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Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...

Given a connected Lie group $G$ with corresponding Lie algebra $\mathfrak{g}$, the adjoint representation/action $\mathrm{Ad} : G \to \mathrm{GL}(\mathfrak{g})$ induces a Lie group homomorphism. It's ...

Given a smooth manifold $M$, one can consider the Lie algebra $\mathcal{X}(M)$ of vector fields equipped with the standard Lie bracket. This is a standard machinery of differential geometry. Gelfand ...

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...