SL2(R)
In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:
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 group action factors through the quotient PSL(2,R) (the 2 × 2 projective special linear group over R). More specifically,
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.
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 generalized 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.
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