The Gabriel-Rosenberg reconstruction theorem, when applied to varieties, states that it is possible to reconstruct any variety $X$ from its category $\mathrm{QCoh}(X)$ of quasi-coherent sheaves. This means that if for two varieties $X$ and $Y$ there is an equivalence of categories $\mathrm{QCoh}...

Simply connected connected compact Lie groups are well-known to be in bijective correspondence with complex semisimple Lie algebras. Now every connected simply connected complex Lie group has a unique compact real form. So it would seem that this gives a bijection between connected simply connect...