In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:
SL ( 2 , R ) = { ( a b c d ) : a , b , c , d ∈ R and a d − b c = 1 } . {\displaystyle {\mbox{SL}}(2,\mathbf {R} )=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\colon a,b,c,d\in \mathbf {R} {\mbox{ and }}ad-bc=1\right\}.}
It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics. SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically,
PSL(2, R) = SL(2, R) / {±I}, where I denotes the 2 × 2 identity matrix. It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL±(2, R), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group, however.
Descriptions SL(2, R) is the group of all linear transformations of R2 that preserve oriented area. It is isomorphic to the symplectic group Sp(2, R) and the special unitary group SU(1, 1). It is also isomorphic to the group of unit-length coquaternions. The group SL±(2, R) preserves unoriented area: it may reverse orientation. The quotient PSL(2, R) has several interesting descriptions, up to Lie group isomorphism:
It is the group of orientation-preserving projective transformations of the real projective line R ∪ {∞}. 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 SL(2, R) is isomorphic to the spin group Spin(2,1)+. Elements of the modular group PSL(2, Z) have additional interpretations, as do elements of the group SL(2, Z) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL(2, R).
Homographies Elements of PSL(2, R) are homographies on the real projective line R ∪ {∞}:
[ x , 1 ] ↦ [ x , 1 ] ( a c b d ) = [ a x + b , c x + d ] = [ a x + b c x + d , 1 ] . {\displaystyle [x,1]\mapsto [x,\ 1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ [ax+b,\ cx+d]\ =\,\left[{\frac {ax+b}{cx+d}},\ 1\right].}
These projective transformations form a subgroup of PSL(2, C), which acts on the Riemann sphere by Möbius transformations. When the real line is considered the boundary of the hyperbolic plane, PSL(2, R) expresses hyperbolic motions.
Möbius transformations Elements of PSL(2, R) act on the complex plane by Möbius transformations:
z ↦ a z + b c z + d (where a , b , c , d ∈ R ) . {\displaystyle z\mapsto {\frac {az+b}{cz+d}}\;\;\;\;{\mbox{ (where }}a,b,c,d\in \mathbf {R} {\mbox{)}}.}
This is precisely the set of Möbius transformations that preserve the upper half-plane. It follows that PSL(2, R) is the group of conformal automorphisms of the upper half-plane. By the Riemann mapping theorem, it is also isomorphic to 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. The above formula can be also used to define Möbius transformations of dual and double (aka split-complex) numbers. The corresponding geometries are in non-trivial relations to Lobachevskian geometry.
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