SL₂(R)

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	Subgroup; Normal subgroup; Quotient group; Group homomorphism; (semi-)direct product;
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	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); G2 F4 E6 E7 E8; Lorentz group; Poincaré group;
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In mathematics, the special linear group SL₂(R) is the group of all real 2 × 2 matrices with determinant one:

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

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 SL(2,R) instead. It is a simple Lie group, and it contains the modular group PSL₂(Z).

Also closely related is the 2-fold covering group, Mp₂(R), a metaplectic group (thinking of SL₂(R) as a symplectic group).

Another related group is $\mathrm{SL}_2^\pm(\mathbf{R})$ the group of 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group, however.

[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 group $\mathrm{SL}_2^\pm(\mathbf{R})$ preserves unoriented area: it may reverse orientation.

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)⁺.

Elements of the modular group PSL₂(Z) have additional interpretations, as do elements of the group SL₂(Z) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL₂(R).

[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, 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 ( $ε = | tr | / 2$ ; dividing by 2 corrects for the effect of dimension), then this yields: $ε < 1$ , elliptic; $ε = 1$ , parabolic; $ε > 1$ , hyperbolic.

The identity element I and negative identity element $- I$ , which correspond to the same element in PSL₂, have trace ±2, and hence by this classification are parabolic elements, though they are often considered separately.

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 subgroups (possibly times ±I), as detailed below.

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

[edit] 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 non-orthogonal basis – and the corresponding element of PSL₂(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, 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.

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 involutions in PSL₂.

Elliptic elements are conjugate into the subgroup of rotations of the Euclidean plane, the special orthogonal group $S O (2)$ ; 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 but not SL.)

[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.

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 SL₂) 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 GL₂ or $\mathrm{SL}_2^\pm$ , the ± can be omitted, but in SL₂ it cannot).

[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.

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°).

[edit] Conjugacy classes

By Jordan normal form, matrices are classified up to conjugacy (in GL_n(C)) by eigenvalues and nilpotence, meaning 1s in the Jordan blocks. Thus elements of SL₂ are classified up to conjugacy in GL₂ (or indeed $\mathrm{SL}_2^\pm$ ) by trace (since determinant is fixed, and trace and determinant determine eigenvalues), except if the eigenvalues are equal, so ±I and the trace +2 and trace $- 2$ parabolic elements are not conjugate (the former have no off-diagonal entries in Jordan form, while the latter do).

Up to conjugacy in SL₂, 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.

[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).

The braid group B₃ is the universal central extension of the modular group.

Under this covering, the preimage of the modular group $\mathrm{PSL}_2(\mathbb{R})$ is the braid group on 3 generators, B₃, 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 Mp₂(R), a metaplectic group, thinking of SL₂(R) as the symplectic group Sp₂(R).

The aforementioned groups together form a sequence:

$\overline{\mathrm{SL}_2(\mathbb{R})} \to \cdots \to \mathrm{Mp}_2(\mathbb{R}) \to \mathrm{SL}_2(\mathbb{R}) \to \mathrm{PSL}_2(\mathbb{R}).$

However, there are other covering groups of $\mathrm{PSL}_2(\mathbb{R})$ corresponding to all n, as $n\mathbb{Z} < \mathbb{Z} \cong \pi_1(\mathrm{PSL}_2(\mathbb{R})),$ which form a lattice by divisibility; these cover SL₂ if and only if n is even.

[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

SL₂(R)

From Wikipedia, the free encyclopedia

Contents

[edit] Descriptions

[edit] Linear fractional transformations

[edit] Möbius transformations

[edit] Adjoint representation

[edit] Classification of elements

[edit] Elliptic elements

[edit] Parabolic elements

[edit] Hyperbolic elements

[edit] Conjugacy classes

[edit] Topology and universal cover

[edit] Algebraic structure

[edit] Representation theory

[edit] See also

[edit] References

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