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I read this article on wikipedia on loops. And I wondered if the loops of loop quantum gravity have the algebraic structure of loops or it's just a coincidence.

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It's just a coincidence, the word loop in loop quantum gravity holds for the use of closed curves to rewrite the degrees of freedom of gravity theory in terms of gauge invariant traces of path ordered exponentials of the gravitational connection (traced holonomies / Wilson loops), although later on people started using open curves in what got to be called spin networks.

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