The aim of this section is to set out the topological concepts in relation to topological bundles, in particular principal bundles. The classical connection is made for interpreting topological concepts in an algebraic setting, providing a manageable methodology for performing calculations. In particular, the connection between principal bundles in topology and the algebraic Hopf–Galois condition is described. The reader familiar with classical theory of bundles can proceed directly to Definition 2.14.
2.1. Topological Aspects of Bundles
As a natural starting point, bundles are defined and topological properties are described. The principal map is defined and shown that injectivity is equivalent to the freeness condition. The image of the canonical map is deduced and necessary conditions are imposed to ensure the bijectivity of this map. The detailed account of the material presented in this section can be found in [
6].
Definition 2.1 A
bundle is a triple
![Axioms 01 00201 i020]()
where
![Axioms 01 00201 i021]()
and
![Axioms 01 00201 i022]()
are topological spaces and
![Axioms 01 00201 i023]()
is a continuous surjective map. Here
![Axioms 01 00201 i022]()
is called the
base space,
![Axioms 01 00201 i021]()
the
total space and
![Axioms 01 00201 i024]()
the
projection of the bundle.
For each
![Axioms 01 00201 i025]()
, the
fibre over
![Axioms 01 00201 i026]()
is the topological space
![Axioms 01 00201 i027]()
,
i.e., the points on the total space which are projected, under
![Axioms 01 00201 i024]()
, onto the point
![Axioms 01 00201 i026]()
in the base space. A bundle whose fibres are homeomorphic which satisfies a condition known as local triviality are known as fibre bundles. This is formally expressed in the next definition.
Definition 2.2 A
fibre bundle is a triple
![Axioms 01 00201 i028]()
where
![Axioms 01 00201 i020]()
is bundle and
![Axioms 01 00201 i029]()
is a topological space such that
![Axioms 01 00201 i027]()
are homeomorphic to
![Axioms 01 00201 i029]()
for each
![Axioms 01 00201 i025]()
. Furthermore,
![Axioms 01 00201 i024]()
satisfies the local triviality condition.
The local triviality condition is satisfied if for each
![Axioms 01 00201 i030]()
, there is an open neighourhood
![Axioms 01 00201 i031]()
such that
![Axioms 01 00201 i032]()
is homeomorphic to the product space
![Axioms 01 00201 i033]()
, in such a way that
![Axioms 01 00201 i024]()
carries over to the projection onto the first factor. That is the following diagram commutes:
The map
![Axioms 01 00201 i036]()
is the natural projection
![Axioms 01 00201 i037]()
and
![Axioms 01 00201 i038]()
is a homeomorphism.
Example 2.3 An example of a fibre bundle which is non-trivial,
i.e., not a global product space, is the Möbius strip. It has a circle that runs lengthwise through the centre of the strip as a base B and a line segment running vertically for the fibre F. The line segments are in fact copies of the real line, hence each
![Axioms 01 00201 i027]()
is homeomorphic to
![Axioms 01 00201 i039]()
and the Mobius strip is a fibre bundle.
Let
![Axioms 01 00201 i040]()
be a topological space which is compact and satisfies the Hausdorff property and G a compact topological group. Suppose there is a right action
![Axioms 01 00201 i041]()
of
![Axioms 01 00201 i042]()
on
![Axioms 01 00201 i040]()
and write
![Axioms 01 00201 i043]()
.
Definition 2.4 An action of
![Axioms 01 00201 i042]()
on
![Axioms 01 00201 i040]()
is said to be
free if
![Axioms 01 00201 i044]()
for any
![Axioms 01 00201 i045]()
implies that
![Axioms 01 00201 i046]()
, the group identity.
With an eye on algebraic formulation of freeness, the
principal map ![Axioms 01 00201 i047]()
is defined as
![Axioms 01 00201 i048]()
.
Proposition 2.5
acts freely on
if and only if
is injective. Proof. “
![Axioms 01 00201 i050]()
" Suppose the action is free, hence
![Axioms 01 00201 i051]()
implies that
![Axioms 01 00201 i052]()
. If
![Axioms 01 00201 i053]()
, then
![Axioms 01 00201 i054]()
and
![Axioms 01 00201 i055]()
. Applying the action of
![Axioms 01 00201 i056]()
to both sides of
![Axioms 01 00201 i055]()
we get
![Axioms 01 00201 i057]()
, which implies
![Axioms 01 00201 i058]()
by the freeness property, concluding
![Axioms 01 00201 i059]()
and
![Axioms 01 00201 i060]()
is injective as required.
“
![Axioms 01 00201 i061]()
" Suppose
![Axioms 01 00201 i060]()
is injective, so
![Axioms 01 00201 i062]()
or
![Axioms 01 00201 i053]()
implies
![Axioms 01 00201 i054]()
and
![Axioms 01 00201 i059]()
. Since
![Axioms 01 00201 i063]()
from the properties of the action, if
![Axioms 01 00201 i064]()
then
![Axioms 01 00201 i052]()
from the injectivity property.
Since
![Axioms 01 00201 i065]()
acts on
![Axioms 01 00201 i066]()
we can define the quotient space
![Axioms 01 00201 i067]()
,
The sets
![Axioms 01 00201 i069]()
are called the
orbits of the points
![Axioms 01 00201 i070]()
. They are defined as the set of elements in
![Axioms 01 00201 i066]()
to which
![Axioms 01 00201 i070]()
can be moved by the action of elements of
![Axioms 01 00201 i065]()
. The set of orbits of
![Axioms 01 00201 i066]()
under the action of
![Axioms 01 00201 i065]()
forms a partition of
![Axioms 01 00201 i066]()
, hence we can define the equivalence relation on
![Axioms 01 00201 i066]()
as,
The equivalence relation is the same as saying
![Axioms 01 00201 i070]()
and
![Axioms 01 00201 i072]()
are in the same orbit,
i.e.,
![Axioms 01 00201 i073]()
. Given any quotient space, then there is a canonical surjective map
which maps elements in
![Axioms 01 00201 i075]()
to their orbits. We define the pull-back along this map
![Axioms 01 00201 i076]()
to be the set
As described above, the image of the principal map
![Axioms 01 00201 i078]()
contains elements of
![Axioms 01 00201 i075]()
in the first leg and the action of
![Axioms 01 00201 i079]()
on
![Axioms 01 00201 i080]()
in the second leg. To put it another way, the image records elements of
![Axioms 01 00201 i081]()
in the first leg and all the elements in the same orbit as this
![Axioms 01 00201 i080]()
in the second leg. Hence we can identify the image of the canonical map as the pull back along
![Axioms 01 00201 i076]()
, namely
![Axioms 01 00201 i082]()
. This is formally proved as a part of the following proposition.
Proposition 2.6
acts freely on
if and only if the map is bijective.
Proof. First note that the map
![Axioms 01 00201 i086]()
is well-defined since the elements
![Axioms 01 00201 i087]()
and
![Axioms 01 00201 i088]()
are in the same orbit and hence map to the same equivalence class under
![Axioms 01 00201 i089]()
. Using Proposition 2.5 we can deduce that the injectivity of
![Axioms 01 00201 i086]()
is equivalent to the freeness of the action. Hence if we can show that
![Axioms 01 00201 i086]()
is surjective the proof is complete.
Take
![Axioms 01 00201 i090]()
. This means
![Axioms 01 00201 i091]()
, which implies
![Axioms 01 00201 i087]()
and
![Axioms 01 00201 i092]()
are in the same equivalence class, which in turn means they are in the same orbit. We can therefore deduce that
![Axioms 01 00201 i093]()
for some
![Axioms 01 00201 i094]()
. So,
![Axioms 01 00201 i095]()
implying
![Axioms 01 00201 i096]()
. Hence
![Axioms 01 00201 i097]()
completing the proof.
Definition 2.7 An action of
![Axioms 01 00201 i083]()
on
![Axioms 01 00201 i084]()
is said to be
principal if the map
![Axioms 01 00201 i098]()
is both injective and continuous (and such that the inverse image of a compact subset is compact in a case of locally compact spaces).
Since the injectivity and freeness condition are equivalent, we can interpret principal actions as both free and continuous actions. We can also deduce that these types of actions give rise to homeomorphisms
![Axioms 01 00201 i086]()
from
![Axioms 01 00201 i099]()
onto the space
![Axioms 01 00201 i100]()
. Principal actions lead to the concept of topological principle bundles.
Definition 2.8 A
principal bundle is a quadruple
![Axioms 01 00201 i101]()
such that
(a)
![Axioms 01 00201 i102]()
is a bundle and
![Axioms 01 00201 i083]()
is a topological group acting continuously on
![Axioms 01 00201 i084]()
with action
![Axioms 01 00201 i103]()
,
![Axioms 01 00201 i104]()
;
(b) the action
![Axioms 01 00201 i105]()
is principal;
(c)
![Axioms 01 00201 i106]()
such that
![Axioms 01 00201 i107]()
;
(d) the induced map
![Axioms 01 00201 i108]()
is a homeomorphism.
The first two properties tell us that principal bundles are bundles admitting a principal action of a group
![Axioms 01 00201 i042]()
on the total space
![Axioms 01 00201 i040]()
,
i.e., principal bundles correspond to principal actions. By Definition
![Axioms 01 00201 i109]()
, principal actions occur when the principal map is both injective and continuous, or equivalently, when the action is free and continuous. The third property ensures that the fibres of the bundle correspond to the orbits coming from the action and the final property implies that the quotient space can topologically be viewed as the base space of the bundle.
Example 2.9 Suppose
![Axioms 01 00201 i040]()
is a topological space and
![Axioms 01 00201 i042]()
a topological group which acts on
![Axioms 01 00201 i040]()
from the right. The triple
![Axioms 01 00201 i110]()
where
![Axioms 01 00201 i111]()
is the orbit space and
![Axioms 01 00201 i024]()
the natural projection is a bundle. A principal action of
![Axioms 01 00201 i042]()
on
![Axioms 01 00201 i040]()
makes the quadruple
![Axioms 01 00201 i112]()
a principal bundle.
We describe a principal bundle
![Axioms 01 00201 i113]()
as a
![Axioms 01 00201 i042]()
-principal bundle over
![Axioms 01 00201 i114]()
, or
![Axioms 01 00201 i040]()
as a
![Axioms 01 00201 i042]()
-principal bundle over
![Axioms 01 00201 i115]()
.
Definition 2.10 A
vector bundle is a bundle
![Axioms 01 00201 i020]()
where each fibre
![Axioms 01 00201 i027]()
is endowed with a vector space structure such that addition and scalar multiplication are continuous maps.
Any vector bundle can be understood as a bundle associated to a principal bundle in the following way. Consider a
![Axioms 01 00201 i042]()
-principal bundle
![Axioms 01 00201 i116]()
and let
![Axioms 01 00201 i117]()
be a representation space of
![Axioms 01 00201 i042]()
,
i.e., a (topological) vector space with a (continuous) left
![Axioms 01 00201 i042]()
-action
![Axioms 01 00201 i118]()
,
![Axioms 01 00201 i119]()
. Then
![Axioms 01 00201 i042]()
acts from the right on
![Axioms 01 00201 i120]()
by
We can define
![Axioms 01 00201 i122]()
and a surjective (continuous map)
![Axioms 01 00201 i123]()
,
![Axioms 01 00201 i124]()
and thus have a fibre bundle
![Axioms 01 00201 i125]()
. In the case where
![Axioms 01 00201 i117]()
is a vector space, we assume that
![Axioms 01 00201 i042]()
acts linearly on
![Axioms 01 00201 i117]()
.
Definition 2.11 A
section of a bundle
![Axioms 01 00201 i126]()
is a continuous map
![Axioms 01 00201 i127]()
such that, for all
![Axioms 01 00201 i128]()
,
i.e., a section is simply a section of the morphism
![Axioms 01 00201 i130]()
. The set of sections of
![Axioms 01 00201 i021]()
is denoted by
![Axioms 01 00201 i131]()
.
Proposition 2.12 Sections in a fibre bundle
associated to a principal
-bundle
are in bijective correspondence with (continuous) maps
such that All such
-equivariant maps are denoted by ![Axioms 01 00201 i134]()
.
Proof. Remember that
![Axioms 01 00201 i135]()
. Given a map
![Axioms 01 00201 i136]()
, define the section
![Axioms 01 00201 i137]()
,
![Axioms 01 00201 i138]()
Conversely, given
![Axioms 01 00201 i139]()
, define
![Axioms 01 00201 i140]()
by assigning to
![Axioms 01 00201 i045]()
a unique
![Axioms 01 00201 i141]()
such that
![Axioms 01 00201 i142]()
. Note that
![Axioms 01 00201 i143]()
is unique, since if
![Axioms 01 00201 i144]()
, then
![Axioms 01 00201 i044]()
and
![Axioms 01 00201 i145]()
. Freeness implies that
![Axioms 01 00201 i046]()
, hence
![Axioms 01 00201 i146]()
. The map
![Axioms 01 00201 i147]()
has the required equivariance property, since the element of
![Axioms 01 00201 i148]()
corresponding to
![Axioms 01 00201 i149]()
is
![Axioms 01 00201 i150]()
.
2.2. Non-Commutative Principal and Associated Bundles
To make the transition from algebraic formulation of principal and associated bundles to non-commutative setup more transparent, we assume that
![Axioms 01 00201 i040]()
is a complex affine variety with an action of an affine algebraic group
![Axioms 01 00201 i042]()
and set
![Axioms 01 00201 i151]()
(all with the usual Euclidean topology). Let
![Axioms 01 00201 i152]()
,
![Axioms 01 00201 i153]()
and
![Axioms 01 00201 i154]()
be the corresponding coordinate rings. Put
![Axioms 01 00201 i155]()
and
![Axioms 01 00201 i156]()
and note the identification
![Axioms 01 00201 i157]()
. Through this identification,
![Axioms 01 00201 i158]()
is a Hopf algebra with comultiplication:
![Axioms 01 00201 i160]()
, counit
![Axioms 01 00201 i161]()
,
![Axioms 01 00201 i162]()
, and the antipode
![Axioms 01 00201 i163]()
,
![Axioms 01 00201 i164]()
.
Using the fact that
![Axioms 01 00201 i042]()
acts on
![Axioms 01 00201 i040]()
we can construct a right coaction of
![Axioms 01 00201 i165]()
on
![Axioms 01 00201 i166]()
by
![Axioms 01 00201 i167]()
,
![Axioms 01 00201 i168]()
. This coaction is an algebra map due to the commutativity of the algebras of functions involved.
We have viewed the spaces of polynomial functions on
![Axioms 01 00201 i066]()
and
![Axioms 01 00201 i065]()
, next we view the space of functions on Y,
![Axioms 01 00201 i169]()
, where
![Axioms 01 00201 i170]()
.
![Axioms 01 00201 i171]()
is a subalgebra of
![Axioms 01 00201 i172]()
by
where
![Axioms 01 00201 i076]()
is the canonical surjection defined above. The map
![Axioms 01 00201 i174]()
is injective, since
![Axioms 01 00201 i175]()
in
![Axioms 01 00201 i176]()
means there exists at least one orbit
![Axioms 01 00201 i177]()
such that
![Axioms 01 00201 i178]()
, but
![Axioms 01 00201 i179]()
, so
![Axioms 01 00201 i180]()
which implies
![Axioms 01 00201 i181]()
. Therefore, we can identify
![Axioms 01 00201 i171]()
with
![Axioms 01 00201 i182]()
. Furthermore,
![Axioms 01 00201 i183]()
if and only if
for all
![Axioms 01 00201 i185]()
,
![Axioms 01 00201 i186]()
. This is the same as
for all
![Axioms 01 00201 i081]()
,
![Axioms 01 00201 i079]()
, where
![Axioms 01 00201 i188]()
is the unit function
![Axioms 01 00201 i189]()
(the identity element of
![Axioms 01 00201 i191]()
). Thus we can identify
![Axioms 01 00201 i192]()
with the
coinvariants of the coaction
![Axioms 01 00201 i193]()
:
Since
![Axioms 01 00201 i192]()
is a subalgebra of
![Axioms 01 00201 i195]()
, it acts on
![Axioms 01 00201 i195]()
via the inclusion map
![Axioms 01 00201 i196]()
,
![Axioms 01 00201 i197]()
. We can identify
![Axioms 01 00201 i198]()
with
![Axioms 01 00201 i199]()
by the map
Note that
![Axioms 01 00201 i201]()
is well defined because
![Axioms 01 00201 i202]()
. Proposition 2.6 immediately yields
Proposition 2.13 The action of
on
is free if and only if
,
is bijective. In view of the definition of the coaction of
![Axioms 01 00201 i205]()
on
![Axioms 01 00201 i206]()
, we can identify
![Axioms 01 00201 i207]()
with the
canonical mapThus the action of
![Axioms 01 00201 i083]()
on
![Axioms 01 00201 i084]()
is free if and only if this purely algebraic map is bijective. In the classical geometry case we take
![Axioms 01 00201 i210]()
,
![Axioms 01 00201 i211]()
and
![Axioms 01 00201 i212]()
, but in general there is no need to restrict oneself to commutative algebras (of functions on topological spaces). In full generality this leads to the following definition.
Definition 2.14 (Hopf–Galois Extensions) Let
![Axioms 01 00201 i165]()
be a Hopf algebra and
![Axioms 01 00201 i206]()
a right
![Axioms 01 00201 i165]()
-comodule algebra with coaction
![Axioms 01 00201 i213]()
. Let
![Axioms 01 00201 i215]()
, the coinvariant subalgebra of
![Axioms 01 00201 i206]()
. We say that
![Axioms 01 00201 i216]()
is a
Hopf–Galois extension if the left
![Axioms 01 00201 i206]()
-module, right
![Axioms 01 00201 i165]()
-comodule map
is an isomorphism.
Proposition 2.13 tells us that when viewing bundles from an algebraic perspective, the freeness condition is equivalent to the Hopf–Galois extension property. Hence, the Hopf–Galois extension condition is a necessary condition to ensure a bundle is principal. Not all information about a topological space is encoded in a coordinate algebra, so to make a fuller reflection of the richness of the classical notion of a principal bundle we need to require conditions additional to the Hopf–Galois property.
Definition 2.15 Let
![Axioms 01 00201 i165]()
be a Hopf algebra with bijective antipode and let
![Axioms 01 00201 i206]()
be a right
![Axioms 01 00201 i165]()
-comodule algebra with coaction
![Axioms 01 00201 i213]()
. Let
![Axioms 01 00201 i219]()
denote the coinvariant subalgebra of
![Axioms 01 00201 i206]()
. We say that
![Axioms 01 00201 i206]()
is a
principal
-comodule algebra if:
(a)
![Axioms 01 00201 i216]()
is a Hopf–Galois extension;
(b) the multiplication map
![Axioms 01 00201 i220]()
,
![Axioms 01 00201 i221]()
, splits as a left
![Axioms 01 00201 i219]()
-module and right
![Axioms 01 00201 i165]()
-comodule map (the equivariant projectivity condition).
As indicated already in [
7,
8,
9], principal comodule algebras should be understood as principal bundles in noncommutative geometry. In particular, if
![Axioms 01 00201 i165]()
is the Hopf algebra associated to a
![Axioms 01 00201 i222]()
-algebra of functions on a quantum group [
10], then the existence of the Haar measure together with the results of [
8] mean that condition (a) in Definition 2.15 implies condition (b) (
i.e., the freeness of the coaction implies its principality).
The following characterisation of principal comodule algebras [
11,
12] gives an effective method for proving the principality of coaction.
Proposition 2.16 A right
-comodule algebra
with coaction
is principal if and only if it admits a strong connection form, that is if there exists a map
such that Here
denotes the multiplication map,
is the unit map,
is the comultiplication,
counit and
the (bijective) antipode of the Hopf algebra
, and
is the flip. Proof. If a strong connection form
![Axioms 01 00201 i234]()
exists, then the inverse of the canonical map
![Axioms 01 00201 i235]()
(see Definition 2.14 ) is the composite
while the splitting of the multiplication map (see Definition 2.15 (b)) is given by
Conversely, if
![Axioms 01 00201 i240]()
is a principal comodule algebra, then
![Axioms 01 00201 i234]()
is the composite
where
![Axioms 01 00201 i243]()
is the left
![Axioms 01 00201 i244]()
-linear right
![Axioms 01 00201 i165]()
-colinear splitting of the multiplication
![Axioms 01 00201 i245]()
.
Example 2.17 Let
![Axioms 01 00201 i166]()
be a right
![Axioms 01 00201 i165]()
-comodule algebra. The space of
![Axioms 01 00201 i246]()
-linear maps
![Axioms 01 00201 i247]()
is an algebra with the
convolution product and unit
![Axioms 01 00201 i249]()
.
![Axioms 01 00201 i172]()
is said to be
cleft if there exists a right
![Axioms 01 00201 i250]()
-colinear map
![Axioms 01 00201 i251]()
that has an inverse in the convolution algebra
![Axioms 01 00201 i252]()
and is normalised so that
![Axioms 01 00201 i253]()
. Writing
![Axioms 01 00201 i254]()
for the convolution inverse of
![Axioms 01 00201 i255]()
, one easily observes that
is a strong connection form. Hence a cleft comodule algebra is an example of a principal comodule algebra. The map
![Axioms 01 00201 i255]()
is called a
cleaving map or a
normalised total integral.
In particular, if
![Axioms 01 00201 i251]()
is an
![Axioms 01 00201 i250]()
-colinear algebra map, then it is automatically convolution invertible (as
![Axioms 01 00201 i257]()
) and normalised. A comodule algebra
![Axioms 01 00201 i172]()
admitting such a map is termed a
trivial principal comodule algebra.
Example 2.18 Let
![Axioms 01 00201 i250]()
be a Hopf algebra of the compact quantum group. By the Woronowicz theorem [
10],
![Axioms 01 00201 i250]()
admits an invariant Haar measure,
i.e., a linear map
![Axioms 01 00201 i258]()
such that, for all
![Axioms 01 00201 i259]()
,
where
![Axioms 01 00201 i261]()
is the Sweedler notation for the comultiplication. Next, assume that the lifted canonical map:
is surjective, and write
for the
![Axioms 01 00201 i265]()
-linear map such that
![Axioms 01 00201 i266]()
, for all
![Axioms 01 00201 i267]()
. Then, by the Schneider theorem [
8],
![Axioms 01 00201 i195]()
is a principal
![Axioms 01 00201 i191]()
-comodule algebra. Explicitly, a strong connection form is
where the coaction is denoted by the Sweedler notation
![Axioms 01 00201 i269]()
; see [
13].
Having described non-commutative principal bundles, we can look at the associated vector bundles. First we look at the classical case and try to understand it purely algebraically. Start with a vector bundle
![Axioms 01 00201 i270]()
associated to a principal
![Axioms 01 00201 i083]()
-bundle
![Axioms 01 00201 i084]()
. Since
![Axioms 01 00201 i271]()
is a vector representation space of
![Axioms 01 00201 i083]()
, also the set
![Axioms 01 00201 i272]()
is a vector space. Consequently
![Axioms 01 00201 i273]()
is a vector space. Furthermore,
![Axioms 01 00201 i274]()
is a left module of
![Axioms 01 00201 i275]()
with the action
![Axioms 01 00201 i276]()
To understand better the way in which
![Axioms 01 00201 i219]()
-module
![Axioms 01 00201 i273]()
is associated to the principal comodule algebra
![Axioms 01 00201 i277]()
we recall the notion of the cotensor product.
Definition 2.19 Given a Hopf algebra
![Axioms 01 00201 i165]()
, right
![Axioms 01 00201 i165]()
-comodule
![Axioms 01 00201 i206]()
with coaction
![Axioms 01 00201 i278]()
and left
![Axioms 01 00201 i165]()
-comodule
![Axioms 01 00201 i271]()
with coaction
![Axioms 01 00201 i279]()
, the
cotensor product is defined as an equaliser:
If
![Axioms 01 00201 i206]()
is an
![Axioms 01 00201 i165]()
-comodule algebra, and
![Axioms 01 00201 i282]()
, the
![Axioms 01 00201 i283]()
is a left
![Axioms 01 00201 i219]()
-module with the action
![Axioms 01 00201 i286]()
In particular, in the case of a principal
![Axioms 01 00201 i083]()
-bundle
![Axioms 01 00201 i084]()
over
![Axioms 01 00201 i287]()
, for any left
![Axioms 01 00201 i288]()
-comodule
![Axioms 01 00201 i271]()
the cotensor product
![Axioms 01 00201 i290]()
is a left
![Axioms 01 00201 i291]()
-module.
The following proposition indicates the way in which cotensor products enter description of associated bundles.
Proposition 2.20 Assume that the fibre
of a vector bundle
associated to a principal
-bundle
is finite dimensional. View
as a left comodule of
with the coaction
(summation implicit) determined by
Then the left
-module of sections
is isomorphic to the left
-module
. Proof. First identify
![Axioms 01 00201 i297]()
with
![Axioms 01 00201 i298]()
. Let
![Axioms 01 00201 i299]()
be a (finite) dual basis. Take
![Axioms 01 00201 i300]()
, and define
![Axioms 01 00201 i302]()
.
In the converse direction, define a left
![Axioms 01 00201 i153]()
-module map
One easily checks that the constructed map are mutual inverses.
Moving away from commutative algebras of functions on topological spaces one uses Proposition 2.20 as the motivation for the following definition.
Definition 2.21 Let
![Axioms 01 00201 i166]()
be a principal
![Axioms 01 00201 i165]()
-comodule algebra. Set
![Axioms 01 00201 i306]()
and let
![Axioms 01 00201 i117]()
be a left
![Axioms 01 00201 i165]()
-comodule. The left
![Axioms 01 00201 i244]()
-module
![Axioms 01 00201 i308]()
is called a
module associated to the principal comodule algebra ![Axioms 01 00201 i166]()
.
![Axioms 01 00201 i309]()
is a projective left
![Axioms 01 00201 i244]()
-module, and if
![Axioms 01 00201 i117]()
is a finite dimensional vector space, then
![Axioms 01 00201 i309]()
is a finitely generated projective left
![Axioms 01 00201 i244]()
-module. In this case it has the meaning of a module of sections over a non-commutative vector bundle. Furthermore, its class gives an element in the
![Axioms 01 00201 i310]()
-group of
![Axioms 01 00201 i244]()
. If
![Axioms 01 00201 i166]()
is a cleft principal comodule algebra, then every associated module is free, since
![Axioms 01 00201 i311]()
as a left
![Axioms 01 00201 i244]()
-module and right
![Axioms 01 00201 i165]()
-comodule, so that