# 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 (with its defining vector space structure), defined on every pair of elements in as the trace 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 Edit

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

- where x,y,z are elements in the algebra and the brackets [] are the Lie brackets

## exercise Edit

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

## references Edit

### on paper Edit

- 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