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The Geometry of Physics by Theodore Frankel

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CHAPTER 15

Lie Groups

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15.1. Lie Groups, Invariant Vector Fields and Forms

Is the unitary group SU (n) connected?

15.1a. Lie Groups

Let M(n × n) be the set of all n × n real matrices. As in Section 1.1d, we shall associate to the matrix x the point in n2-dimensional euclidean space whose coordinates are x11, x12,…, xnn. Topologically then, M(n × n) is simply euclidean n2 space! The general linear group Gl(n, image) is the group of all real n × n matrices x = (xij) with determinant det x ≠ 0. Since det x is an nth-degree polynomial in the coordinates, ...

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