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.

Read more about this topic:  Killing Form

Famous quotes containing the word properties:

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)