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

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III.75 Quantum Groups

  Shahn Majid

There are at least three different paths that lead to the objects known today as quantum groups. They could be summarized briefly as quantum geometry, quantum symmetry, and self-duality. Any one of them would be a great reason to invent quantum groups and each of them had a role in the development of the modern theory.

1 Quantum Geometry

One of the great discoveries in physics in the last century was that classical mechanics should be replaced by quantum mechanics, in which the space of possible positions and momenta of a particle is replaced by the formulation of position and momentum as mutually noncommuting operators. This noncommutativity underlies Heisenberg’s “uncertainty principle,” but it also suggests ...

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