KillingCartan

KillingCartan is a term that refers to a specific type of geometrical object or concept arising from the study of Lie groups and Lie algebras. It is named after the mathematicians Elie Cartan and Wilhelm Killing, who made foundational contributions to these fields. The Killing form is a bilinear form defined on a Lie algebra. This form is invariant under the adjoint representation of the Lie algebra. In simpler terms, it's a way to measure how elements of the Lie algebra interact with each other in a way that respects the structure of the algebra.

The Killing form is a crucial tool in the classification of Lie algebras. Its properties, such as its definiteness, can reveal important information about the structure of the Lie algebra. For instance, a simple Lie algebra is semisimple if and only if its Killing form is non-degenerate. This means that if the Killing form is non-degenerate, then there are no non-trivial ideals that are orthogonal to all other elements. The Killing form also plays a role in understanding the representations of Lie algebras and Lie groups, which are fundamental to many areas of physics, including quantum mechanics and particle physics. The concept is a cornerstone in the theoretical framework of modern mathematics and physics.