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Lie Groups, Physics, and Geometry by Robert Gilmore

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2

Lie groups

 

 

Lie groups are beautiful, important, and useful because they have one foot in each of the two great divisions of mathematics – algebra and geometry. Their algebraic properties derive from the group axioms. Their geometric properties derive from the identification of group operations with points in a topological space. The rigidity of their structure comes from the continuity requirements of the group composition and inversion maps. In this chapter we present the axioms that define a Lie group.

2.1 Algebraic properties

The algebraic properties of a Lie group originate in the axioms for a group.

Definition   A set gi, gj, gk, . . . (called group elements or group operations) together with a combinatorial operation (called group ...

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