In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the
special linear group
In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...
SL(2, R) or SL
2(R) is the
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 iden ...
of 2 × 2
real matrices
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the ...
with
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
one:
:
It is a
connected non-compact simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by John ...
real
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
of dimension 3 with applications in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
,
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, and
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
.
SL(2, R) acts on the
complex upper half-plane by
fractional linear transformations. The
group action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
factors through the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
PSL(2, R) (the 2 × 2
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 associa ...
over R). More specifically,
:PSL(2, R) = SL(2, R) / ,
where ''I'' denotes the 2 × 2
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
. It contains the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
PSL(2, Z).
Also closely related is the 2-fold
covering group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous (topology), continuous group homomorphism. The map ''p'' is called the c ...
, Mp(2, R), a
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, ...
(thinking of SL(2, R) as a
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 gr ...
).
Another related group is SL
±(2, R), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
, however.
Descriptions
SL(2, R) is the group of all
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s of R
2 that preserve
oriented area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
. It is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to 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 gr ...
Sp(2, R) and the special unitary group
SU(1, 1). It is also isomorphic to the group of unit-length
coquaternions. The group SL
±(2, R) preserves unoriented area: it may reverse orientation.
The quotient PSL(2, R) has several interesting descriptions, up to Lie group isomorphism:
* It is the group of
orientation-preserving
projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
s of the
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...
* It is the group of
conformal automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s of the
unit disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
.
* It is the group of
orientation-preserving
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the
hyperbolic plane.
* It is the restricted
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physi ...
of three-dimensional
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
. Equivalently, it is isomorphic to the
indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension (vector space), dimensional real number, real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bi ...
SO
+(1,2). It follows that SL(2, R) is isomorphic to the
spin group
In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathbb_2 \to \o ...
Spin(2,1)
+.
Elements of the modular group PSL(2, Z) have additional interpretations, as do elements of the group SL(2, Z) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL(2, R).
Homographies
Elements of PSL(2, R) are
homographies on the
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...
:
:
These projective transformations form a subgroup of PSL(2, C), which acts on the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
by
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
s.
When the real line is considered the boundary of the
hyperbolic plane, PSL(2, R) expresses
hyperbolic motions.
Möbius transformations
Elements of PSL(2, R) act on the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
by Möbius transformations:
:
This is precisely the set of Möbius transformations that preserve the
upper half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
. It follows that PSL(2, R) is the group of conformal automorphisms of the upper half-plane. By the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...
, it is also isomorphic to the group of conformal automorphisms of the unit disc.
These Möbius transformations act as the
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the
upper half-plane model of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
.
The above formula can be also used to define Möbius transformations of
dual and
double (aka split-complex) numbers. The corresponding geometries are in non-trivial relations
to
Lobachevskian geometry.
Adjoint representation
The group SL(2, R) acts on its Lie algebra sl(2, R) by
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
*Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change o ...
(remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear
representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of
quadratic forms
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
on R
2. The result is the following representation:
:
The
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
on sl(2, R) has
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
(2,1), and induces an isomorphism between PSL(2, R) and the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physi ...
SO
+(2,1). This action of PSL(2, R) on
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
restricts to the isometric action of PSL(2, R) on the
hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
of the hyperbolic plane.
Classification of elements
The
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s of an element ''A'' ∈ SL(2, R) satisfy the
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
:
and therefore
:
This leads to the following classification of elements, with corresponding action on the Euclidean plane:
* If
, then ''A'' is called elliptic, and is conjugate to a
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
.
* If
, then ''A'' is called parabolic, and is a
shear mapping
In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance function, signed distance from a given straight line, line parallel (geometry), paral ...
.
* If
, then ''A'' is called hyperbolic, and is a
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation (mathematics), rotation or shear mapping.
For a fixed p ...
.
The names correspond to the classification of
conic section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
s by
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
: if one defines eccentricity as half the absolute value of the trace (ε = , tr, ; dividing by 2 corrects for the effect of dimension, while absolute value corresponds to ignoring an overall factor of ±1 such as when working in PSL(2, R)), then this yields:
, elliptic;
, parabolic;
, hyperbolic.
The identity element 1 and negative identity element −1 (in PSL(2, R) they are the same), have trace ±2, and hence by this classification are parabolic elements, though they are often considered separately.
The same classification is used for SL(2, C) and PSL(2, C) (
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
s) and PSL(2, R) (real Möbius transformations), with the addition of "loxodromic" transformations corresponding to complex traces;
analogous classifications are used elsewhere.
A subgroup that is contained with the elliptic (respectively, parabolic, hyperbolic) elements, plus the identity and negative identity, is called an elliptic subgroup (respectively,
parabolic subgroup, hyperbolic subgroup).
The trichotomy of SL(2, R) into elliptic, parabolic, and hyperbolic elements is a classification into ''subsets,'' not ''subgroups:'' these sets are not closed under multiplication (the product of two parabolic elements need not be parabolic, and so forth). However, each element is conjugate to a member of one of 3 standard
one-parameter subgroups (possibly times ±1), as detailed below.
Topologically, as trace is a continuous map, the elliptic elements (excluding ±1) form an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
, as do the hyperbolic elements (excluding ±1). By contrast, the parabolic elements, together with ±1, form a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
that is not open.
Elliptic elements
The
eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
for an elliptic element are both complex, and are
conjugate values on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. Such an element is conjugate to a
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
of the Euclidean plane – they can be interpreted as rotations in a possibly non-orthogonal basis – and the corresponding element of PSL(2, R) acts as (conjugate to) a
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
of the hyperbolic plane and of
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
.
Elliptic elements of the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
must have eigenvalues , where ''ω'' is a primitive 3rd, 4th, or 6th
root of unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
. These are all the elements of the modular group with finite
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
, and they act on the
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
as periodic diffeomorphisms.
Elements of trace 0 may be called "circular elements" (by analogy with eccentricity) but this is rarely done; they correspond to elements with eigenvalues ±''i'', and are conjugate to rotation by 90°, and square to -''I'': they are the non-identity
involution
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
s in PSL(2).
Elliptic elements are conjugate into the subgroup of rotations of the Euclidean plane, 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. ...
SO(2); the angle of rotation is
arccos of half of the trace, with the sign of the rotation determined by orientation. (A rotation and its inverse are conjugate in GL(2) but not SL(2).)
Parabolic elements
A parabolic element has only a single eigenvalue, which is either 1 or -1. Such an element acts as a
shear mapping
In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance function, signed distance from a given straight line, line parallel (geometry), paral ...
on the Euclidean plane, and the corresponding element of PSL(2, R) acts as a
limit rotation of the hyperbolic plane and as a
null rotation of
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
.
Parabolic elements of the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
act as
Dehn twists of the torus.
Parabolic elements are conjugate into the 2 component group of standard shears × ±''I'':
. In fact, they are all conjugate (in SL(2)) to one of the four matrices
,
(in GL(2) or SL
±(2), the ± can be omitted, but in SL(2) it cannot).
Hyperbolic elements
The
eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
for a hyperbolic element are both real, and are reciprocals. Such an element acts as a
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation (mathematics), rotation or shear mapping.
For a fixed p ...
of the Euclidean plane, and the corresponding element of PSL(2, R) acts as a
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
of the hyperbolic plane and as a
Lorentz boost
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
on
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
.
Hyperbolic elements of the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
act as
Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
s of the torus.
Hyperbolic elements are conjugate into the 2 component group of standard squeezes × ±''I'':
; the
hyperbolic angle of the hyperbolic rotation is given by
arcosh of half of the trace, but the sign can be positive or negative: in contrast to the elliptic case, a squeeze and its inverse are conjugate in SL₂ (by a rotation in the axes; for standard axes, a rotation by 90°).
Conjugacy classes
By
Jordan normal form
\begin
\lambda_1 1\hphantom\hphantom\\
\hphantom\lambda_1 1\hphantom\\
\hphantom\lambda_1\hphantom\\
\hphantom\lambda_2 1\hphantom\hphantom\\
\hphantom\hphantom\lambda_2\hphantom\\
\hphantom\lambda_3\hphantom\\
\hphantom\ddots\hphantom\\
...
, matrices are classified up to conjugacy (in GL(''n'', C)) by eigenvalues and nilpotence (concretely, nilpotence means where 1s occur in the Jordan blocks). Thus elements of SL(2) are classified up to conjugacy in GL(2) (or indeed SL
±(2)) by trace (since determinant is fixed, and trace and determinant determine eigenvalues), except if the eigenvalues are equal, so ±I and the parabolic elements of trace +2 and trace -2 are not conjugate (the former have no off-diagonal entries in Jordan form, while the latter do).
Up to conjugacy in SL(2) (instead of GL(2)), there is an additional datum, corresponding to orientation: a clockwise and counterclockwise (elliptical) rotation are not conjugate, nor are a positive and negative shear, as detailed above; thus for absolute value of trace less than 2, there are two conjugacy classes for each trace (clockwise and counterclockwise rotations), for absolute value of the trace equal to 2 there are three conjugacy classes for each trace (positive shear, identity, negative shear), and for absolute value of the trace greater than 2 there is one conjugacy class for a given trace.
Iwasawa or KAN decomposition
The
Iwasawa decomposition of a group is a method to construct the group as a product of three Lie subgroups ''K'', ''A'', ''N''.
For
these three subgroups are
:
:
:
These three elements are the generators of the Elliptic, Hyperbolic, and Parabolic subsets respectively.
Topology and universal cover
As a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, PSL(2, R) can be described as the
unit tangent bundle of the hyperbolic plane. It is a
circle bundle, and has a natural
contact structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...
induced by the
symplectic structure
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
on the hyperbolic plane. SL(2, R) is a 2-fold cover of PSL(2, R), and can be thought of as the bundle of
spinor
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
s on the hyperbolic plane.
The fundamental group of SL(2, R) is the infinite
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
Z. The
universal covering group, denoted
, is an example of a finite-dimensional Lie group that 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 fai ...
. That is,
admits no
faithful, finite-dimensional
representation.
As a topological space,
is a line bundle over the hyperbolic plane. When imbued with a left-invariant
metric
Metric or metrical may refer to:
Measuring
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
...
, the
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
becomes one of the
eight Thurston geometries. For example,
is the universal cover of the unit tangent bundle to any
hyperbolic surface. Any manifold modeled on
is orientable, and is a
circle bundle over some 2-dimensional hyperbolic
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space.
D ...
(a
Seifert fiber space).

Under this covering, the preimage of the modular group PSL(2, Z) is the
braid group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
on 3 generators, ''B''
3, which is the
universal 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 \op ...
of the modular group. These are lattices inside the relevant algebraic groups, and this corresponds algebraically to the universal covering group in topology.
The 2-fold covering group can be identified as Mp(2, R), a
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, ...
, thinking of SL(2, R) as the symplectic group Sp(2, R).
The aforementioned groups together form a sequence:
:
However, there are other covering groups of PSL(2, R) corresponding to all ''n'', as ''n'' Z < Z ≅ π
1 (PSL(2, R)), which form a
lattice of covering groups by divisibility; these cover SL(2, R)
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''n'' is even.
Algebraic structure
The
center of SL(2, R) is the two-element group , and the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
PSL(2, R) is
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by John ...
.
Discrete subgroups of PSL(2, R) are called
Fuchsian groups. These are the hyperbolic analogue of the Euclidean
wallpaper group
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...
s and
Frieze groups. The most famous of these is the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
PSL(2, Z), which acts on a tessellation of the hyperbolic plane by ideal triangles.
The
circle group
In mathematics, the circle group, denoted by \mathbb T or , 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 = \.
The circle g ...
SO(2)
In mathematics, the circle group, denoted by \mathbb T or , 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 = \.
The circle g ...
is a
maximal compact subgroup
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
of SL(2, R), and the circle SO(2) / is a maximal compact subgroup of PSL(2, R).
The
Schur multiplier
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 \ope ...
of the
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 ...
PSL(2, R) is much larger than Z, and the universal
central extension is much larger than the universal covering group. However these large central extensions do not take the
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
into account and are somewhat pathological.
Representation theory
SL(2, R) is a real, non-compact
simple Lie group, and is the split-real form of the
complex Lie group SL(2, C). The
Lie algebra
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 ident ...
of SL(2, R), denoted sl(2, R), is the algebra of all real,
traceless 2 × 2 matrices. It is the
Bianchi algebra of type VIII.
The finite-dimensional representation theory of SL(2, R) is equivalent to the
representation theory of SU(2), which is the compact real form of SL(2, C). In particular, SL(2, R) has no nontrivial finite-dimensional unitary representations. This is a feature of every connected simple non-compact Lie group. For outline of proof, see
non-unitarity of representations.
The infinite-dimensional representation theory of SL(2, R) is quite interesting. The group has several families of unitary representations, which were worked out in detail by
Gelfand and
Naimark (1946),
V. Bargmann (1947), and
Harish-Chandra
Harish-Chandra (né Harishchandra) FRS (11 October 1923 – 16 October 1983) was an Indian-American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
Early ...
(1952).
See also
*
Linear 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 ...
*
Special linear group
In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...
*
Projective 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 associa ...
*
Modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
*
SL(2, C) (Möbius transformations)
*
Projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
*
Fuchsian group
*
Table of Lie groups
*
Anosov flow
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
References
*
*
*
*
* {{cite book, first=William, last=Thurston, title=Three-dimensional geometry and topology. Vol. 1, mr=1435975 , editor=Silvio Levy , series=Princeton Mathematical Series , volume=35 , publisher=Princeton University Press , location=Princeton, NJ , year=1997 , isbn=0-691-08304-5
Group theory
Lie groups
Projective geometry
Hyperbolic geometry
3-manifolds