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.
The quotient PSL(2,R) has several interesting descriptions:
- It is the group of orientation-preserving projective transformations of the real projective line R∪{∞}.
- It is the group of conformal automorphisms of the unit disc.
- It is the group of orientation-preserving isometries of the hyperbolic plane.
- It is the restricted Lorentz group of three-dimensional Minkowski space. Equivalently, it is isomorphic to the indefinite orthogonal group SO+(1,2). It follows that SL(2,R) is isomorphic to the spin group Spin(2,1)+.
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).
Read more about this topic: SL2(R)
Famous quotes containing the word descriptions:
“The fundamental laws of physics do not describe true facts about reality. Rendered as descriptions of facts, they are false; amended to be true, they lose their explanatory force.”
—Nancy Cartwright (b. 1945)
“Our Lamaze instructor . . . assured our class . . . that our cervix muscles would become “naturally numb” as they swelled and stretched, and deep breathing would turn the final explosions of pain into “manageable discomfort.” This descriptions turned out to be as accurate as, say a steward advising passengers aboard the Titanic to prepare for a brisk but bracing swim.”
—Mary Kay Blakely (20th century)
“Matter-of-fact descriptions make the improbable seem real.”
—Mason Cooley (b. 1927)