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However, if A is a field with more than 2 elements, then E(2, A) = [GL(2, A), GL(2, A)], and if A is a field with more than 3 elements, E(2, A) = [SL(2, A), SL(2, A)]. [ dubious – discuss ] In some circumstances these coincide: the special linear group over a field or a Euclidean domain is generated by transvections, and the stable special ...
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.
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 ...
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.
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.
In the case of the complex special linear group SL(n,C), the compact real form is the special unitary group SU(n) and the split real form is the real special linear group SL(n,R). The classification of real forms of semisimple Lie algebras was accomplished by Élie Cartan in the context of Riemannian symmetric spaces. In general, there may be ...
Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, F). When n = 1, the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that Sp(2, F) = SL(2, F). For n > 1, there are additional conditions, i.e. Sp(2n, F) is then a proper subgroup of SL ...
The group GL(2, Z) is the linear maps preserving the standard lattice Z 2, and SL(2, Z) is the orientation-preserving maps preserving this lattice; they thus descend to self-homeomorphisms of the torus (SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) mapping class group of the torus, meaning that ...