Search results
Results from the WOW.Com Content Network
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 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.
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 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.
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.
Tesla’s global vehicle sales rose 2.3% in the final quarter of 2024 thanks to 0% financing, free charging and low-priced leases. Tesla sales dropped 1.1% in 2024, its first annual decline in a ...
Z n=2 Z 2 n>2 SL(1,R) is a single point and therefore compact and simply connected sl(n,R) n 2 −1 SL(2,R) Orientation-preserving isometries of the Poincaré half-plane, isomorphic to SU(1,1), isomorphic to Sp(2,R). N 0 Z: The universal cover has no finite-dimensional faithful representations. sl(2,R) 3 O(n) orthogonal group: real orthogonal ...
2. In a medium bowl, stir the flour, pudding mix, cream of tartar, baking soda and salt. With the mixer on low, gradually add the dry ingredients to the butter mixture, beating just until ...