In mathematics, the special linear group SL(2,R) or SL_{2}(R) is the group of all real 2 × 2 matrices with determinant one:

\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 }adbc=1\right\}.
It is a simple real Lie group with applications in geometry, topology, representation theory, and physics.
SL(2,R) acts on the complex upper halfplane by fractional linear transformations. The group action factors through the quotient PSL(2,R) (the 2 × 2 projective special linear group over R). More specifically,

PSL(2,R) = SL(2,R)/{±I},
where I denotes the 2 × 2 identity matrix. It contains the modular group PSL(2,Z).
Also closely related is the 2fold covering group, Mp(2,R), a metaplectic group (thinking of SL(2,R) as a symplectic group).
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, however.
Contents

Descriptions 1

Linear fractional transformations 1.1

Möbius transformations 1.2

Adjoint representation 1.3

Classification of elements 2

Elliptic elements 2.1

Parabolic elements 2.2

Hyperbolic elements 2.3

Conjugacy classes 2.4

Topology and universal cover 3

Algebraic structure 4

Representation theory 5

See also 6

References 7
Descriptions
SL(2,R) is the group of all linear transformations of R^{2} that preserve oriented area. It is isomorphic to the symplectic group Sp(2,R) and the generalized special unitary group SU(1,1). It is also isomorphic to the group of unitlength coquaternions. The group SL^{±}(2,R) preserves unoriented area: it may reverse orientation.
The quotient PSL(2,R) has several interesting descriptions:
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 R∪{∞} as linear fractional transformations:

x \mapsto \frac{ax+b}{cx+d}.
This is analogous to the action of PSL(2,C) on the Riemann sphere by Möbius transformations. 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:

z \mapsto \frac{az+b}{cz+d}\;\;\;\;\mbox{ (where }a,b,c,d\in\mathbf{R}\mbox{)}.
This is precisely the set of Möbius transformations that preserve the upper halfplane. It follows that PSL(2,R) is the group of conformal automorphisms of the upper halfplane. 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 upper halfplane 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 and double (aka splitcomplex) numbers. The corresponding geometries are in nontrivial relations^{[1]} to Lobachevskian geometry.
Adjoint representation
The group SL(2,R) acts on its Lie algebra sl(2,R) by conjugation (remember that the Lie algebra elements are also 2 by 2 matrices), yielding a faithful 3dimensional linear 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^{2}. The result is the following representation:

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto \begin{bmatrix} a^2 & 2ac & c^2 \\ ab & ad+bc & cd \\ b^2 & 2bd & d^2 \end{bmatrix}.
The Killing form on sl(2,R) has signature (2,1), and induces an isomorphism between PSL(2,R) and the Lorentz group SO^{+}(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 eigenvalues of an element A ∈ SL(2,R) satisfy the characteristic polynomial

\lambda^2 \,\, \mathrm{tr}(A)\,\lambda \,+\, 1 \,=\, 0
and therefore

\lambda = \frac{\mathrm{tr}(A) \pm \sqrt{\mathrm{tr}(A)^2  4}}{2}.
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.

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 sections by 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: \epsilon < 1, elliptic; \epsilon = 1, parabolic; \epsilon > 1, 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 transformations) 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).
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 oneparameter subgroups (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 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 nonorthogonal 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^{−1}}, where ω is a primitive 3rd, 4th, or 6th root of unity. These are all the elements of the modular group with finite 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 nonidentity involutions 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 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 null rotation of Minkowski space.
Parabolic elements of the modular group act as Dehn twists of the torus.
Parabolic elements are conjugate into the 2 component group of standard shears × ±I: \left(\begin{smallmatrix}1 & \lambda \\ & 1\end{smallmatrix}\right) \times \{\pm I\}. In fact, they are all conjugate (in SL(2)) to one of the four matrices \left(\begin{smallmatrix}1 & \pm 1 \\ & 1\end{smallmatrix}\right), \left(\begin{smallmatrix}1 & \pm 1 \\ & 1\end{smallmatrix}\right) (in GL(2) or SL^{±}(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 of the hyperbolic plane and as a Lorentz boost on Minkowski space.
Hyperbolic elements of the modular group act as Anosov diffeomorphisms of the torus.
Hyperbolic elements are conjugate into the 2 component group of standard squeezes × ±I: \left(\begin{smallmatrix}\lambda \\ & \lambda^{1}\end{smallmatrix}\right) \times \{\pm 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, 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 offdiagonal 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 the negatives of these), 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 structure on the hyperbolic plane. SL(2,R) is a 2fold cover of PSL(2,R), and can be thought of as the bundle of spinors on the hyperbolic plane.
The fundamental group of SL(2,R) is the infinite cyclic group Z. The universal covering group, denoted \overline{\mbox{SL}(2,\mathbf{R})}, is an example of a finitedimensional Lie group that is not a matrix group. That is, \overline{\mbox{SL}(2,\mathbf{R})} admits no faithful, finitedimensional representation.
As a topological space, \overline{\mbox{SL}(2,\mathbf{R})} is a line bundle over the hyperbolic plane. When imbued with a leftinvariant metric, the 3manifold \overline{\mbox{SL}(2,\mathbf{R})} becomes one of the eight Thurston geometries. For example, \overline{\mbox{SL}(2,\mathbf{R})} is the universal cover of the unit tangent bundle to any hyperbolic surface. Any manifold modeled on \overline{\mbox{SL}(2,\mathbf{R})} is orientable, and is a circle bundle over some 2dimensional hyperbolic orbifold (a Seifert fiber space).
Under this covering, the preimage of the modular group PSL(2,Z) is the braid group on 3 generators, B_{3}, 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 2fold 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:

\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}).
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 n is even.
Algebraic structure
The center (group theory) of SL(2,R) is the twoelement group {±1}, and the quotient PSL(2,R) is simple.
Discrete subgroups of PSL(2,R) are called Fuchsian groups. These are the hyperbolic analogue of the Euclidean wallpaper groups and Frieze groups. 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 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 Z, and the universal 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
SL(2,R) is a real, noncompact simple Lie group, and is the splitreal 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, traceless 2 × 2 matrices. It is the Bianchi algebra of type VIII.
The finitedimensional 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 finitedimensional unitary representations.
The infinitedimensional 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 HarishChandra (1952).
See also
References

V. Bargmann, Irreducible Unitary Representations of the Lorentz Group, The Annals of Mathematics, 2nd Ser., Vol. 48, No. 3 (Jul., 1947), pp. 568–640

Gelfand, I.; Neumark, M. Unitary representations of the Lorentz group. Acad. Sci. USSR. J. Phys. 10, (1946), pp. 93–94

HarishChandra, Plancherel formula for the 2×2 real unimodular group. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), pp. 337–342

Serge Lang, SL2(R). Graduate Texts in Mathematics, 105. SpringerVerlag, New York, 1985. ISBN 0387961984

William Thurston. Threedimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0691083045

^ Kisil, Vladimir V. (2012). Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R). London: Imperial College Press. p. xiv+192.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.