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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2017 • Question 3

Question

Math 6510 • Differentiable Manifolds Spring 2017 • Question 3

Let \(\Phi :\mathfrak {sl}_2(\mathbb {R})\to \mathbb {R}\) be a Lie algebra homomorphism, where \(\mathfrak {sl}_2(\mathbb {R})\) is the Lie algebra of \(\mathrm {SL}_2(\mathbb {R})\) and \(\mathbb {R}\) is the Lie algebra of the Lie group \(\mathbb {R}\). Prove that \(\Phi =0\).

Raw LaTeX

Let \(\Phi:\mathfrak{sl}_2(\R)\to\R\) be a Lie algebra homomorphism, where
\(\mathfrak{sl}_2(\R)\) is the Lie algebra of \(\mathrm{SL}_2(\R)\) and \(\R\) is the
Lie algebra of the Lie group \(\R\). Prove that \(\Phi=0\).

Topic Tags

Fundamental Groups Lie Algebras and Exponential Maps Lie Group Actions and Homomorphisms Lie Groups and Matrix Lie Groups