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The Geometry of Physics by Theodore Frankel

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APPENDIX E

Orbits and Morse-Bott Theory in Compact Lie Groups

image

There once was a real classy Groupie

Who longed from the homotopyists to mut’nie

Bott appeared, it was Fate,

Made her period 8

By applying Morse Code to her Loopie.

E.a. The Topology of Conjugacy Orbits

We now wish to study in more detail the topology of conjugacy orbits in a compact Lie group G with given maximal torus T. But first we present an example (more complicated than the SO(3) case of Figure D.1) to keep in mind.

Let G be the nine-dimensional unitary group U(3). The subgroup of diagonal matrices T = {diag[exp(1), exp(2), exp(3)]} is a three-dimensional maximal torus. ...

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