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:

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.

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 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.