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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.48   Lie Theory

   Mark Ronan

1   Lie Groups

Why are groups important in mathematics? One major reason is that it is often possible to understand a mathematical structure by understanding its symmetries, and the symmetries of a given mathematical structure form a group. Some mathematical structures are so symmetrical that they have not just a finite number of symmetries, but a continuous family of them. When this is the case, we find ourselves in the realms of Lie groups and Lie theory.

One of the simplest “continuous” groups is the group SO(2), which consists of all rotations of the plane Image about the origin. With each element of SO(2) one ...

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