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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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Algebra of regular functions on the quadratic cone and SU(2) representations
Apply the Borel-Weil theorem to the Lie group $G={\rm SL}_2({\mathbb C})$, with Borel $B$, the complexification of ${\rm SU}(2)$. The flag variety $G/B={\mathbb P}^1$, and we have $V_m=\Gamma({\mathb …