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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.
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.
One important subset of the modular group is the dyadic monoid, which is the monoid of all strings of the form ST n 1 ST n 2 ST n 3... for positive integers n i. This monoid occurs naturally in the study of fractal curves , and describes the self-similarity symmetries of the Cantor function , Minkowski's question mark function , and the Koch ...
In mathematics, a modular form is a (complex) analytic function on the upper half-plane, ... For G = Γ(1) = SL(2, Z), this gives back the afore-mentioned definitions.
The center Z of the group SL(2, R) is a cyclic group {I, −I} of order 2, consisting of the identity matrix and its negative. On any irreducible representation, the center either acts trivially, or by the nontrivial character of Z , which represents the matrix - I by multiplication by -1 in the representation space.
In 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z). [2]In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms M(Γ) is generated in weight at most 3 when is the congruence subgroup () of prime level N in SL(2, Z) using the theory of toric modular forms. [3]
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.
For example, SL(2,R)/SO(2) is the hyperbolic plane, and SL(2,C)/SU(2) is hyperbolic 3-space. For a reductive group G over a field k that is complete with respect to a discrete valuation (such as the p-adic numbers Q p), the affine building X of G plays the role of the symmetric space.