SL2(R)

In mathematics, the special linear group SL₂(R) is the group of all real 2 × 2 matrices with determinant one:

:

It is a real Lie group with important applications in geometry, topology, representation theory, and physics.

Closely related to SL₂(R) is the projective linear group PSL₂(R). This is the quotient of SL₂(R) obtained by identifying each element with its negative:

: $mbox\{PSL\}\_2(mathbb\{R\})\; =\; mbox\{SL\}\_2(mathbb\{R\})/\{-1,+1\; \}.$

Some authors denote this group by SL₂(R) instead. It is a simple Lie group, and it contains the modular group PSL₂(Z).

Descriptions

SL₂(R) is the group of all linear transformations of R² that preserve oriented area. It is isomorphic to the symplectic group Sp₂(R) and the generalized special unitary group SU(1,1). It is also isomorphic to the group of unit-length coquaternions.

The quotient PSL₂(R) has several interesting descriptions:
* It is the group of orientation-preserving projective transformations of the real projective line $mathbb\{R\}cup\{infty\}$.
* It is the group of conformal automorphisms of the unit disc.
* It is the group of orientation-preserving isometries of the hyperbolic plane.
* It is the restricted Lorentz group of three-dimensional Minkowski space. Equivalently, it is isomorphic to the indefinite orthogonal group SO⁺(1,2). It follows that SL₂(R) is isomorphic to the spin group Spin(2,1)⁺.

Linear fractional transformations

Elements of PSL₂(R) act on the real projective line $mathbb\{R\}cup\{infty\}$ as linear fractional transformations:

: $x\; mapsto\; frac\{ax+b\}\{cx+d\}.$

This is analogous to the action of PSL₂(C) on the Riemann sphere by Möbius transformations. It is the restriction of the action of PSL₂(R) on the hyperbolic plane to the boundary at infinity.

Möbius transformations

Elements of PSL₂(R) act on the complex plane by Möbius transformations:

: $z\; mapsto\; frac\{az+b\}\{cz+d\};;;;mbox\{\; (where\; \}a,b,c,dinmathbb\{R\}mbox\{)\}.$

This is precisely the set of Möbius transformations that preserve the upper half-plane. It follows that PSL₂(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 upper half-plane model of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the Poincaré disk model.

Adjoint representation

The group SL₂(R) acts on its Lie algebra sl₂(R) by conjugation, yielding a faithful 3-dimensional linear representation of PSL₂(R). This can alternatively be described as the action of PSL₂(R) on the space of quadratic forms on R². The result is the following representation:

:

The Killing form on sl₂(R) has signature (2,1), and induces an isomorphism between PSL₂(R) and the Lorentz group SO⁺(2,1). This action of PSL₂(R) on Minkowski space restricts to the isometric action of PSL₂(R) on the hyperboloid model of the hyperbolic plane.

Classification of elements

The eigenvalues of an element "A" ∈ SL₂(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:

* If | tr("A") | < 2, then "A" is called elliptic.

* If | tr("A") | = 2, then "A" is called parabolic.

* If | tr("A") | > 2, then "A" is called hyperbolic.

Elliptic elements

The eigenvalues for an elliptic element are both complex, and are conjugate values on the unit circle. Such an element acts as a rotation of the Euclidean plane, and the corresponding element of PSL₂(R) acts as a rotation of the hyperbolic plane and of Minkowski space.

Elliptic elements of the modular group must have eigenvalues { "&omega;", 1/"&omega;" }, where "&omega;" 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.

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 PSL₂(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.

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 PSL₂(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.

Topology and universal cover

As a topological space, PSL₂("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. SL₂("R") is a 2-fold cover of PSL₂("R"), and can be thought of as the bundle of spinors on the hyperbolic plane.

The fundamental group of SL₂("R") is the infinite cyclic group Z. The universal covering group, denoted $overline\{mbox\{SL\}\_2(mathbb\{R\})\}$, is an example of a finite-dimensional Lie group that is not a matrix group. That is, $overline\{mbox\{SL\}\_2(mathbb\{R\})\}$ admits no faithful, finite-dimensional representation.

As a topological space, $overline\{mbox\{SL\}\_2(mathbb\{R\})\}$ is a line bundle over the hyperbolic plane. When imbued with a left-invariant metric, the 3-manifold $overline\{mbox\{SL\}\_2(mathbb\{R\})\}$ becomes one of the eight Thurston geometries. For example, $overline\{mbox\{SL\}\_2(mathbb\{R\})\}$ is the universal cover of the unit tangent bundle to any hyperbolic surface. Any manifold modeled on $overline\{mbox\{SL\}\_2(mathbb\{R\})\}$ is orientable, and is a circle bundle over some 2-dimensional hyperbolic orbifold (a Seifert fiber space).

Algebraic structure

The center of SL₂(R) is the two-element group {-1,1}, and the quotient PSL₂(R) is simple.

Discrete subgroups of PSL₂(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 PSL₂(Z), which acts on a tesselation of the hyperbolic plane by ideal triangles.

The circle group SO(2) is a maximal compact subgroup of SL₂(R), and the circle SO(2)/{-1,+1} is a maximal compact subgroup of PSL₂(R).

The Schur multiplier of PSL₂(R) is Z, and the universal central extension is the same as the universal covering group.

Representation theory

SL₂(R) is a real, non-compact simple Lie group, and is the split-real form of the complex Lie group SL₂(C). The Lie algebra of SL₂(R), denoted sl₂(R), is the algebra of all real, traceless 2 &times; 2 matrices. It is the Bianchi algebra of type VIII.

The finite-dimensional representation theory of SL₂(R) is equivalent to the representation theory of SU(2), which is the compact real form of SL₂(C). In particular, SL₂(R) has no nontrivial finite-dimensional unitary representations.

The infinite-dimensional representation theory of SL₂(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 (1952).

ee also

* linear group
* special linear group
* projective linear group
* hyperbolic isometry
* modular group
* Möbius transformation
* projective transformation
* Fuchsian group
* Table of Lie groups

References

*V. Bargmann, [http://links.jstor.org/sici?sici=0003-486X%28194707%292%3A48%3A3%3C568%3AIUROTL%3E2.0.CO%3B2-Z, "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
* Harish-Chandra, "Plancherel formula for the 2&times;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. "Springer-Verlag, New York", 1985. ISBN 0-387-96198-4
* William Thurston. "Three-dimensional geometry and topology. Vol. 1". Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5