How Cartan-Killing form categorized the subalgebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:16:30Z http://mathoverflow.net/feeds/question/95071 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95071/how-cartan-killing-form-categorized-the-subalgebra How Cartan-Killing form categorized the subalgebra ciksalma 2012-04-24T21:01:16Z 2012-04-24T21:01:16Z <p>Let $X_{i}, i=1,2,3,4$ being the generators of my Lie algebra. $X_{i}$ (i.e. taking the adjoint representations) generates the algebra $X=aX_{1}+bX_{2}+ cX_{3}+dX_{4}$ with four parameters. Using the Cartan-Killing form, $\kappa(X_{i},X_{i})= Trace(X_{i}^{2})$, we can check if the subalgebra having a compact or non-compact generators. Compact generator if $\kappa(X_{i},X_{i})$ is negative-definite and non-compact if $\kappa(X_{i},X_{i})$ being positive definite. </p> <p>My question is, what happen if one of the subalgebra involved indefinite answers. Meaning for example, $\kappa(X_{1},X_{1})= -a^{2} + \alpha a^{2}$ for some constant alpha. Is this acceptable? If yes how do I categorized the generators? </p>