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Abstract: Using quantum-group methods and noncommutative geometry
we show that even finite sets and finite-dimensional algebras can be
endowed with rich `differentiable manifold' structures and associated
geometry. In the case of finite groups the translation-invariant
differentiable structures are classified by conjugacy classes and there is
an induced `braided Lie algebra' and Killing form metric. Permutation
groups are looked at in detail, with S_3 as a concrete example. We obtain
it's de Rahm cohomology, moduli space of flat connections and Ricci
curvature (it turns out to be an `Einstein space'). The relations
of the group appear as cycles and we show that the Yang-Mills action
has a natural interpretation in terms of their holonomies. We
also indicate some connections with representation theory.
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