂S2
dzdx = −2(2x− 1)zdx+ 2x(2x− 1)dz,(
z/δ1
X+
)
d
(
z∗/δ2
X+
)
= 1
2(z/X+)d(z∗/H)− 1
2(z/H)d(z∗/X+) = 2zdx,
from which we see that the covariance condition holds.
It remains to explain how the proposed semiclassical transversality condition (4) relates to
the quantum transversality. This should be considered purely as motivation in that we have not
constructed the assumed ∗-product quantisations.
Remark 5.5. We suppose ∗-product quantisations C∞(X)[[λ]], C∞(M)[[λ]] and C∞(G)[[λ]]
forming a quantum principal bundle when working over C[[λ]], at least to O
(
λ2
)
. Here X, G are
a Poisson manifold and a PLG respectively, and ∇̂G is bicovariant as defined by Ξ∗ while ∇̂X is
covariant for the differentials of the total space of the quantum bundle. We compute ver(p • dq)
in (2.1) working at the quantum group level. We assume that the coproduct is undeformed and
Poisson Principal Bundles 17
write $•πε(h) = S•h(1) • dh(2) for S• the deformed antipode. Then
λver
(
∇̂Xdpdq
)
+O
(
λ2
)
= ver(p • dq − (dq) • p) = ver(p • dq − d(q • p) + q • dp)
= p ∗ q ¯(0) ⊗$
•πε
(
q ¯(1)
)
− q ¯(0) ∗ p ¯(0) ⊗ S
•p ¯(1)(1) •$•πε
(
q ¯(1)
)
• p ¯(1)(2)
= p ∗ q ¯(0) ⊗$
•πε
(
q ¯(1)
)
− q ¯(0) ∗ p ¯(0) ⊗ S
•p ¯(1)(1) ∗ p ¯(1)(2) •$•πε
(
q ¯(1)
)
− q ¯(0) ∗ p ¯(0) ⊗ S
•p ¯(1)(1) •
[
$•πε(q ¯(1)), p ¯(1)(2)
]
•
=
[
p, q ¯(0)
]
∗ ⊗$
•πε
(
q ¯(1)
)
− q ¯(0) ∗ p ¯(0) ⊗ S
•p ¯(1)(1) •
[
$•πε
(
q ¯(1)
)
, p ¯(1)(2)
]
•
= λ
({
p, q ¯(0)
}
⊗$πε
(
q ¯(1)
)
+ q ¯(0)p ¯(0) ⊗ Sp ¯(1)(1)∇̂Gdp ¯(1)(2)
$πε
(
q ¯(1)
))
,
where we dropped the bullets in the last line as there is already a λ out front and we are working
to O
(
λ2
)
. Hence at order λ we must have that
ver∇̂Xdpdq =
{
p, q ¯(0)
}
⊗$πε
(
q ¯(1)
)
+ q ¯(0)p ¯(0) ⊗ ∇̂
G
$πε(p ¯(1)
)$πε
(
q ¯(1)
)
=
{
p, ver(dq)1
}
⊗ ver(dq)2 + ver(dp)1ver(dq)1 ⊗ ∇̂Gver(dp)2ver(dq)2,
where we brought Sp ¯(1)(1) into the subscript of ∇̂G. We then recognised q ¯(0) ⊗ $πε(q ¯(1)) =
ver(dq) = ver(dq)1 ⊗ ver(dq)2, say.
Next, recall that ∇̂G is defined by Ξ: g∗ ⊗ g∗ → g∗, with dual Ξ∗ : g → g ⊗ g. We also note
that ver2 is a left-invariant 1-form which, under the identification of Ω1(G) with C∞(G, g∗),
corresponds to a constant function. Hence, our condition on ver∇̂X is equivalent to
iξ̃
(
∇̂Xdpdq
)
= (id⊗ ξ)ver
(
∇̂Xdpdq
)
= ver(dp)1ver(dq)1 ⊗
〈
Ξ∗(ξ)1, ver(dp)2
〉〈
Ξ∗(ξ)2, ver(dq)2
〉
+ {p, verξ(dq)}
for all ξ ∈ g, where (id⊗ξ)ver = iξ̃. Finally, ver(dp)1⊗
〈
Ξ∗1, ver(dp)2
〉
= verΞ∗1(dp) = i
Ξ̃∗1
(dp) =
Ξ̃∗1(p), and similarly for the other factor, gives us the condition
iξ̃
(
∇̂Xdpdq
)
= Ξ̃∗1ξ (p)Ξ̃∗2ξ (q) +
{
p, ξ̃(q)
}
X
for all p, q ∈ C∞(X), ξ ∈ g. It is then straightforward to extend this to 1-forms τ, η ∈ Ω1(X) to
find the condition (4) in Definition 5.1.
6 Connections on Poisson principal bundles
In noncommutative differential geometry, once we have a quantum principal bundle P , the next
order of business is to find a ‘spin connection’ ω : Λ1
H → Ω1
P which is equivariant (where H
coacts on Λ1
H by a right adjoint coaction that follows from the assumed Ω1
H bicovariance) and
such that ver(ω(v)) = 1 ⊗ v for all v ∈ Λ1
H . This provides a splitting of Ω1
P in the form of
an associated projection Πω defined by Πω(dp) = p ¯(0)ω
(
$πεp ¯(1)
)
, which is equivariant for the
right coaction of H, a left P -module map, idempotent and has kerΠω = PΩ1
AP , see [7] and [4,
Chapter 5]. We also require ω to be such that (id − Πω)dP ⊆ Ω1
AP , in which case it defines
a connection on P itself by
∇P : P → Ω1
AP = Ω1
A ⊗A P, ∇P p = dp− p ¯(0)ω($πεp ¯(1)).
The first equality is canonically afforded by viewing Ω1
A ⊆ Ω1
P and multiplying there. It is
known[4, Proposition 5.50] that this is a bimodule connection if and only if
∑
aiω
(
Λ1
H
)
bi = 0
18 S. Majid and L. Williams
for all
∑
ai ⊗ bi ∈ A⊗ A such that
∑
aibi = 0 and
∑
(dai)bi = 0. In this case, the generalised
braiding is
σP (p⊗ (da)b) = p(da)b− p ¯(0)
[
a, ω
(
$πεp ¯(1)
)]
b.
Moreover, ∇P commutes with the right coaction of H under our assumptions. As a consequence,
for any H-comodule V , we can define E = (P ⊗V )H as a noncommutative analogue of the space
of sections of the associated bundle (the superscript H indicates the invariant subspace under
the tensor product coaction) and a connection [4, 7]
∇E : (P ⊗ V )H = E →
(
Ω1
AP ⊗ V
)H
= Ω1
A ⊗A E, ∇E = ∇P ⊗ id.
We see that the key here for the theory of associated bundles and ‘spin connections’ is the
construction of an equivariant ∇P . Its properties of being a bimodule connection when viewing
A ⊆ P and working inside Ω1
P come down to
∇P : P → Ω1
AP, σP : PΩ1
A → Ω1
AP,
∇P (a.p) = (da).p+ a.∇P (p), ∇P (p.a) = (∇P p).a+ σP (p.da), (6.1)
a.σP (p.db) = σP (a.p.db), (σP (p.db)).a = σP (p.(db).a) (6.2)
for all a, b ∈ A, p ∈ P . We constructed ∇P from ω above with conditions to ensure that σP is
well-defined by (6.1), after which it automatically obeys the A-bimodule map conditions (6.2).
We emphasise the product of P and Ω1
P for clarity. There are exactly similar inherited formulae
for ∇E , now with σE :
(
PΩ1
A ⊗ V
)H → (
Ω1
AP ⊗ V
)H
. The point is that by working ‘upstairs’
in Ω1
P , we avoid mention of ⊗A.
What this comes down to for a classical principal bundle P = C∞(X), Lie structure group G
and Lie algebra g with basis {ei}, say, (with Λ1
H replaced now by g∗) is that a spin connection
is an equivariant Lie-algebra valued 1-form on X,
ω = ωi ⊗ ei, ξ.ωi + ωi ⊗ [ξ, ei] = 0, iξ̃(ω
i)ei = ξ (6.3)
for all ξ ∈ g and sum over i understood. Then
∇P : P → Ω1
hor, ∇P p = dp− ωiẽi(p), σP (pda) = (da)p (6.4)
obeys the above Leibniz rules with everything commutative (so σP = id on Ω1
hor). From this,
restricting to E = C∞G (X,V ) (the sections of the vector bundle associated to a representation V
of G) gives the induced connection ∇E on the associated bundle. Interior product with a vector
field gives the covariant derivative along that vector field in the usual sense.
We now suppose that we have such a classical principal bundle with spin connection ω and
seek to deformation-quantise everything. In fact, we are only going to look at this up to and
including first order in λ, what could be called the ‘semiclassical’ level. Formally, this can be
done by working over the ring C[λ]/
(
λ2
)
in place of C[[λ]] as explained in [3], but for continuity
with the general scheme, we will use the deformation point of view but ignore O
(
λ2
)
as errors
to be corrected in a 2nd order theory. We can assume that C∞(X)[[λ]] has an associative
∗-product but we only assume this O
(
λ2
)
for the bimodule product • of Ω1(X)[[λ]], since we
allow that ∇̂X could have curvature. We do, however, fix the antisymmetric form of the leading
products as per (2.5) applied now on X as a Poisson manifold. This means that our theory has
a slightly different flavour from previous sections, being specific to the leading part of a preferred
deformation scheme.
Poisson Principal Bundles 19
Lemma 6.1. Let X, G be a Poisson principal bundle as in Definition 5.1 and suppose we are
given a connection in the sense
∇P : C∞(X)→ Ω1
hor, ∇P (ap) = (da)p+ a∇P p
for all p ∈ C∞(X), a ∈ C∞(M). Consider Ω1(X)[[λ]] quantised as in (2.5) and ∇•P p in here of
the form
∇•P p = ∇P p+
λ
2
Γ(p) +O
(
λ2
)
, Γ: C∞(X)→ Ω1
hor.
Then the first of (6.1) holds with ∗ and • products to O
(
λ2
)
if and only if
Γ(ap) = aΓ(p) + ∇̂Xda∇P p− ∇̂Xdpda−∇P {a, p}X , (6.5)
while the second of (6.1) holding requires
σ•P (pda) = pda+ λς(pda) +O
(
λ2
)
, ς : Ω1
hor → Ω1
hor,
ς(pda) = ∇̂Xda∇P p− ∇̂Xdpda−∇P {a, p}X .
If this is well-defined then (6.2) hold with ∗ and • products to O
(
λ2
)
.
Proof. This is simply a matter of putting the form of the ∗, • products and ∇•P into (6.1)–(6.2)
and discarding anything O
(
λ2
)
. For example,
∇•P (a ∗ p) = ∇P (ap) +
λ
2
∇P {a, p}X +
λ
2
Γ(ap) +O
(
λ2
)
,
(da) • p+ a • ∇•P p = (da)p− λ
2
∇̂Xdpda+ a∇P p+
λ
2
∇̂Xda∇P p+ a
λ
2
Γ(p) +O
(
λ2
)
gives the first result. For σ•P we first establish that σ•P (pda) = pda+O(λ) by computing the right
hand side at classical order, then σ(p•da) = σ•P
(
pda+ λ
2 ∇̂
X
dpda
)
= σ•P (pda) + λ
2 ∇̂
X
dpda to O
(
λ2
)
and we equate this to ∇•P (p ∗ a)− (∇•P ) • a, from which Γ conveniently cancels given our result
for Γ(ap). That (6.2) holds to O
(
λ2
)
is a tedious calculation. E.g., a • σ(pdb) = σ(a • (pdb)),
comes down to ∇̂Xdadb−∇̂Xdbda = d{a, b}X on expanding both sides and using the stated formula
for σ•P . Details are omitted. Finally, note that η ∈ Ω1
hor is characterised by iξ̃η = 0 for all ξ ∈ g
and the transversality condition in Definition 5.1 then ensures that ∇̂Xτ preserves Ω1
hor for all
τ ∈ Ω1(X) (because iξ̃∇̂
X
τ η = 0 for all η ∈ Ω1
hor). It is then clear that the image of the stated
formula for ς is in Ω1
hor. �
This lemma clarifies what we mean by quantizing a spin connection in the ∇P form to
semiclassical order as a bimodule connection at this level. The order λ data for the associated
bundle connection is just Γ restricted to f ∈ E = C∞G (V ) with ∇•Ef = ∇Ef + λ
2 Γ(f) +O
(
λ2
)
.
Our approach to construct Γ and ς is to start with a deformed ‘spin connection’ which, at
order λ, can have an additional component
ω• = ω +
λ
2
αi ⊗ ei +O
(
λ2
)
, ξ.αi ⊗ ei + αi ⊗ [ξ, ei] = 0, (6.6)
where αi ∈ Ω1
hor, and follow the lines of a deformed version of (6.4). We can take α = 0 as
a canonical choice here.
20 S. Majid and L. Williams
Theorem 6.2. Let X, G be a Poisson principal bundle as in Definition 5.1, ω a classical ‘spin
connection’ on X with corresponding ∇P in (6.4), and α a horizontal equivariant g-valued 1-form
on X as in (6.6). Then
Γ(p) = Ξ̃∗1ei
(
Ξ̃∗2ei (p)
)
ωi − ∇̂Xdẽi(p)ω
i − ẽi(p)αi, ς(τ) = −∇̂Xei.τω
i (6.7)
for all p ∈ C∞(X) and τ ∈ Ω1
hor constructs ∇•P = ∇P + λ
2 Γ +O
(
λ2
)
and σ•P = id + λς +O
(
λ2
)
as a bimodule connection at semiclassical order by Lemma 6.1. Moreover, Γ and hence ∇•P
to O
(
λ2
)
are equivariant.
Proof. We look at Γ first. Using the bundle transversality condition, we have
iξ̃∇
X
dẽi(p)
ωi = Ξ̃∗1ξ (ẽi(p))iΞ̃∗2ξ
(
ωi
)
+
{
ẽi(p), iξ̃
(
ωi
)}
X
= Ξ̃∗1ξ
(
Ξ̃∗2ξ (p)
)
= iξ̃
(
Ξ̃∗1ei
(
Ξ̃∗2ei (p)
)
ωi
)
,
where iξ̃
(
ωi
)
= ξi for a spin connection as in (6.3), these being constants as ξ = ξiei ∈ g. Hence
the first two terms of Γ are horizontal. The last term of Γ is horizontal as the αi are assumed
so. In fact, to be horizontal, we just need ẽi(p)iξ̃
(
αi
)
= 0 for all p. But since the classical action
is free, ver is surjective and it follows that we need αi ∈ Ω1
hor if we want the image of Γ to be
horizontal.
Next, for a ∈ C∞(M), we have ξ̃(ap) = aξ̃(p) for all ξ ∈ g from which it is clear that
Γ(ap)− aΓ(p) = −∇̂Xd(aẽi(p))
ωi + a∇̂Xdẽi(p)ω
i = −ẽi(p)∇̂Xdaωi,
while
∇̂Xda∇P p− ∇̂Xdpda−∇P {a, p}X
= ∇̂Xdadp− ∇̂Xda(ẽi(p)ωi)− ∇̂Xdpda− d{a, p}X + ei.{a, p}Xωi
= −ẽi(p)∇̂Xdaωi − {a, ẽi(p)}ωi + ei.{a, p}Xωi = −ẽi(p)∇̂Xdaωi
also, by Lemma 3.1 and the Poisson-compatibility of ∇̂X . This also gives us
ς(pda) = −ẽi(p)∇̂Xdaωi = −∇̂Xẽi(p)daω
i = −∇̂Xei.(pda)ω
i
since ei. is the Lie derivative along ẽi and acts trivially on 1-forms on the base. Hence, this is
well-defined on Ω1
hor and as stated. Since it is also Γ(ap) − aΓ(p), it follows that it lies in Ω1
hor
also, as one can also directly check using the bundle transversality condition.
So far, we have not used the assumed equivariance properties of ω and α. We now assume
these. Then
ξ.Γ(p) =
(
ξ̃Ξ̃∗1ei Ξ̃∗2ei (p)
)
ωi +
(
Ξ̃∗1ei Ξ̃∗2ei (p)
)
(ξ.ωi)− (ξei.p)α
i − (ei.p)
(
ξ.αi
)
− ξ.∇̂Xdei.pω
i
=
(
ξ̃Ξ̃∗1ei Ξ̃∗2ei (p)
)
ωi −
(
Ξ̃∗1[ξ,ei]
Ξ̃∗2[ξ,ei]
(p)
)
ωi − (eiξ.p)α
i
− ∇̂Xdξei.pω
i − ∇̂Xdei.p
(
ξ.ωi
)
−
(
δ1
ξei.p
)(
δ2
ξ.ω
i
)
=
(
ξΞ∗1ei Ξ∗2ei .p
)
ωi −
(
Ξ∗1[ξ,ei]
Ξ∗2[ξ,ei]
.p
)
ωi − (eiξ.p)α
i − ∇̂Xdeiξ.pω
i +
(
δ1
ξ [δ
2
ξ , ei].p
)
ωi
=
(
Ξ∗1ei Ξ∗2ei ξ.p
)
ωi − (eiξ.p)α
i − ∇̂Xdeiξ.pω
i = Γ(ξ.p)
as required. We used equivariance of α to transfer ξ.αi to [ξ, ei], then covariance of ∇̂X in our
Poisson sense of Definition 4.2 to expand ξ.∇̂X and equivariance of the classical ωi as in (6.3) to
transfer any actions ξ.ωi and δ2
ξ.ω
i to the relevant ei. We finally used the bicovariance condition
in terms of Ξ in Theorem 3.5 to recognise the answer. We already have ∇P equivariant, so ∇•P
would be quantum group covariant to O
(
λ2
)
similarly to Remark 4.3. �
Poisson Principal Bundles 21
Example 6.3. Our theme of the q-Hopf fibration gives a modest example. Indeed, the q-
monopole connection is known at the quantum group level [4, 7] and in the classical limit
constructs the classical monopole by the choice ω
(
t−1dt
)
= e0, or in our terms ω = e0 ⊗H. We
have seen that Ξ∗H = −H ⊗H in Example 4.6, so
Γ(p) = H̃(H̃(p))e0 − ∇̂X
dH̃(p)
e0 − H̃(p)α = −|p|pα
for any α ∈ Ω1
hor and p of homogenous degree. Here the first two terms cancel using our results in
Example 3.6. Representations of S1 are labelled by n ∈ Z and the associated ‘charge n monopole’
bundle has sections En = C∞S1(SU2) with dense subspace given by restricting to elements p of
homogeneous degree −n. We see that the correction to the classical monopole ∇En at first order
is just to add npα.
Finally, we outline how the formula (6.5) could be obtained from a deformation-quantisation.
This should be seen purely as motivation.
Remark 6.4. We return to the setting of Remark 5.5, where we suppose for the sake of dis-
cussion that ∗-product quantisations C∞(X)[[λ]], C∞(M)[[λ]] and C∞(G)[[λ]] form a quan-
tum principal bundle when working over C[[λ]], at least to O
(
λ2
)
. We assume as there that
the coproduct and d are undeformed and note that the deformed antipode is S•h = Sh −
λ
2{Sh(1), h(2)}Sh(3) +O
(
λ2
)
. Then
$•(v) := S•v(1) • dv(2) = Sv(1) • dv(2) −
λ
2
{Sv(1), v(2)}(Sv(3))dv(4)
= $(v) +
λ
2
(
∇̂GdSv(1)
dv(2) − {Sv(1), v(2)}$πεv(3)
)
= $(v) +
λ
2
v(2)∇̂GdSv(1)
$πεv(3) = $(v)− λ
2
∇̂G$πεv(1)
$πεv(2)
to errors O
(
λ2
)
, where v is a function on G vanishing at e, viewed in H+. For the third equality,
we used dv(2) = v(2)$πεv(3), that ∇̂G is a contravariant connection, the Leibniz rule and S2 = id
on the classical group to identify its subscript. We now view this as a map
$• : H+ → g∗[[λ]], 〈ξ,$•(v)〉 = ξ̃(v)(e)− λ
2
Ξ̃∗1ξ (Ξ̃∗2ξ (v))(e) +O
(
λ2
)
, (6.8)
where e is the group identity, on noting that
〈ξ,$πε(v(1))〉〈η,$πε(v(2))〉 =
d
dt
∣∣∣
0
d
ds
∣∣∣
0
v
(
etξesη
)
= ξ̃(η̃(v))(e)
and 〈ξ,$(v)〉 = d
dt
∣∣
0
v
(
etξ
)
= ξ̃(v)(e) for any functions v and ξ, η ∈ g (we apply the first
observation to the output of Ξ∗ ∈ g⊗ g). In the quantum differential calculus, we identify left-
invariant 1-forms Λ1
H with H+ modulo the kernel of $• and on the quantum bundle we suppose
a spin connection ω• : Λ1
H → Ω1(X)[[λ]] which we take as given by (6.8) followed by (6.6) viewed
as a map from g∗, to give
∇•P p = dp− p ¯(0) • ω
i〈ei, $•(p ¯(1))〉 = ∇P p+
λ
2
Γ(p) +O
(
λ2
)
for Γ(p) as in (6.7), after a calculation to expand the bullets to semiclassical order.
22 S. Majid and L. Williams
7 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. The
PLG condition itself is viewed as a certain covariance of the Poisson bracket { , }G and extends
to the notion of a PLG action on { , }X . We extended these covariance notions to Poisson-level
quantum differential structures in the sense of Poisson-compatible contravariant connections ∇̂G
and ∇̂X respectively in the sense of [1, 17, 20, 21, 30] and characterised bicovariance of the
former. Section 5 introduced a further transversality condition expressing that X →M = X/G
is a Poisson-level version of a quantum principal bundle. This is such thatM is not only a Poisson
manifold but inherits a Poisson-level quantum differential structure in the sense of ∇̂M . We were
then able to formulate ‘spin connections’ and connections on associated bundles at semiclassical
level. Results were illustrated on the Poisson level of the q-Hopf fibration deforming SU2 → S2
with S1 fibre, the latter as a PLG with zero { , }G but nonzero ∇̂G. It remains to compute
more complicated examples.
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 differential 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 2-parameter Podleś spheres [32] is unclear since a direct quantum
principal bundle approach would seem to require coalgebra bundles [8], for which the notion
of differential calculus on the fibre is not known. 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 was not a problem in our approach at
the Poisson level as we were careful not to assume flatness of ∇̂G, but quantisation would be
a challenge. It should be possible, for example, to construct a nonassociative deformation of the
next Hopf fibration S7 → S4 with SU2 fibre, combining the Hopf-quasigroup methods in [22]
and the nonassociative bicovariant calculi by twisting in [1, 2].
A third interesting direction would be quantum group frame bundles as in [26] at the Poisson
level whereby Ω1(M) is an associated bundle to a Poisson principal bundle. In this context, it
could make sense to ask that ∇̂ comes from a contravariant connection on the underlying classical
principal bundle in the sense of [17]. The framed theory would, moreover, extend the above
to a PLG-covariant version of Riemannian geometry on M as a somewhat different approach
to [3]. Indeed, one expects the quantum metrics at semiclassical order to be different in the two
approaches even for the Poisson level of the metric for the q-sphere. It could also be of interest to
consider quantum fibrations more generally, and to relate to integrability of Lie algebroids [11].
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1 Introduction
2 Preliminaries
2.1 Elements of noncommutative differential geometry
2.2 Elements of deformation theory
2.3 Elements of Poisson–Lie theory
3 Poisson–Lie groups and their semiclassical calculi revisited
4 PLG actions on Poisson manifolds
5 Formulation of Poisson principal bundles
6 Connections on Poisson principal bundles
7 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-211182 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T06:13:47Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Majid, Shahn Williams, Liam 2025-12-25T13:24:16Z 2021 Poisson Principal Bundles. Shahn Majid and Liam Williams. SIGMA 17 (2021), 006, 23 pages 1815-0659 2020 Mathematics Subject Classification: 58B32; 53D17; 17B37; 17B62 arXiv:1903.12006 https://nasplib.isofts.kiev.ua/handle/123456789/211182 https://doi.org/10.3842/SIGMA.2021.006 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-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the 𝑞-Hopf fibration on the standard 𝑞-sphere. We also construct the Poisson level of the spin connection on a principal bundle. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Poisson Principal Bundles Article published earlier |
| spellingShingle | Poisson Principal Bundles Majid, Shahn Williams, Liam |
| title | Poisson Principal Bundles |
| title_full | Poisson Principal Bundles |
| title_fullStr | Poisson Principal Bundles |
| title_full_unstemmed | Poisson Principal Bundles |
| title_short | Poisson Principal Bundles |
| title_sort | poisson principal bundles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211182 |
| work_keys_str_mv | AT majidshahn poissonprincipalbundles AT williamsliam poissonprincipalbundles |