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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2010 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2010 • Question 4

Given a Lie group \(G\) with Lie algebra \(\mathfrak {g}\) and some \(v\in \mathfrak {g}\), let \(\{\theta _t^v\}_{t\in \mathbb {R}}\) be the corresponding 1-paramater group of diffeomorphisms of \(G\). Prove that \(\theta _t^v(gh)=g\theta _t^v(h)\) for any \(t\in \mathbb {R}\) and \(g,h\in G\).

Raw LaTeX

Given a Lie group \(G\) with Lie algebra \(\mathfrak{g}\) and some
\(v\in\mathfrak{g}\), let \(\{\theta_t^v\}_{t\in\R}\) be the corresponding
1-paramater group of diffeomorphisms of \(G\). Prove that
\(\theta_t^v(gh)=g\theta_t^v(h)\) for any \(t\in\R\) and \(g,h\in G\).

Topic Tags

Diffeomorphisms and Isotopy Methods Lie Algebras and Exponential Maps Lie Groups and Matrix Lie Groups