SL2(R)

SL2(R)

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

$\mbox{SL}(2,\mathbf{R}) = \left\{ \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) : a,b,c,d\in\mathbf{R}\mbox{ and }ad-bc=1\right\}.$

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

SL(2,R) acts on the complex upper half-plane by fractional linear transformations. The 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.