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The finite-dimensional representation theory of SL(2, R) is equivalent to the representation theory of SU(2), which is the compact real form of SL(2, C). In particular, SL(2, R) has no nontrivial finite-dimensional unitary representations. This is a feature of every connected simple non-compact Lie group.
The algebra plays an important role in the study of chaos and fractals, as it generates the Möbius group SL(2,R), which describes the automorphisms of the hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL(2,C) describes the automorphisms of the hyperbolic 3-dimensional ball.
These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When R is the finite field of order q, the notation SL(n, q) is sometimes used.
It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL(2, R). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ 2. Thus in the case of the Lie algebra sl 2, the infinitesimal character of an irreducible representation is specified by one complex number.
of the Lie algebra sl 2 of 2 by 2 matrices with zero trace. It follows that sl 2-triples in g are in a bijective correspondence with the Lie algebra homomorphisms from sl 2 into g. The alternative notation for the elements of an sl 2-triple is {H, X, Y}, with H corresponding to h, X corresponding to e, and Y corresponding to f. H is called a ...
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The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL 2 (R) can also be interpreted as the symplectic group Sp 2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the ...
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