Can you give a source for where you "often" see it? I'm having trouble understanding what you want. Are you asking if the Killing form is a metric?center o bass said:The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by
##K_{ab} = k \delta_{ab}## for some proportionality constant ##k##. How is this statement proved?
A compact simple group is a type of mathematical group that is both compact (meaning it has a finite or countably infinite number of elements) and simple (meaning it has no nontrivial normal subgroups).
A Killing metric is a type of metric used to measure distances between elements of a compact simple group. It is defined by the group's inner product and is often used to study the geometry and topology of the group.
The Killing metric is closely related to the structure of compact simple groups. It can be used to understand the group's symmetries, subgroups, and other important characteristics. In fact, the Killing metric is often used as a tool to classify and study different types of compact simple groups.
The Killing metric is important in mathematics because it allows us to study and understand the properties of compact simple groups. These groups have many applications in various areas of mathematics, including algebra, geometry, and physics. By using the Killing metric, we can gain insight into the structure and behavior of these groups, leading to a deeper understanding of mathematical concepts.
The Killing metric is calculated using the group's inner product, which is a type of binary operation that takes two elements of the group and produces a real number. This inner product is then used to define a distance function, which is used to measure distances between elements of the group. The specific calculation of the Killing metric may vary depending on the specific group and its structure.