SL(2,R)

The special linear group SL (2, R) or SL2 (R ) is the group of real 2 × 2 matrices with determinant 1:

It is a Lie group with a variety of applications in geometry, topology, representation theory, harmonic analysis, number theory, modular forms and physics.

Representation theory

For every natural number d there is one irreducible up to isomorphism unique, ( d 1) - dimensional representation of SL (2, R). An explicit implementation of this irreducible representation is as follows. Be

The vector space of homogeneous polynomials of degree d in two variables. This vector space is ( d 1) -dimensional and acts by

The infinite-dimensional representations of are described by the Langlands classification.

Lie algebra

Is a Lie group, its Lie algebra is the Lie algebra of traceless matrices

A vector space based on the 3 - dimensional vector space, for example,

With the commutator relations

This Lie algebra is simple, it has two non- conjugate Cartan subalgebras -: one generated by the other from.

The Killing form is. It is negative definite on the subspace generated by positive definite on the subspace generated by and.

Linear Algebra

Matrices of corresponding invertible linear transformations of the vector space. The matrix acts by

For volume matrices of the form, but generally not the Euclidean metric.

Classification of the 2 × 2 matrices

The eigenvalues ​​of a matrix A ∈ SL (2, R) are the zeros of the characteristic polynomial

And can be calculated by the solution formula for quadratic equations as

It then classifies the matrices according to the following classification:

  • If | tr (A) | <2, then A is an elliptic matrix.
  • If | tr (A) | = 2, then A is a parabolic matrix.
  • If | tr (A) | > 2, then A is a hyperbolic matrix.

Elliptic elements

Elliptical elements are of the form

With and.

The matrix acts on the Euclidean plane as a rotation with fixed point 0 and angle of rotation.

Parabolic elements

Parabolic elements are of the form

With and.

The matrix acts on the Euclidean plane as shear.

Hyperbolic elements

Hyperbolic elements are of the form

With and.

The matrix acts as Dehnstauchung, that is, it expands in the direction of an eigenvector, shrinks toward the other eigenvector, will receive a total but the surface area.

Hyperbolic geometry

Matrices from acting on the upper half plane

By

They act as isometries of the hyperbolic metric.

Because ± I acts as an identity mapping, factored this effect of SL (2, R)

Projective geometry and fractional- linear transformations

The projective line is the set of all lines through the origin in. The effect of on is on a well - defined effect of.

Through a bijection is defined between and. After this identification of and acts on broken - linear transformations

The Veronese embedding is equivariant with respect to the irreducible representation.

Also the rim at infinity, the hyperbolic plane. The effect of on the compactification of the hyperbolic plane by fractional- linear transformations is continuous. Elliptic elements have a fixed point in parabolic elements have a fixed point in, hyperbolic elements have two fixed points in.

Fuchsian groups

Discrete subgroups of are called Fuchsian groups.

The limit set of a Fuchsian group Γ is the average of the completion of a web, and the definition of the limit set is independent of the chosen point.

A Fuchsian group is called Fuchsian group first kind if the limit set is all about. Otherwise, it is a Fuchsian group 2 Article

Fuchsian groups first type are the so-called lattice in, ie discrete subgroups for which there is a fundamental domain finite volume in the hyperbolic plane.

An example of a lattice in the modular group in the theory of modular forms play a central role with many number-theoretic applications, among others.

If a Fuchsian group contains no elements of order 2, then it is the projection of a discrete subgroup of. ( Set of Culler )

Topology

The circle group is a maximal compact subgroup of. The sub-group is a deformation retract of, in particular, the two spaces are homotopy equivalent.

The fundamental group of is isomorphic to the higher homotopy groups are trivial.

The universal cover of is an example of a Lie group has no faithful finite- dimensional representation, so no subgroup of a general linear group is isomorphic.

The quotient is diffeomorphic to Einheitstangentialbündel the hyperbolic plane.