SL2(R)

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

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

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.