SL2(R)

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The correct title of this article is SL₂(R). It features superscript or subscript characters that are substituted or omitted because of technical limitations.

Groups

Group theory

Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product direct sum

Discrete groups
Classification of finite simple groups Cyclic group Z_n Alternating group A_n Sporadic groups Mathieu group M_11..12,M_22..24 Conway group Co_1..3 Janko group J_1..4 Fischer group F_22..24 Baby Monster group B Monster group M

Continuous groups
Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) E₆ E₇ E₈ F₄ G₂ SL₂ Lorentz group Poincaré group

Infinite dimensional groups
conformal group diffeomorphism group Quantum group O(∞) SU(∞) Sp(∞)

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

$\mbox{SL}_2(\mathbb{R}) = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a,b,c,d\in\mathbb{R}\mbox{ and }ad-bc=1\right\}.$

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).

1 Descriptions
2 Classification of elements
3 Topology and universal cover
4 Algebraic structure
5 Representation theory
6 See also
7 References

[edit] 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)⁺.

[edit] 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.

[edit] Möbius transformations

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

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

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.

[edit] 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:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto \begin{bmatrix} a^2 & 2ac & c^2 \\ ab & ad+bc & cd \\ b^2 & 2bd & d^2 \end{bmatrix}.$

The Killing form on 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.

[edit] 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.

[edit] 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 { ω, 1/ω }, 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.

[edit] 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.

[edit] 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.

[edit] 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).

[edit] 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.

[edit] Representation theory

Main article: Representation theory of SL2(R)

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 × 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).

[edit] See also

[edit] 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
Harish-Chandra, 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. 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