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The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of quadratic forms on R 2. The result is the following representation:
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 braid group B 3 is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group SL 2 (R) → PSL 2 (R). Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of B 3 modulo its center ; equivalently, to the ...
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.
SL British/French mandatory allocation SL [28] 1947 1959 D French protectorate allocation Saint Kitts and Nevis: VP-LK, VP-LL [28] V4 British colonial allocation Saint Lucia: VQ-L [28] 1965 J6 British colonial allocation Saint Vincent and the Grenadines: VP-V [28] 1959 J8 British colonial allocation Sarawak: VR-W [28] 1929 British colonial ...
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.
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.
"On a Ragga Tip" is a song by British breakbeat hardcore group SL2, released as a single in 1992. The song contains samples of Jah Screechy's "Walk and Skank" and Kid 'n Play's "Gittin' Funky (UK Remix)". The song was a success, peaking at No. 2 in the United Kingdom and the Netherlands, No. 3 in Ireland and No. 42 in Belgium. It remains the ...