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The Lie algebra of SL(2, R), denoted sl(2, R), is the algebra of all real, traceless 2 × 2 matrices. It is the Bianchi algebra of type VIII. 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).
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 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 ...
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.
6.1 sl 2 (C) 6.2 sl 3 (C) 7 Examples. ... Download QR code; Print/export Download as PDF; ... The basic yet nontrivial facts [14] then are (1) ...
The element h of the sl 2-triple is semisimple, with the simple eigenvalues j, j − 2, ..., −j on a submodule of g isomorphic to V j. The elements e and f move between different eigenspaces of h, increasing the eigenvalue by 2 in case of e and decreasing it by 2 in case of f. In particular, e and f are nilpotent elements of the Lie algebra g.