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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 Killing form was essentially introduced into Lie algebra theory by Élie Cartan () in his thesis.In a historical survey of Lie theory, Borel (2001) has described how the term "Killing form" first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. [2]
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.
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.
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.
Assume that g is a finite dimensional Lie algebra over a field of characteristic zero.From the representation theory of the Lie algebra sl 2, one concludes that the Lie algebra g decomposes into a direct sum of finite-dimensional subspaces, each of which is isomorphic to V j, the (j + 1)-dimensional simple sl 2-module with highest weight j.
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]
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, ... For G = Γ(1) = SL(2, Z), this gives back the afore-mentioned definitions.