8/11/24
Welcome to the first `quippet', this is largely a collection of little facts to practice writing
about little facts, but also a bit of a history for me to look back on. Of course, if anyone else
ever learns something from them, that would also be a plus!
In a nutshell: the invariant symmetric bilinear form of the adjoint representation of a Lie Algebra.
Given a Lie Algebra g, one of the canonical representations of the Lie Algebra is the adjoint representation. This is obtained by the map on the elements of the lie algebra rho = x \mapsto [x, -].
Given a representation of a Lie Algebra V obtained by a map rho, the symmetric bilinear form is a bilinear map B_V: V \times V \rightarrow k. which takes the trace of the product of pairs of traces of elements in the representation.
B_V(x,y) = tr(rho(x)\rho(y)).
The Killing form is useful for classifying Lie algebras! But more on that later.