Here, \ mathcal { K } is the Killing form on " g ", and thus the first term is the standard kinetic term of quantum field theory.
22.
For example, the Killing form on the Lie algebra of the Heisenberg group is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group.
23.
In mathematics, the "'Killing form "', named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
24.
This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors.
25.
Where K is an invariant nondegenerate bilinear form over \ mathfrak { g } ( if G is semisimple, the Killing form will do ) and "'F "'is the curvature form
26.
Similarly, the Killing form of a Lie algebra need not restrict to the Killing form of a subalgebra ( this will happen for the Cartan subalgebra in the semisimple case, but need not in general ).
27.
Similarly, the Killing form of a Lie algebra need not restrict to the Killing form of a subalgebra ( this will happen for the Cartan subalgebra in the semisimple case, but need not in general ).
28.
The definition of compactness in terms of negative definiteness of the Killing form is more restrictive, since using this definition it can be shown that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.
29.
Since central elements can be identified with invariant inner products on the finite type Lie algebra \ mathfrak { g }, one typically normalizes the level so that the Killing form has level twice the dual Coxeter number.
30.
When the level is non-critical, i . e ., the inner product is not minus one half of the Killing form, the vacuum representation has a conformal element, given by the Sugawara construction.
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