Killing form

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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

on line[edit]