SL (2, R)

SL (2, R) or SL ₂ (R) is a group of real 2 × 2 matrices with unit determinant :

{\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\}.

{\ displaystyle {\ 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 \}.}

{\ displaystyle {\ 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 \}.}

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

SL (2, R ) acts on the linear fractional transformations. The action of the group is factorized onto the factor group PSL (2, R) ( $2\times 2$ ${\ displaystyle 2 \ times 2}$ ${\ displaystyle 2 \ times 2}$ projective special linear group over R ). More precisely,

\mathrm {PSL} (2,\mathbf {R} )=\mathrm {SL} (2,\mathbf {R} )/\{{\pm }E\}

{\ displaystyle \ mathrm {PSL} (2, \ mathbf {R}) = \ mathrm {SL} (2, \ mathbf {R}) / \ {{\ pm} E \}}

{\ displaystyle \ mathrm {PSL} (2, \ mathbf {R}) = \ mathrm {SL} (2, \ mathbf {R}) / \ {{\ pm} E \}}

,

where E is $2\times 2$ ${\ displaystyle 2 \ times 2}$ ${\ displaystyle 2 \ times 2}$ unit matrix . SL (2, R ) contains the modular group PSL (2, Z ).

Also, the group SL (2, R ) is closely connected with the 2-fold Mp (2, R ), the (if we consider SL (2, R ) as a symplectic group ).

Another related group is $\mathrm {SL} ^{\pm }(2,\mathbf {R} )$ ${\ displaystyle \ mathrm {SL} ^ {\ pm} (2, \ mathbf {R})}$ ${\ displaystyle \ mathrm {SL} ^ {\ pm} (2, \ mathbf {R})}$ group of real $2\times 2$ ${\ displaystyle 2 \ times 2}$ ${\ displaystyle 2 \ times 2}$ matrices with determinant $\pm 1$ ${\ displaystyle \ pm 1}$ $\ pm 1$ . However, this group is most often used in the context of a modular group .

Description

SL (2, R ) is the group of all linear transformations of the space R ² that preserve the oriented area . The group is isomorphic to the symplectic group Sp (2, R ) and the generalized special unitary group SU (1,1). The group is also isomorphic to the group of unit length . Group $\mathrm {SL} ^{\pm }(2,\mathbf {R} )$ ${\ displaystyle \ mathrm {SL} ^ {\ pm} (2, \ mathbf {R})}$ $\mathrm {SL} ^{\pm }(2,\mathbf {R} )$ maintains a non-oriented area - it can maintain orientation.

The PSL (2, R ) factor has several interesting descriptions:

This is a group of orientation- preserving projective transformations of a $\mathbf {R} \cup \{\infty \}$ ${\ displaystyle \ mathbf {R} \ cup \ {\ infty \}}$ $\mathbf {R} \cup \{\infty \}$ .
This is the group of conformal automorphisms of the unit circle .
This is a group of orientation- preserving movements of a hyperbolic plane .
This is a limited Lorentz group of the three- dimensional Minkowski space . Equivalently, it is isomorphic to the indefinite orthogonal group SO ⁺ (1,2). It follows that SL (2, R ) is isomorphic to the spinor group Spin (2,1) ⁺ .

Elements of the modular group PSL (2, Z ) have additional interpretations as elements of the group SL (2, Z ) (as linear transformations of the torus), and these representations can also be considered in the light of the general theory of the group SL (2, R ).

Linear fractional

Elements of the group PSL (2, R ) act on the $\mathbf {R} \cup \{\infty \}$ ${\ displaystyle \ mathbf {R} \ cup \ {\ infty \}}$ as linear fractional transformations :

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

{\ displaystyle x \ mapsto {\ frac {ax + b} {cx + d}}.}

This action is similar to the action of PSL (2, C ) on the Riemann sphere by Mobius transformations . The action is a restriction of the action of the group PSL (2, R ) on the hyperbolic plane at the boundary of infinity.

Mobius Transformation

Elements of the group PSL (2, R ) act on the complex plane by the Möbius transformation:

z\mapsto {\frac {az+b}{cz+d}}\;\;\;\;(a,b,c,d\in \mathbf {R} )

{\ displaystyle z \ mapsto {\ frac {az + b} {cz + d}} \; \; \; \; (a, b, c, d \ in \ mathbf {R})}

.

This is exactly the set of Mobius transformations that preserve the upper half of the plane . It follows that PSL (2, R ) is a group of conformal automorphisms of the upper half of the plane. By the Riemann mapping theorem, this group is a group of conformal automorphisms of the unit disk.

These Mobius transformations act as isometries of the model of the upper half of the plane of the hyperbolic space, and the corresponding Mobius transformations of the disk are hyperbolic isometries of the Poincare disk model .

The formula above can also be used to determine the Möbius transform of dual and double numbers . The corresponding geometries are in nontrivial connection ^[1] with the Lobachevsky geometry .

Attached View

The group SL (2, R ) acts on its Lie algebras sl (2, R ) by conjugation (recall that the elements of the Lie algebra are also 2 x 2 matrices), giving a strict 3-dimensional linear representation of the group PSL (2, R ). This can alternatively be described as the action of the group PSL (2, R ) on the surface of quadratic forms on R ² . The result is the following view:

{\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}}.

{\ displaystyle {\ 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}}.}

The Killing form on sl (2, R ) has signature (2,1) and generates an isomorphism between PSL (2, R ) and the Lorentz group SO ⁺ (2,1). This action of the group PSL (2, R ) in Minkowski space is limited to the isometric action of the group PSL (2, R ) on the hyperboloid model of the hyperbolic plane.

Classification of Elements

Element eigenvalues $A\in \mathrm {SL} (2,\mathbf {R} )$ ${\ displaystyle A \ in \ mathrm {SL} (2, \ mathbf {R})}$ satisfy the equation for the characteristic polynomial

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

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

But because

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

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

This leads to the following classification of elements with the corresponding action on the Euclidean plane:

If | tr ( A ) | <2, then the element (matrix) A is called elliptic and it is a rotation .
If | tr ( A ) | = 2, then the element A is called parabolic and it is an extension of space.
If | tr ( A ) |> 2, then the element A is called hyperbolic and is .

The names correspond to the classification of conical sections by eccentricity - if you define the eccentricity as half the value of the trace ( $\epsilon ={\tfrac {1}{2}}tr$ ${\ displaystyle \ epsilon = {\ tfrac {1} {2}} tr}$ . Dividing by 2 corrects the dimension effect, while the absolute value corresponds to ignoring the sign (factor $\pm 1$ ${\ displaystyle \ pm 1}$ ) when we work with PSL (2, R )), whence it follows: $\epsilon <1$ ${\ displaystyle \ epsilon <1}$ for an elliptical element, $\epsilon =1$ ${\ displaystyle \ epsilon = 1}$ for parabolic, $\epsilon >1$ ${\ displaystyle \ epsilon> 1}$ for hyperbolic.

The unit element 1 and the negative element −1 (they coincide in PSL (2, R )) have the following $\pm 2$ ${\ displaystyle \ pm 2}$ , and therefore, according to this classification, they are parabolic elements, although they are often considered separately.

The same classification is used for SL (2, C ) and PSL (2, C ) ( Mobius transformations ) and PSL (2, R ) (real Mobius transformations) with the addition of “loxodromic” transformations corresponding to complex traces. Similar classifications are used in many other places.

A subgroup containing elliptic (respectively, parabolic and hyperbolic) elements, plus a unit element and negative for it, is called an elliptic subgroup (respectively, a , a hyperbolic subgroup ).

This classification is by subsets , not by subgroups - these sets are not closed by multiplication (the product of two parabolic elements will not necessarily be parabolic, for example). However, all elements are combined into 3 standard , as described below.

Topologically, since the trace is a continuous mapping, elliptic elements (without $\pm 1$ ${\ displaystyle \ pm 1}$ ) are an open set , as well as hyperbolic elements (without $\pm 1$ ${\ displaystyle \ pm 1}$ ), while parabolic elements (including $\pm 1$ ${\ displaystyle \ pm 1}$ ) are a closed set .

Elliptical Elements

The eigenvalues for an elliptic element are both complex and are conjugate values on the unit circle . Such an element is conjugated with a rotation of the Euclidean plane - they can be interpreted as rotations in a (possibly) non-orthogonal basis and the corresponding element of the group PSL (2, R ) acts as a (conjugated) rotation of the hyperbolic plane and Minkowski space .

Elliptic elements of the modular group must have eigenvalues $\{\omega ,\omega ^{-1}\}$ ${\ displaystyle \ {\ omega, \ omega ^ {- 1} \}}$ where $\omega$ ${\ displaystyle \ omega}$ is a primitive 3rd, 4th or 6th root of unity . They are all elements of a modular group with finite order and they act on the torus as periodic diffeomorphisms.

Elements with trace 0 can be called “circular elements” (similar to eccentricity), but this is rarely used. These traces correspond to elements with eigenvalues. $\pm i$ ${\ displaystyle \ pm i}$ and correspond to rotations on $90^{\circ }$ ${\ displaystyle 90 ^ {\ circ}}$ , and the square corresponds to - E - they are non-identical involutions in PSL (2).

Elliptic elements are conjugated inside a subgroup of rotations of the Euclidean plane, the orthogonal group SO (2). The rotation angle is equal to the arccos half of the trace with the rotation sign (rotation and its inverse are conjugated in GL (2), but not in SL (2).)

Parabolic Elements

A parabolic element has only one eigenvalue, which is either 1 or −1. Such an element acts as an extension of space on the Euclidean plane, and the corresponding element of the group PSL (2, R ) acts as a restriction of rotation of the hyperbolic plane and as zero rotation of the Minkowski space .

The parabolic elements of the modular group act as twists of the torus.

Parabolic elements are conjugated in a 2-component group of standard shifts $\times {\pm }E$ ${\ displaystyle \ times {\ pm} E}$ : $\left({\begin{smallmatrix}1&\lambda \\&1\end{smallmatrix}}\right)\times \{\pm E\}$ ${\ displaystyle \ left ({\ begin {smallmatrix} 1 & \ lambda \\ & 1 \ end {smallmatrix}} \ right) \ times \ {\ pm E \}}$ . In fact, they are all conjugate (in SL (2)) to one of the four matrices $\left({\begin{smallmatrix}1&\pm 1\\&1\end{smallmatrix}}\right)$ ${\ displaystyle \ left ({\ begin {smallmatrix} 1 & \ pm 1 \\ & 1 \ end {smallmatrix}} \ right)}$ , $\left({\begin{smallmatrix}-1&\pm 1\\&-1\end{smallmatrix}}\right)$ ${\ displaystyle \ left ({\ begin {smallmatrix} -1 & \ pm 1 \\ & - 1 \ end {smallmatrix}} \ right)}$ (in GL (2) or $\mathrm {SL} ^{\pm }(2)$ ${\ displaystyle \ mathrm {SL} ^ {\ pm} (2)}$ , $\pm$ ${\ displaystyle \ pm}$ can be omitted, but cannot be omitted in SL (2)).

Hyperbolic Elements

The eigenvalues for a hyperbolic element are real and opposite. Such an element acts as Euclidean plane, and the corresponding element PSL (2, R ) acts as a parallel transfer of the hyperbolic plane and as a Lorentz boost in Minkowski space .

The hyperbolic elements of the modular group act as diffeomorphisms of the Anosov torus.

Hyperbolic elements fall into a 2-component group of standard compressions $\times {\pm }E$ ${\ displaystyle \ times {\ pm} E}$ : $\left({\begin{smallmatrix}\lambda \\&\lambda ^{-1}\end{smallmatrix}}\right)\times \{\pm E\}$ ${\ displaystyle \ left ({\ begin {smallmatrix} \ lambda \\ & \ lambda ^ {- 1} \ end {smallmatrix}} \ right) \ times \ {\ pm E \}}$ ; the hyperbolic angle of the hyperbolic rotation is defined as an arcosh of half the trace, but the sign can be either positive or negative, in contrast to the elliptical case. Compression and its inverse transformation are conjugated in SL₂ (by rotation in the axes, for standard axes, rotation is carried out on $90^{\circ }$ ${\ displaystyle 90 ^ {\ circ}}$ )

Conjugation classes

By the Jordan normal form, matrices are classified up to conjugation (in GL ( n , C )) by eigenvalues and nilpotency (specifically, nilpotency means where the units are in the Jordan cells). Such elements of the group SL (2) are classified up to conjugacy in GL (2) ( $\mathrm {SL} ^{\pm }(2)$ ${\ displaystyle \ mathrm {SL} ^ {\ pm} (2)}$ ) on the trace (since the determinant is fixed, and the trace and determinant are determined by eigenvalues), unless the eigenvalues are equal, so the elements are equal $\pm E$ ${\ displaystyle \ pm E}$ and the parabolic elements of trace +2 and trace −2 are not conjugate (the first has no off-diagonal elements in the form of Jordan, while the second has).

Up to conjugation in SL (2) (instead of GL (2)), there is additional information corresponding to the orientation - (elliptical) clockwise and counterclockwise rotations are not conjugated, are not a positive or negative shift, as described above. Then, for an absolute value of a trace less than 2, there are two conjugate classes for each trace (clockwise or counterclockwise rotation). For an absolute trace value of 2, there are three conjugate classes for each trace (positive shift, zero shift, negative shift). For the absolute value of a trace greater than 2, there is one conjugacy class for a given trace.

Topological and universal cover

As a topological space , PSL (2, R ) can be described as the hyperbolic plane. It is a fibration on a circle and has a natural contact structure generated by a symplectic structure on the hyperbolic plane. The group SL (2, R ) is a 2-fold covering of the group PSL (2, R ) and it can be considered as a bundle of spinors on the hyperbolic plane.

The fundamental group of the group SL (2, R ) is a finite cyclic group Z. , denoted by ${\overline {{\mbox{SL}}(2,\mathbf {R} )}}$ ${\ displaystyle {\ overline {{\ mbox {SL}} (2, \ mathbf {R})}}}$ , is an example of a finite-dimensional Lie group that is not a . I.e ${\overline {{\mbox{SL}}(2,\mathbf {R} )}}$ ${\ displaystyle {\ overline {{\ mbox {SL}} (2, \ mathbf {R})}}}$ does not allow an finite-dimensional representation .

Like a topological space ${\overline {{\mbox{SL}}(2,\mathbf {R} )}}$ ${\ displaystyle {\ overline {{\ mbox {SL}} (2, \ mathbf {R})}}}$ is a line bundle over a hyperbolic plane. If a space is endowed with a left-invariant metric , a ${\overline {{\mbox{SL}}(2,\mathbf {R} )}}$ ${\ displaystyle {\ overline {{\ mbox {SL}} (2, \ mathbf {R})}}}$ becomes one of Thurston 's eight geometries . For example, ${\overline {{\mbox{SL}}(2,\mathbf {R} )}}$ ${\ displaystyle {\ overline {{\ mbox {SL}} (2, \ mathbf {R})}}}$ is a universal covering of the unit tangent bundle for any hyperbolic surface . Any variety modeled on ${\overline {{\mbox{SL}}(2,\mathbf {R} )}}$ ${\ displaystyle {\ overline {{\ mbox {SL}} (2, \ mathbf {R})}}}$ , is orientable and is a bundle on a circle over some two-dimensional hyperbolic orbifold ( Seifert bundle ).

The braid group B ₃ is a universal central extension of the modular group .

Given this covering, the inverse image of the modular group PSL (2, Z ) is a braid group on 3 generators, B ₃ , which is a universal central extension of the modular group. They are lattices inside the corresponding algebraic groups and this corresponds to an algebraically universal covering group in topology.

The 2-fold covering group can be called Mp (2, R ), the , if we understand SL (2, R ) as the symplectic group Sp (2, R ).

The above groups form the 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} ).

{\ displaystyle {\ 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 groups covering the group PSL (2, R ) corresponding to all n , such that $n\mathbf {Z} <\mathbf {Z} \cong \pi _{1}(\mathrm {PSL} (2,\mathbf {R} ))$ ${\ displaystyle n \ mathbf {Z} <\ mathbf {Z} \ cong \ pi _ {1} (\ mathrm {PSL} (2, \ mathbf {R}))}$ so that they form a . They are a covering of SL (2, R ) if and only if n is even.

Algebraic structure

The center of the group SL (2, R ) is a two-element group $\{\pm 1\}$ ${\ displaystyle \ {\ pm 1 \}}$ and the PSL factor (2, R ) is a simple group.

Discrete subgroups of the group PSL (2, R ) are called Fuchsian groups . They are a hyperbolic analogue of Euclidean wallpaper groups and border groups . The most famous of them is the modular group PSL (2, Z ), which acts on the tiling of the hyperbolic plane by ideal triangles .

The group U (1) , which can be regarded as SO (2) , is a maximal compact subgroup of the group SL (2, R ) and the circle $\mathrm {SO} (2)/\{\pm 1\}$ ${\ displaystyle \ mathrm {SO} (2) / \ {\ pm 1 \}}$ is a maximal compact subgroup of the group PSL (2, R ).

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

Representation Theory

SL (2, R ) is a real noncompact simple Lie group and is a split real form of the complex Lie group SL (2, C ). The Lie algebra of the group SL (2, R ), denoted by sl (2, R ), is the algebra of all real, traceless ^[2] $2\times 2$ ${\ displaystyle 2 \ times 2}$ matrices. This is type VIII.

The finite-dimensional theory of representations of the group SL (2, R ) is equivalent to the , which is a compact real form of the group SL (2, C ). In particular, SL (2, R ) does not have non-trivial finite-dimensional unitary representations. This is a property of any connected simple noncompact Lie group. For a sketch of the proof, see the article .

The infinite-dimensional representation theory of the group SL (2, R ) is very interesting. The group has several families of unitary representations, which were developed in detail by Gelfand and Naimark (1946), V. Bargman (1947) and Harish-Chandra (1952).

Notes

↑ Kisil, 2012 , p. xiv + 192.
↑ A traceless matrix is a matrix whose trace is 0.

Literature

Valentine Bargmann. Irreducible Unitary Representations of the Lorentz Group // Annals of Mathematics . - 1947. - T. 48 , no. 3 . - S. 568-640 . - DOI : 10.2307 / 1969129 .
Gelfand I.M., Naimark M.A. Unitary representations of the Lorentz group // Izv. USSR Academy of Sciences. Ser. Mat .. - 1947. - T. 11 , no. 5 . - S. 411-504 .
Harish-chandra. Plancherel formula for the $2\times 2$ ${\ displaystyle 2 \ times 2}$ real unimodular group // Proc. Natl. Acad. Sci. USA - 1952.- T. 38 . - S. 337–342 . - DOI : 10.1073 / pnas . 38.4.337 . - PMID 16589101 .
Serge Lang. $S$ $L$ $2$ $($ $R$ $)$ ${\ displaystyle \ mathrm {SL} _ {2} (\ mathbf {R})}$ . - New York: Springer-Verlag, 1985.- T. 105. - ISBN 0-387-96198-4 . - DOI : 10.1007 / 978-1-4612-5142-2 .
- Leng S. $SL_{2}(R)$ ${\ displaystyle SL_ {2} (R)}$ / Translation from English V.I. Vasyunina and M.A. Semenov-Tian-Shansky; Edited by A.A. Kirillova. - Moscow: Mir, 1977.
William Thurston Three-dimensional geometry and topology. Vol. 1.- Princeton, NJ: Princeton University Press, 1997.- T. 35.- ISBN 0-691-08304-5 .
Vladimir V. Kisil. Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL (2, R). - London: Imperial College Press, 2012 .-- ISBN 978-1-84816-858-9 . - DOI : 10.1142 / p835 .

[_114aa42ac9999b3c-1] Kisil, 2012 , p. xiv + 192.

[2] A traceless matrix is a matrix whose trace is 0.