In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct sum of a commutative summand ( for which the corresponding subgroup is a torus ) and a summand on which the Killing form is negative definite.
32.
It is important to note that the converse of the first result above is false : Even if the Killing form of a Lie algebra is negative semidefinite, this does not mean that the Lie algebra is the Lie algebra of some compact group.
33.
Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal group ), differential geometry ( Riemannian metric ), differential topology ( intersection forms of four-manifolds ), and Lie theory ( the Killing form ).
34.
Gave a very short proof that if a finite-dimensional Lie algebra ( in any characteristic ) has a non-degenerate invariant bilinear form and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.
35.
If the inner product on \ mathfrak { g }, the Lie algebra of G in which F takes values, is given by the Killing form on \ mathfrak { g }, then this may be denoted as \ int _ M \ mathrm { Tr } ( F \ wedge * F ), since
36.
If \ mathfrak { g } is a linear Lie algebra ( a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space " V " ) over an algebraically closed field, then any Cartan subalgebra of \ mathfrak { g } is the centralizer of a maximal nilpotent ( Engel's theorem ), but then its Killing form is identically zero, contradicting semisimplicity.
37.
Here " x " and " t " are spacetime coordinates, (, ) is the Killing form of a real r-dimensional Cartan algebra \ mathfrak { h } of a Kac Moody algebra over \ mathfrak { h }, & alpha; i is the i th simple root in some root basis, n i is the Coxeter number, m is the mass ( or bare mass in the quantum field theory version ) and & beta; is the coupling constant.
