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 covering group of a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
''H'' is a
covering space 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 ...
''G'' of ''H'' such that ''G'' is a topological group and the covering map is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
. The map ''p'' is called the covering homomorphism. A frequently occurring case is a double covering group, a
topological double cover in which ''H'' has
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
2 in ''G''; examples include the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
s,
pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
s, and
metaplectic group
In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, ...
s.
Roughly explained, saying that for example the metaplectic group Mp
2''n'' is a ''double cover'' of the
symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...
Sp
2''n'' means that there are always two elements in the metaplectic group representing one element in the symplectic group.
Properties
Let ''G'' be a covering group of ''H''. 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 ...
''K'' of the covering homomorphism is just the fiber over the identity in ''H'' and is a
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 ...
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 ...
of ''G''. The kernel ''K'' is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
in ''G'' if and only if ''G'' is
Hausdorff (and if and only if ''H'' is Hausdorff). Going in the other direction, if ''G'' is any topological group and ''K'' is a discrete normal subgroup of ''G'' then the quotient map ''p'' : ''G'' → ''G''/''K'' is a covering homomorphism.
If ''G'' is
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 ...
then ''K'', being a discrete normal subgroup, necessarily lies in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of ''G'' and is therefore
abelian. In this case, the center of ''H'' = ''G''/''K'' is given by
:
As with all covering spaces, the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of ''G'' injects into the fundamental group of ''H''. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space. In particular, if ''G'' is
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
then the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
is isomorphic to ''K''. The group ''K''
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
simply transitively on the fibers (which are just left
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s) by right multiplication. The group ''G'' is then a
principal ''K''-bundle over ''H''.
If ''G'' is a covering group of ''H'' then the groups ''G'' and ''H'' are
locally isomorphic. Moreover, given any two connected locally isomorphic groups ''H''
1 and ''H''
2, there exists a topological group ''G'' with discrete normal subgroups ''K''
1 and ''K''
2 such that ''H''
1 is isomorphic to ''G''/''K''
1 and ''H''
2 is isomorphic to ''G''/''K''
2.
Group structure on a covering space
Let ''H'' be a topological group and let ''G'' be a covering space of ''H''. If ''G'' and ''H'' are both
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
and
locally path-connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness a ...
, then for any choice of element ''e''* in the fiber over ''e'' ∈ ''H'', there exists a unique topological group structure on ''G'', with ''e''* as the identity, for which the covering map ''p'' : ''G'' → ''H'' is a homomorphism.
The construction is as follows. Let ''a'' and ''b'' be elements of ''G'' and let ''f'' and ''g'' be
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
s in ''G'' starting at ''e''* and terminating at ''a'' and ''b'' respectively. Define a path ''h'' : ''I'' → ''H'' by ''h''(''t'') = ''p''(''f''(''t''))''p''(''g''(''t'')). By the path-lifting property of covering spaces there is a unique lift of ''h'' to ''G'' with initial point ''e''*. The product ''ab'' is defined as the endpoint of this path. By construction we have ''p''(''ab'') = ''p''(''a'')''p''(''b''). One must show that this definition is independent of the choice of paths ''f'' and ''g'', and also that the group operations are continuous.
Alternatively, the group law on ''G'' can be constructed by lifting the group law ''H'' × ''H'' → ''H'' to ''G'', using the lifting property of the covering map ''G'' × ''G'' → ''H'' × ''H''.
The non-connected case is interesting and is studied in the papers by Taylor and by Brown-Mucuk cited below. Essentially there is an obstruction to the existence of a universal cover which is also a topological group such that the covering map is a morphism: this obstruction lies in the third cohomology group of the group of components of ''G'' with coefficients in the fundamental group of ''G'' at the identity.
Universal covering group
If ''H'' is a path-connected, locally path-connected, and
semilocally simply connected
In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space ''X'' is semi-locally simply connected if ...
group then it has a
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 ...
. By the previous construction the universal cover can be made into a topological group with the covering map a continuous homomorphism. This group is called the universal covering group of ''H''. There is also a more direct construction which we give below.
Let ''PH'' be the
path group
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
of ''H''. That is, ''PH'' is the space of
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
s in ''H'' based at the identity together with 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 ...
. The product of paths is given by pointwise multiplication, i.e. (''fg'')(''t'') = ''f''(''t'')''g''(''t''). This gives ''PH'' the structure of a topological group. There is a natural group homomorphism ''PH'' → ''H'' which sends each path to its endpoint. The universal cover of ''H'' is given as the quotient of ''PH'' by the normal subgroup of
null-homotopic
In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed ...
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
s. The projection ''PH'' → ''H'' descends to the quotient giving the covering map. One can show that the universal cover is
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 ...
and the kernel is just the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of ''H''. That is, we have a
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
:
where
is the universal cover of ''H''. Concretely, the universal covering group of ''H'' is the space of homotopy classes of paths in ''H'' with pointwise multiplication of paths. The covering map sends each path class to its endpoint.
Lattice of covering groups
As the above suggest, if a group has a universal covering group (if it is path-connected, locally path-connected, and semilocally simply connected), with discrete center, then the set of all topological groups that are covered by the universal covering group form a lattice, corresponding to the lattice of subgroups of the center of the universal covering group: inclusion of subgroups corresponds to covering of quotient groups. The maximal element is the universal covering group
while the minimal element is the universal covering group mod its center,
.
This corresponds algebraically to the
universal perfect central extension
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
(called "covering group", by analogy) as the maximal element, and a group mod its center as minimal element.
This is particularly important for Lie groups, as these groups are all the (connected) realizations of a particular Lie algebra. For many Lie groups the center is the group of scalar matrices, and thus the group mod its center is the projectivization of the Lie group. These covers are important in studying
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group
\mathrm(V) = \mathrm(V) / F^*,
where GL(' ...
s of Lie groups, and
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups ...
s lead to the discovery of
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
s: a projective representation of a Lie group need not come from a linear representation of the group, but does come from a linear representation of some covering group, in particular the universal covering group. The finite analog led to the covering group or Schur cover, as discussed above.
A key example arises from
SL2(R), which has center and fundamental group Z. It is a double cover of the centerless
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
PSL
2(R), which is obtained by taking the quotient by the center. By
Iwasawa decomposition In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a conseq ...
, both groups are circle bundles over the complex upper half-plane, and their universal cover
is a real line bundle over the half-plane that forms one of
Thurston's eight geometries. Since the half-plane is contractible, all bundle structures are trivial. The preimage of SL
2(Z) in the universal cover is isomorphic to the
braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
on three strands.
Lie groups
The above definitions and constructions all apply to the special case of
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. In particular, every covering of a
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 ...
is a manifold, and the covering homomorphism becomes a
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
. Likewise, given any discrete normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism.
Two Lie groups are locally isomorphic if and only if their
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
are isomorphic. This implies that a homomorphism φ : ''G'' → ''H'' of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras
:
is an isomorphism.
Since for every Lie algebra
there is a unique simply connected Lie group ''G'' with Lie algebra
, from this follows that the universal covering group of a connected Lie group ''H'' is the (unique) simply connected Lie group ''G'' having the same Lie algebra as ''H''.
Examples
* The universal covering group of the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
T is the additive group of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s R with the covering homomorphism given by the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
exp: R → T. The kernel of the exponential map is isomorphic to Z.
* For any integer ''n'' we have a covering group of the circle by itself T → T which sends ''z'' to ''z''
''n''. The kernel of this homomorphism is the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
consisting of the ''n''th
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
.
* The rotation group
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
has as a universal cover the group
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
which is isomorphic to the group of
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
s in the quaternions. This is a double cover since the kernel has order 2. (cf the
tangloids
Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.
A description of the game appeared in the book ''"Martin Gardner's New Mathematical Diversions from Scientific American"'' by Martin Gardner ...
.)
* The
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an ...
U(''n'') is covered by the compact group T × SU(''n'') with the covering homomorphism given by ''p''(''z'', ''A'') = ''zA''. The universal cover is R × SU(''n'').
* The
special orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
SO(''n'') has a double cover called the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
Spin(''n''). For ''n'' ≥ 3, the spin group is the universal cover of SO(''n'').
* For ''n'' ≥ 2, the universal cover of the
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
SL(''n'', R) is ''not'' a
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fait ...
(i.e. it has no faithful finite-dimensional
representations).
References
*
*
* {{cite journal , last=Brown , first=R. , last2=Mucuk , first2=O. , title=Covering groups of nonconnected topological groups revisited , journal=Math. Proc. Cambridge Philos. Soc. , volume=115 , issue=1 , pages=97–110 , year=1994 , doi=10.1017/S0305004100071942 , arxiv=math/0009021 , bibcode=2000math......9021B , citeseerx=10.1.1.236.9436
Topological groups
Lie groups