Invariant forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:14:19Z http://mathoverflow.net/feeds/question/25432 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25432/invariant-forms Invariant forms Mariano Suárez-Alvarez 2010-05-20T22:41:57Z 2010-05-20T23:09:40Z <p>Is there a classification of pairs $(\mathfrak g,V)$, with $\mathfrak g$ a Lie algebra and $V$ a $\mathfrak g$-module, such that $(\Lambda^3V)^{\mathfrak g}\neq0$?</p> http://mathoverflow.net/questions/25432/invariant-forms/25435#25435 Answer by JosÃ© Figueroa-O'Farrill for Invariant forms JosÃ© Figueroa-O'Farrill 2010-05-20T23:09:40Z 2010-05-20T23:09:40Z <p>Not a full answer, but you can get plenty of examples by taking $V$ to be the coadjoint representation of a Lie algebra with nonzero Killing form. Then $\omega \in \Lambda^3\mathfrak{g}^*$, defined by $$\omega(X,Y,Z) = \mathrm{Tr}~\mathrm{ad}([X,Y])\mathrm{ad}(Z),$$ is invariant and nonzero.</p> <p>Similarly, any metric Lie algebra with $V$ the adjoint representation also works. The invariant 3-form is then $$\omega(X,Y,Z) = \langle [X,Y], Z\rangle,$$ with $\langle-,-\rangle$ the ad-invariant inner product.</p>