faithful adjoint representation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:31:49Z http://mathoverflow.net/feeds/question/45281 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45281/faithful-adjoint-representation faithful adjoint representation unknown (google) 2010-11-08T09:41:43Z 2010-11-08T13:59:41Z <p>When $n>3$ is even, how can I show that $PGL(n,\mathbb{R})$ has a faithful adjoint representation? Of course when n is even, $PGL(n,\mathbb{R})$ is not connected.</p> http://mathoverflow.net/questions/45281/faithful-adjoint-representation/45306#45306 Answer by S. Carnahan for faithful adjoint representation S. Carnahan 2010-11-08T13:59:41Z 2010-11-08T13:59:41Z <p>Suppose $A \in PGL_n(\mathbb{R})$ lies in the kernel of the adjoint representation. Then for any lift $\tilde{A}$ of $A$ in $GL_n(\mathbb{R})$, and any traceless matrix $B$, we have $\tilde{A}B = B\tilde{A}$. This implies $\tilde{A}$ is scalar, and hence $A$ is the identity.</p>