Killing form

The Killing form or Cartan-Killing form(wikipedia), named after the mathematician Wilhelm Killing, is an invariant bilinear form $(\_,\_)$on a Lie algebra ${\mathfrak {g}}$ (with its defining vector space structure), defined on every pair $(x,y)$ of elements in ${\mathfrak {g}}$ as the trace $(\_,\_):=tr(ad(x)ad(y))$of the matrix product for the adjoint representation of x and y. For a simple Lie algebra, the invariant bilinear form is unique up to scaling. A Lie algebra is semi-simple if and only if its Killing form is non-degenerate.

invariance

The Killing form has an invariance (or associative) property:

• $([x,y],z)=(x,[y,z])$ where x,y,z are elements in the algebra and the brackets [] are the Lie brackets

exercise

• Write out the killing form for sl2, with its usual generators e,f and h.

references

on paper

• J.E.Humphreys, Introduction to Lie algebras and representation theory,ISBN 9780387900537, pp.21-
• A.Knapp: Representation theory of semisimple groups, ISBN 0691090890, p.7