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:

Algebraic structure → Group theory
Group theory
Basic notions
Group homomorphisms
Finite groups
Classification of finite simple groups
Modular groups
  • PSL(2, )
  • SL(2, )
Topological and Lie groups
Infinite dimensional Lie group
  • O(∞)
  • SU(∞)
  • Sp(∞)
Algebraic groups

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.

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