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: 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 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, 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 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. The quotient PSL(2,R) has several interesting descriptions: Elements of the modular group PSL(2,Z) have additional interpretations, as do elements of the group SL(2,Z) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL(2,R).