In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Lie groupoid is a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*''Group'' with a partial functi ...
where the set
of
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
s and the set
of
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are both
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, all the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
:
are
submersions.
A Lie groupoid can thus be thought of as a "many-object generalization" of a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, just as a groupoid is a many-object generalization of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. Accordingly, while Lie groups provide a natural model for (classical)
continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the
correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of
Lie algebroid In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought of ...
s.
Lie groupoids were introduced by
Charles Ehresmann under the name ''differentiable groupoids''.
Definition and basic concepts
A Lie groupoid consists of
* two smooth manifolds
and
* two
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
submersions (called, respectively, source and target projections)
* a map
(called multiplication or composition map), where we use the notation
* a map
(called unit map or object inclusion map), where we use the notation
* a map
(called inversion), where we use the notation
such that
* the composition satisfies
and
for every
for which the composition is defined
*the composition is
associative, i.e.
for every
for which the composition is defined
*
works as an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
, i.e.
for every
and
and
for every
*
works as an
inverse, i.e.
and
for every
.
Using the language of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a Lie groupoid can be more compactly defined as a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*''Group'' with a partial functi ...
(i.e. a
small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
where all the morphisms are invertible) such that the sets
of objects and
of morphisms are manifolds, the maps
,
,
,
and
are smooth and
and
are submersions. A Lie groupoid is therefore not simply a
groupoid object In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined.
Defin ...
in the
category of smooth manifolds: one has to ask the additional property that
and
are submersions.
Lie groupoids are often denoted by
, where the two arrows represent the source and the target. The notation
is also frequently used, especially when stressing the simplicial structure of the associated
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system.
A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...
.
In order to include more natural examples, the manifold
is not required in general to be
Hausdorff or
second countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
(while
and all other spaces are).
Alternative definitions
The original definition by Ehresmann required
and
to possess a smooth structure such that only
is smooth and the maps
and
are subimmersions (i.e. have locally constant
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
). Such definition proved to be too weak and was replaced by Pradines with the one currently used.
While some authors introduced weaker definitions which did not require
and
to be submersions, these properties are fundamental to develop the entire Lie theory of groupoids and algebroids.
First properties
The fact that the source and the target map of a Lie groupoid
are smooth submersions has some immediate consequences:
* the
-fibres
, the
-fibres
, and the set of composable morphisms
are
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s;
* the inversion map
is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
;
*the unit map
is a
smooth embedding;
*the
isotropy groups are
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s;
* the orbits
are
immersed submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
s;
* the
-fibre
at a point
is a
principal -bundle over the orbit
at that point.
Subobjects and morphisms
A Lie subgroupoid of a Lie groupoid
is a
subgroupoid (i.e. a
subcategory of the category
) with the extra requirement that
is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if
. Any Lie groupoid
has two canonical wide subgroupoids:
* the unit/identity Lie subgroupoid
;
* the inner subgroupoid
, i.e. the bundle of isotropy groups (which however may fail to be smooth in general).
A normal Lie subgroupoid is a wide Lie subgroupoid
inside
such that, for every
with
, one has
. The isotropy groups of
are therefore
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
s of the isotropy groups of
.
A Lie groupoid morphism between two Lie groupoids
and
is a groupoid morphism
(i.e. a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
between the categories
and
), where both
and
are smooth. The
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
of a morphism between Lie groupoids over the same base manifold is automatically a normal Lie subgroupoid.
The
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
has a natural groupoid structure such that the projection
is a groupoid morphism; however, unlike
quotients of Lie groups,
may fail to be a Lie groupoid in general. Accordingly, the
isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist ...
for groupoids cannot be specialised to the entire category of Lie groupoids, but only to special classes.
A Lie groupoid is called abelian if its isotropy Lie groups are
abelian. For similar reasons as above, while the definition of
abelianisation
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
of a group extends to set-theoretical groupoids, in the Lie case the analogue of the quotient
may not exist or be smooth.
Bisections
A bisection of a Lie groupoid
is a smooth map
such that
and
is a diffeomorphism of
. In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold
such that
and
are diffeomorphisms; the relation between the two definitions is given by
.
The set of bisections forms a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, with the multiplication
defined as
and inversion defined as
Note that the definition is given in such a way that, if
and
, then
and
.
The group of bisections can be given the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and ...
, as well as an (infinite-dimensional) structure of
Fréchet manifold In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.
More precisely, a Fréchet manifold consists of a Haus ...
compatible with the group structure, making it into a Fréchet-Lie group.
A local bisection
is defined analogously, but the multiplication between local bisections is of course only partially defined.
Examples
Trivial and extreme cases
*Lie groupoids
with one object are the same thing as Lie groups.
*Given any manifold
, there is a Lie groupoid
called the pair groupoid, with precisely one morphism from any object to any other.
*The two previous examples are particular cases of the trivial groupoid
, with structure maps
,
,
,
and
.
*Given any manifold
, there is a Lie groupoid
called the unit groupoid, with precisely one morphism from one object to itself, namely the identity, and no morphisms between different objects.
*More generally, Lie groupoids with
are the same thing as bundle of Lie groups (not necessarily locally trivial). For instance, any vector bundle is a bundle of abelian groups, so it is in particular a(n abelian) Lie groupoid.
Constructions from other Lie groupoids
*Given any Lie groupoid
and a surjective submersion
, there is a Lie groupoid
, called its pullback groupoid or induced groupoid, where
contains triples
such that
and
, and the multiplication is defined using the multiplication of
. For instance, the pullback of the pair groupoid of
is the pair groupoid of
.
*Given any two Lie groupoids
and
, there is a Lie groupoid
, called their
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
, such that the groupoid morphisms
and
are surjective submersions.
*Given any Lie groupoid
, there is a Lie groupoid
, called its tangent groupoid, obtained by considering the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of
and
and the
differential of the structure maps.
*Given any Lie groupoid
, there is a Lie groupoid
, called its cotangent groupoid obtained by considering the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
of
, the
dual of the Lie algebroid
(see below), and suitable structure maps involving the differentials of the left and right translations.
*Given any Lie groupoid
, there is a Lie groupoid
, called its jet groupoid, obtained by considering the
k-jets of the local bisections of
(with smooth structure inherited from the
jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Je ...
of
) and setting
,
,
,
and
.
Examples from differential geometry
*Given a submersion
, there is a Lie groupoid
, called the submersion groupoid or fibred pair groupoid, whose structure maps are induced from the pair groupoid
(the condition that
is a submersion ensures the smoothness of
). If
is a point, one recovers the pair groupoid.
*Given a Lie group
acting
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad r ...
on a manifold
, there is a Lie groupoid
, called the action groupoid or translation groupoid, with one morphism for each triple
with
.
*Given any
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
, there is a Lie groupoid
, called the general linear groupoid, with morphisms between
being linear isomorphisms between the fibres
and
. For instance, if
is the trivial vector bundle of rank
, then
is the action groupoid.
*Any
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
with structure group ''
'' defines a Lie groupoid
, where ''
'' acts on the pairs
componentwise, called the gauge groupoid. The multiplication is defined via compatible representatives as in the pair groupoid.
*Any
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
on a manifold
defines two Lie groupoids,
(or
) and
, called respectively the monodromy/homotopy/fundamental groupoid and the holonomy groupoid of
, whose morphisms consist of the
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
, respectively
holonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
, equivalence classes of paths entirely lying in a leaf of
. For instance, when
is the trivial foliation with only one leaf, one recovers, respectively, the fundamental groupoid and the pair groupoid of
. On the other hand, when
is a simple foliation, i.e. the foliation by (connected) fibres of a submersion
, its holonomy groupoid is precisely the submersion groupoid
but its monodromy groupoid may even fail to be Hausdorff, due to a general criterion in terms of vanishing cycles. In general, many elementary foliations give rise to monodromy and holonomy groupoids which are not Hausdorff.
*Given any
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
, there is a Lie groupoid
, called its germ groupoid, endowed with the sheaf topology and with structure maps analogous to those of the jet groupoid. This is another natural example of Lie groupoid whose arrow space is not Hausdorff nor secound countable.
Important classes of Lie groupoids
Note that some of the following classes make sense already in the category of set-theoretical or
topological groupoids.
Transitive groupoids
A Lie groupoid is transitive (in older literature also called connected) if it satisfies one of the following equivalent conditions:
* there is only one orbit;
* there is at least a morphism between any two objects;
* the map
(also known as the anchor of
) is surjective.
Gauge groupoids constitute the prototypical examples of transitive Lie groupoids: indeed, any transitive Lie groupoid is isomorphic to the gauge groupoid of some principal bundle, namely the
-bundle
, for any point
. For instance:
* the trivial Lie groupoid
is transitive and arise from the trivial principal
-bundle
. As particular cases, Lie groups
and pair groupoids
are trivially transitive, and arise, respectively, from the principal
-bundle
, and from the principal
-bundle
;
* an action groupoid
is transitive if and only if the group action is
transitive, and in such case it arises from the principal bundle
with structure group the isotropy group (at an arbitrary point);
* the general linear groupoid of
is transitive, and arises from the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
;
*pullback groupoids, jet groupoids and tangent groupoids of
are transitive if and only if
is transitive.
As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a to ...
of a (connected) smooth manifold
. This is naturally a topological groupoid, which is moreover transitive; one can see that
is isomorphic to the gauge groupoid of the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of
. Accordingly,
inherits a smooth structure which makes it into a Lie groupoid.
Submersions groupoids
are an example of non-transitive Lie groupoids, whose orbits are precisely the fibres of
.
A stronger notion of transitivity requires the anchor
to be a surjective submersion. Such condition is also called local triviality, because
becomes locally isomorphic (as Lie groupoid) to a trivial groupoid over any open
(as a consequence of the local triviality of principal bundles).
When the space
is second countable, transitivity implies local triviality. Accordingly, these two conditions are equivalent for many examples but not for all of them: for instance, if
is a transitive pseudogroup, its germ groupoid
is transitive but not locally trivial.
Proper groupoids
A Lie groupoid is called proper if
is a
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
map. As a consequence
* all isotropy groups of
are
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
;
* all orbits of
are closed submanifolds;
* the orbit space
is
Hausdorff.
For instance:
* a Lie group is proper if and only if it is compact;
* pair groupoids are always proper;
*unit groupoids are always proper;
* an action groupoid is proper if and only if the action is
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
;
*the fundamental groupoid is proper if and only if the fundamental groups are
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
.
As seen above, properness for Lie groupoids is the "right" analogue of compactness for Lie groups. One could also consider more "natural" conditions, e.g. asking that the source map
is proper (then
is called s-proper), or that the entire space
is compact (then
is called compact), but these requirements turns out to be too strict for many examples and applications.
Étale groupoids
A Lie groupoid is called étale if it satisfies one of the following equivalent conditions:
* the dimensions of
and
are equal;
*
is a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Form ...
;
* all the
-fibres are
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
As a consequence, also the
-fibres, the isotropy groups and the orbits become discrete.
For instance:
* a Lie group is étale if and only if it is discrete;
*pair groupoids are never étale;
*unit groupoids are always étale;
*an action groupoid is étale if and only if
is discrete;
* fundamental groupoids are always étale (but fundamental groupoids of a foliations are not);
* germ groupoids of pseudogroups are always étale.
Effective groupoids
An étale groupoid is called effective if, for any two local bisections
, the condition
implies
. For instance:
* Lie groups are effective if and only if are trivial;
*unit groupoids are always effective;
*an action groupoid is effective if the
-action is
free and
is discrete.
In general, any effective étale groupoid arise as the germ groupoid of some pseudogroup. However, a (more involved) definition of effectiveness, which does not assume the étale property, can also be given.
Source-connected groupoids
A Lie groupoid is called
-connected if all its
-fibres are
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. Similarly, one talks about
-simply connected groupoids (when the
-fibres are
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
) or source-k-connected groupoids (when the
-fibres are
k-connected
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concep ...
, i.e. the first
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s are trivial).
Note that the entire space of arrows
is not asked to satisfy any connectedness hypothesis. However, if
is a source-
-connected Lie groupoid over a
-connected manifold, then
itself is automatically
-connected.
For instanceː
* Lie groups are source
-connected if and only are
-connected;
* a pair groupoid is source
-connected if and only if
is
-connected;
* unit groupoids are always source
-connected;
* action groupoids are source
-connected if and only if
is
-connected.
* monodromy groupoids (hence also fundamental groupoids) are source simply connected.
Further related concepts
Actions and principal bundles
Recall that an action of a groupoid
on a set
along a function
is defined via a collection of maps
for each morphism
between
. Accordingly, an action of a Lie groupoid
on a manifold
along a smooth map
consists of a groupoid action where the maps
are smooth. Of course, for every
there is an induced smooth action of the isotropy group
on the fibre
.
Given a Lie groupoid
, a principal
-bundle consists of a
-space
and a
-invariant surjective submersion
such that
is a diffeomorphism. Equivalent (but more involved) definitions can be given using
-valued cocycles or local trivialisations.
When
is a Lie groupoid over a point, one recovers, respectively, standard
Lie group action In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.
__TOC__
Definition and first properties
Let \sigma: G \times M \to M, ( ...
s and
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
s.
Representations
A representation of a Lie groupoid
consists of a Lie groupoid action on a vector bundle
, such that the action is fibrewise linear, i.e. each bijection
is a linear isomorphism. Equivalently, a representation of
on
can be described as a Lie groupoid morphism from
to the general linear groupoid
.
Of course, any fibre
becomes a representation of the isotropy group
. More generally, representations of transitive Lie groupoids are uniquely determined by representations of their isotropy groups, via the construction of the
associated vector bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
.
Examples of Lie groupoids representations include the following:
* representations of Lie groups
recover standard
Lie group representations
* representations of pair groupoids
are trivial vector bundles
* representations of unit groupoids
are vector bundles
* representations of action groupoid
are
-
equivariant vector bundle In mathematics, given an action \sigma: G \times_S X \to X of a group scheme ''G'' on a scheme ''X'' over a base scheme ''S'', an equivariant sheaf ''F'' on ''X'' is a sheaf of \mathcal_X-modules together with the isomorphism of \mathcal_-modules
...
s
* representations of fundamental groupoids
are vector bundles endowed with
flat connections
The set
of isomorphism classes of representations of a Lie groupoid
has a natural structure of
semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
, with direct sums and tensor products of vector bundles.
Differentiable cohomology
The notion of differentiable cohomology for Lie groups generalises naturally also to Lie groupoids: the definition relies on the
symplicial structure of the
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system.
A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...
of
, viewed as a category.
More precisely, recall that the space
consists of strings of
composable morphisms, i.e.
and consider the map
.
A differentiable ''
''-cochain of
with coefficients in some representation
is a smooth section of the pullback vector bundle
. One denotes by
the space of such ''
''-cochains, and considers the differential
, defined as
Then
becomes a
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
and its cohomology, denoted by
, is called the differentiable cohomology of
with coefficients in
. Note that, since the differential at degree zero is
, one has always
.
Of course, the differentiable cohomology of
as a Lie groupoid coincides with the standard differentiable cohomology of
as a Lie group (in particular, for
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
s one recovers the usual
group cohomology). On the other hand, for any ''proper'' Lie groupoid
, one can prove that
for every
.
The Lie algebroid of a Lie groupoid
Any Lie groupoid
has an associated
Lie algebroid In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought of ...
, obtained with a construction similar to the one which associates a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
to any Lie groupː
* the vector bundle
is the vertical bundle with respect to the source map, restricted to the elements tangent to the identities, i.e.
;
* the Lie bracket is obtained by identifying
with the left-invariant vector fields on
, and by transporting their Lie bracket to
;
* the anchor map
is the differential of the target map
restricted to
.
The
Lie group–Lie algebra correspondence In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are Isomorphism, isomorphic to each other have Lie algebras ...
generalises to some extends also to Lie groupoids: the first two Lie's theorem (also known as the subgroups–subalgebras theorem and the homomorphisms theorem) can indeed be easily adapted to this setting.
In particular, as in standard Lie theory, for any s-connected Lie groupoid
there is a unique (up to isomorphism) s-simply connected Lie groupoid
with the same Lie algebroid of
, and a local diffeomorphism
which is a groupoid morphism. For instance,
* given any connected manifold
its pair groupoid
is s-connected but not s-simply connected, while its fundamental groupoid
is. They both have the same Lie algebroid, namely the tangent bundle
, and the local diffeomorphism
is given by
.
* given any foliation
on
, its holonomy groupoid
is s-connected but not s-simply connected, while its monodromy groupoid
is. They both have the same Lie algebroid, namely the foliation algebroid
, and the local diffeomorphism
is given by