Killing Form - Properties

Properties

  • The Killing form B is bilinear and symmetric.
  • The Killing form is an invariant form, in the sense that it has the 'associativity' property
B(, z) = B(x, ),
where is the Lie bracket.
  • If g is a simple Lie algebra then any invariant symmetric bilinear form on g is a scalar multiple of the Killing form.
  • The Killing form is also invariant under automorphisms s of the algebra g, that is,
B(s(x), s(y)) = B(x, y)
for s in Aut(g).
  • The Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
  • The Killing form of a nilpotent Lie algebra is identically zero.
  • If I, J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form.
  • If a given Lie algebra g is a direct sum of its ideals I1,...,In, then the Killing form of g is the direct sum of the Killing forms of the individual summands.

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