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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact?
The answer to this question can be found in Lubos Motl's answer to this question of mine on Physics.SE.
The key here are the weight lattices bosonic representations $\Gamma$ of these gauge groups.
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