The anchor to cap map ca replaces exp for SL(2)! | Classical to Quantum | Wild Egg Maths
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After a review of the Dihedron algebra and its subspace of anchors (sl(2)) and its subgroup of unit dihedrons (SL(2)) we concentrate on the true relation between sl(2) and SL(2).
In the Lie group setting this relation is invariably described in terms of the "exponential map" which is a transcendental function which cannot in fact be evaluated. This fact is conveniently ignored in virtually all treatments of the subject.
But we are going to replace the "exp map" with a natural algebraic variant: the cap map between anchors and elements of SL(2). This has the potential to provide a broad new approach to simple Lie groups and their harmonic analysis and representation theory. The fundamental result connects this with the Cayley transform that we met earlier in this series when discussing the rotation group in 3D.
We illustrate with a simple example.
Here is a link to the pdf of this lecture: https://www.dropbox.com/scl/fi/jn18by...
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