View source for SL2(R)

{{Short description|Group of real 2×2 matrices with unit determinant}}
{{DISPLAYTITLE:SL<sub>2</sub>('''R''')}}
{{Group theory sidebar |Topological}}

In [[mathematics]], the [[special linear group]] '''SL(2,R)''' or '''SL<sub>2</sub>(R)''' is the [[Group (mathematics)|group]] of [[2 × 2 real matrices]] with [[determinant]] one:
: <math>\mbox{SL}(2,\mathbf{R}) = \left\{ \left( \begin{matrix}
a & b \\
c & d
\end{matrix} \right) : a,b,c,d\in\mathbf{R}\mbox{ and }ad-bc=1\right\}.</math>

It is a [[Connected_space|connected]] [[noncompact]] [[simple Lie group|simple]] [[real Lie group]] with applications in [[geometry]], [[topology]], [[representation theory]], and [[physics]].

SL(2,'''R''') acts on the [[complex upper half-plane]] by fractional linear transformations. The [[Group action (mathematics)|group action]] factors through the [[quotient group|quotient]] '''PSL(2,R)''' (the 2&nbsp;×&nbsp;2 [[projective special linear group]] over '''R'''). More specifically,
:PSL(2,'''R''')&nbsp;=&nbsp;SL(2,'''R''')/{±''I''},
where ''I'' denotes the 2&nbsp;×&nbsp;2 [[identity matrix]]. It contains the [[modular group]] PSL(2,'''Z''').

Also closely related is the 2-fold [[covering group]], Mp(2,'''R'''), a [[metaplectic group]] (thinking of SL(2,'''R''') as a [[symplectic group]]).

Another related group is SL<sup>±</sup>(2,'''R''') the group of real 2&nbsp;×&nbsp;2 matrices with determinant ±1; this is more commonly used in the context of the [[modular group]], however.

==Descriptions==

SL(2,'''R''') is the group of all [[linear transformation]]s of '''R'''<sup>2</sup> that preserve [[Orientation (mathematics)|oriented]] [[area (geometry)|area]]. It is [[Group isomorphism|isomorphic]] to the [[symplectic group]] Sp(2,'''R''') and the special unitary group [[SU(1,1)]].  It is also isomorphic to the group of unit-length [[split-quaternion|coquaternions]]. The group SL<sup>±</sup>(2,'''R''') preserves unoriented area: it may reverse orientation.

The quotient PSL(2,'''R''') has several interesting descriptions:
* It is the group of [[orientation (mathematics)|orientation]]-preserving [[projective transformation]]s of the [[real projective line]] {{nowrap|'''R''' ∪ {∞}.}}
* It is the group of [[conformal map|conformal]] [[automorphism]]s of the [[unit disc]].
* It is the group of [[orientation (mathematics)|orientation]]-preserving [[isometry|isometries]] of the [[Hyperbolic space|hyperbolic plane]].
* It is the restricted [[Lorentz group]] of three-dimensional [[Minkowski space]].  Equivalently, it is isomorphic to the [[indefinite orthogonal group]] SO<sup>+</sup>(1,2).  It follows that SL(2,'''R''') is isomorphic to the [[spin group]] Spin(2,1)<sup>+</sup>.

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''').

===Linear fractional transformations===
Elements of PSL(2,'''R''') act on the [[real projective line]] {{nowrap|'''R''' ∪ {∞}{{null}}}} as [[linear fractional transformation]]s:
: <math>x \mapsto \frac{ax+b}{cx+d}. </math>

This is analogous to the action of PSL(2,'''C''') on the [[Riemann sphere]] by [[Möbius transformation]]s. It is the restriction of the action of PSL(2,'''R''') on the [[hyperbolic plane]] to the boundary at infinity.

===Möbius transformations===
Elements of PSL(2,'''R''') act on the complex plane by Möbius transformations:
: <math>z \mapsto \frac{az+b}{cz+d}\;\;\;\;\mbox{ (where }a,b,c,d\in\mathbf{R}\mbox{)}.</math>

This is precisely the set of Möbius transformations that preserve the [[upper half-plane]].  It follows that PSL(2,'''R''') is the group of conformal automorphisms of the upper half-plane.  By the [[Riemann mapping theorem]], it is also the group of conformal automorphisms of the unit disc.

These Möbius transformations act as the [[isometries]] of the [[Poincaré half-plane model|upper half-plane model]] of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the [[Poincaré disk model]].

The above formula can be also used to define Möbius transformations of [[dual number|dual]] and [[split-complex number|double (aka split-complex) numbers]]. The corresponding geometries are in non-trivial relations<ref name="raw">{{cite book |last=Kisil |first=Vladimir V. |year=2012 |title=Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R) | location=London |publisher=Imperial College Press|page=xiv+192 |isbn=978-1-84816-858-9 | mr=2977041 | doi=10.1142/p835}}</ref> to [[hyperbolic geometry|Lobachevskian geometry]].

===Adjoint representation===
The group SL(2,'''R''') acts on its Lie algebra sl(2,'''R''') by [[conjugation (group theory)|conjugation]] (remember that the Lie algebra elements are also 2 by 2 matrices), yielding a faithful 3-dimensional linear [[representation theory|representation]] of PSL(2,'''R''').  This can alternatively be described as the action of PSL(2,'''R''') on the space of [[quadratic forms]] on '''R'''<sup>2</sup>.  The result is the following representation:
:<math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \mapsto \begin{bmatrix}
a^2 & 2ab & b^2 \\
ac & ad+bc & bd \\
c^2 & 2cd & d^2
\end{bmatrix}.</math>

The [[Killing form]] on sl(2,'''R''') has [[metric signature|signature]] (2,1), and induces an isomorphism between PSL(2,'''R''') and the [[Lorentz group]] SO<sup>+</sup>(2,1).  This action of PSL(2,'''R''') on [[Minkowski space]] restricts to the isometric action of PSL(2,'''R''') on the [[hyperboloid model]] of the hyperbolic plane.

==Classification of elements==
The [[eigenvalue]]s of an element ''A'' ∈ SL(2,'''R''') satisfy the [[characteristic polynomial]]
:<math> \lambda^2 \,-\, \mathrm{tr}(A)\,\lambda \,+\, 1 \,=\, 0</math>

and therefore
:<math> \lambda = \frac{\mathrm{tr}(A) \pm \sqrt{\mathrm{tr}(A)^2 - 4}}{2}. </math>

This leads to the following classification of elements, with corresponding action on the Euclidean plane:
* If | tr(''A'') | < 2, then ''A'' is called '''elliptic,''' and is conjugate to a [[rotation (mathematics)|rotation]].
* If | tr(''A'') | = 2, then ''A'' is called '''parabolic,''' and is a [[shear mapping]].
* If | tr(''A'') | > 2, then ''A'' is called '''hyperbolic,''' and is a [[squeeze mapping]].

The names correspond to the classification of [[conic section]]s by [[Eccentricity (mathematics)|eccentricity]]: if one defines eccentricity as half the absolute value of the trace (ε = ½ 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: <math>\epsilon < 1</math>, elliptic; <math>\epsilon = 1</math>, parabolic; <math>\epsilon > 1</math>, 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.

[[Möbius transformation#Classification|The same classification]] is used for SL(2,'''C''') and PSL(2,'''C''') ([[Möbius transformation]]s) and PSL(2,'''R''') (real Möbius transformations), with the addition of "loxodromic" transformations corresponding to complex traces; [[Eccentricity (mathematics)#Analogous classifications|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''').

This 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, all elements are conjugate into one of 3 standard [[one-parameter subgroup]]s (possibly times ±1), as detailed below.

Topologically, as trace is a continuous map, the elliptic elements (excluding ±1) are an [[open set]], as are the hyperbolic elements (excluding ±1), while the parabolic elements (including ±1) are a [[closed set]].

===Elliptic elements===
The [[eigenvalues]] for an elliptic element are both complex, and are [[complex conjugate|conjugate]] values on the [[unit circle]].  Such an element is conjugate to a [[rotation]] 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]] of the hyperbolic plane and of [[Minkowski space]].

Elliptic elements of the [[modular group]] must have eigenvalues {ω, ω<sup>−1</sup>}, where ''ω'' is a primitive 3rd, 4th, or 6th [[root of unity]].  These are all the elements of the modular group with finite [[order (group theory)|order]], and they act on the [[torus]] 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 (mathematics)|involution]]s in PSL(2).

Elliptic elements are conjugate into the subgroup of rotations of the Euclidean plane, the [[special orthogonal group]] SO(2); the angle of rotation is [[Arccos (trigonometry)|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]] on the Euclidean plane, and the corresponding element of PSL(2,'''R''') acts as a [[limit rotation]] of the hyperbolic plane and as a [[Lorentz group#Conjugacy classes|null rotation]] of [[Minkowski space]].

Parabolic elements of the [[modular group]] act as [[Dehn twist]]s of the torus.

Parabolic elements are conjugate into the 2 component group of standard shears × ±''I'': <math>\left(\begin{smallmatrix}1 & \lambda \\ & 1\end{smallmatrix}\right) \times \{\pm I\}</math>. In fact, they are all conjugate (in SL(2)) to one of the four matrices <math>\left(\begin{smallmatrix}1 & \pm 1 \\ & 1\end{smallmatrix}\right)</math>, <math>\left(\begin{smallmatrix}-1 & \pm 1 \\ & -1\end{smallmatrix}\right)</math> (in GL(2) or SL<sup>±</sup>(2), the ± can be omitted, but in SL(2) it cannot).

===Hyperbolic elements===
The [[eigenvalues]] for a hyperbolic element are both real, and are reciprocals.  Such an element acts as a [[squeeze mapping]] of the Euclidean plane, and the corresponding element of PSL(2,'''R''') acts as a [[Translation (geometry)|translation]] of the hyperbolic plane and as a [[Lorentz transformation|Lorentz boost]] on [[Minkowski space]].

Hyperbolic elements of the [[modular group]] act as [[Anosov diffeomorphism]]s of the torus.

Hyperbolic elements are conjugate into the 2 component group of standard squeezes × ±''I'': <math>\left(\begin{smallmatrix}\lambda \\ & \lambda^{-1}\end{smallmatrix}\right) \times \{\pm I\}</math>; 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]], 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<sup>±</sup>(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.

==Topology and universal cover==
As a [[topological space]], 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]] induced by the [[Symplectic manifold|symplectic structure]] 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]]s on the hyperbolic plane.

The fundamental group of SL(2,'''R''') is the infinite [[cyclic group]] '''Z'''.  The [[covering group#Universal covering group|universal covering group]], denoted <math>\overline{\mbox{SL}(2,\mathbf{R})}</math>, is an example of a finite-dimensional Lie group that is not a [[matrix group]].  That is, <math>\overline{\mbox{SL}(2,\mathbf{R})}</math> admits no [[faithful representation|faithful]], finite-dimensional [[group representation|representation]].

As a topological space, <math>\overline{\mbox{SL}(2,\mathbf{R})}</math> is a line bundle over the hyperbolic plane.  When imbued with a left-invariant [[Riemannian metric|metric]], the [[3-manifold]] <math>\overline{\mbox{SL}(2,\mathbf{R})}</math> becomes one of the [[Geometrization conjecture#The eight Thurston geometries|eight Thurston geometries]].  For example, <math>\overline{\mbox{SL}(2,\mathbf{R})}</math> is the universal cover of the unit tangent bundle to any [[Riemann surface|hyperbolic surface]].  Any manifold modeled on <math>\overline{\mbox{SL}(2,\mathbf{R})}</math> is orientable, and is a [[circle bundle]] over some 2-dimensional hyperbolic [[orbifold]] (a [[Seifert fiber space]]).

[[File:Braid-modular-group-cover.svg|thumb|376px|The [[braid group]] ''B''<sub>3</sub> is the [[universal central extension]] of the [[modular group]].]]
Under this covering, the preimage of the modular group PSL(2,'''Z''') is the [[braid group]] on 3 generators, ''B''<sub>3</sub>, which is the [[universal central extension]] 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]], thinking of SL(2,'''R''') as the symplectic group Sp(2,'''R''').

The aforementioned groups together form a sequence:
:<math>\overline{\mathrm{SL}(2,\mathbf{R})} \to \cdots \to \mathrm{Mp}(2,\mathbf{R})
\to \mathrm{SL}(2,\mathbf{R}) \to \mathrm{PSL}(2,\mathbf{R}).</math>
However, there are other covering groups of PSL(2,'''R''') corresponding to all ''n'', as ''n'' '''Z''' < '''Z''' ≅ π<sub>1</sub> (PSL(2,'''R''')), which form a [[lattice of covering groups]] by divisibility; these cover SL(2,'''R''') if and only if ''n'' is even.

==Algebraic structure==
The [[center (group theory)|center]] of SL(2,'''R''') is the two-element group {±1}, and the [[quotient]] PSL(2,'''R''') is [[simple group|simple]].

Discrete subgroups of PSL(2,'''R''') are called [[Fuchsian group]]s.  These are the hyperbolic analogue of the Euclidean [[wallpaper group]]s and [[Frieze group]]s.  The most famous of these is the [[modular group]] PSL(2,'''Z'''), which acts on a tessellation of the hyperbolic plane by ideal triangles.

The [[circle group]] [[orthogonal group|SO(2)]] is a [[maximal compact subgroup]] of SL(2,'''R'''), and the circle SO(2)/{±1} is a maximal compact subgroup of PSL(2,'''R''').

The [[Schur multiplier]] of the discrete group PSL(2,'''R''') is much larger than '''[[integer|Z]]''', and the universal [[Group extension#Central extension|central extension]] is much larger than the universal covering group. However these large central extensions do not take the topology into account and are somewhat pathological.

==Representation theory==
{{main article|Representation theory of SL2(R)}}
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]] of SL(2,'''R'''), denoted sl(2,'''R'''), is the algebra of all real, [[trace (linear algebra)|traceless]] 2 × 2 matrices.  It is the [[Bianchi classification|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 [[representation theory of the Lorentz group#Non-unitarity|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 [[Israel Gelfand|Gelfand]] and [[Mark Naimark|Naimark]] (1946), [[V. Bargmann]] (1947), and [[Harish-Chandra]] (1952).

==See also==
{{colbegin}}
* [[linear group]]
* [[special linear group]]
* [[projective linear group]]
* [[modular group]]
* [[SL(2,C)|SL(2,'''C''') (Möbius transformations)]]
* [[projective transformation]]
* [[Fuchsian group]]
* [[Table of Lie groups]]
* [[Anosov flow]]
{{colend}}

==References==
{{Reflist}}
* {{cite journal|first=Valentine|last=Bargmann|mr=0021942|title=Irreducible Unitary Representations of the Lorentz Group|journal=[[Annals of Mathematics]]| volume=48 |issue=3 |year=1947 |pages=568–640 |jstor=1969129 |doi=10.2307/1969129}}
* {{cite journal|last1=Gelfand | first1=Israel Moiseevich| last2=Neumark |first2=Mark Aronovich |title=Unitary representations of the Lorentz group | journal=Acad. Sci. USSR. J. Phys. | volume=10 | year=1946 | pages=93–94 | mr=0017282}}
* {{cite journal|author=Harish-Chandra|title=Plancherel formula for the 2×2 real unimodular group|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=38|year=1952|pages=337–342|mr=0047055|pmc=1063558|doi=10.1073/pnas.38.4.337|pmid=16589101}}
* {{cite book|first=Serge|last=Lang|title=<math>\mathrm{SL}_2(\mathbf{R})</math>|series=Graduate Texts in Mathematics | volume=105 | publisher=Springer-Verlag | location=New York | year=1985 | isbn=0-387-96198-4 | doi=10.1007/978-1-4612-5142-2 | mr=0803508}}
* {{cite book|first=William|last=Thurston|title=Three-dimensional geometry and topology. Vol. 1| mr=1435975 | others=Edited by Silvio Levy | series=Princeton Mathematical Series | volume=35 | publisher=Princeton University Press | location=Princeton, NJ | year=1997 | isbn=0-691-08304-5}}

[[Category:Group theory]]
[[Category:Lie groups]]
[[Category:Projective geometry]]
[[Category:Hyperbolic geometry]]