sl2 or sl3

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\{\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 connected noncompact 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 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.

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