StructurallyConstantBilinearForm - crowlogic/arb4j GitHub Wiki

The Killing form, also proposed to be called the structurally constant bilinear form, is a bilinear form defined in the context of a Lie algebra L over a field F. Here's a detailed mathematical definition:

Given a Lie algebra L, the Killing form B is defined for any x, y in L by

$$B(x, y) = tr(ad(x) \cdot ad(y))$$

where ad(x) represents the adjoint representation of x, which is a linear map from L to L given by ad(x)(y) = [x, y]. Here, [x, y] denotes the Lie bracket of x and y. The trace of a linear map, represented by tr, is the sum of the diagonal entries in the matrix representation of the linear map relative to some basis.

An equivalent definition of the Killing form is:

$$B(x, y) = \sum_i \sum_j c^k_{ij} c^l_{kj} g_{il}$$

Here, c^k_{ij} are the structure constants of the Lie algebra in some basis, and g_{il} is the metric tensor.

The Killing form is characterized by several significant properties:

  1. It is symmetric: $$B(x, y) = B(y, x)$$ for all x, y in L.
  2. It is invariant under the adjoint action: $$B([x, y], z) + B(y, [x, z]) = 0$$ for all x, y, z in L.
  3. A finite-dimensional Lie algebra is semisimple if and only if the Killing form is nondegenerate.

The term "Structurally Constant Bilinear Form" would be an appropriate alternative name for the Killing form, as it highlights the key mathematical aspects of the form without referencing its eponymous origin.

As previously explained, a Lie algebra's structure constants are coefficients that arise when the Lie bracket operation is expanded in terms of a basis for the Lie algebra. They are the components of the Lie bracket in the chosen basis. Since the Killing form is defined by means of these structure constants, calling it the "Structurally Constant Bilinear Form" would indeed be a meaningful choice.

Still, the usage of this term would require explaining its connection to the traditional term "Killing form", especially when communicating with mathematicians who are accustomed to the standard terminology.