CartanKilling

The Cartan-Killing form, also known as the Killing form, is a fundamental bilinear form in the theory of Lie algebras, named after Élie Cartan and Wilhelm Killing. It plays a crucial role in the classification and study of Lie algebras, particularly in understanding their structure and representations.

Given a Lie algebra **g** over a field (typically the real or complex numbers), the Killing form is defined as a symmetric bilinear form on the dual space **g***. For any two elements **X** and **Y** in **g**, the Killing form is given by the trace of the composition of the adjoint representations of **X** and **Y**, denoted as **B(X, Y) = tr(ad(X) ∘ ad(Y))**, where **ad(X)** is the adjoint operator associated with **X**. The adjoint operator acts on **g** by the Lie bracket: **ad(X)(Y) = [X, Y]**.

The Killing form is invariant under the adjoint action of the Lie algebra, meaning that for any **X, Y, Z** in **g**, the following holds: **B([X, Y], Z) = B(X, [Y, Z])**. This property makes it a natural candidate for studying the structure of **g**. The Killing form is also non-degenerate if and only if the Lie algebra is semisimple, a key class of Lie algebras with no non-trivial solvable ideals.

The Killing form is closely related to the concept of semisimplicity in Lie algebras. A Lie algebra is semisimple if and only if its Killing form is non-degenerate. This criterion provides a powerful tool for distinguishing between solvable and semisimple Lie algebras. Additionally, the Killing form can be used to define an inner product on the Lie algebra, which facilitates the study of its geometric and algebraic properties.

In representation theory, the Killing form helps classify irreducible representations of semisimple Lie algebras, particularly through the use of highest weight theory. It also appears in the context of symmetric spaces and the classification of compact Lie groups. The Killing form remains a central object in the study of Lie algebras and their applications in mathematics and physics.