Killing form

Definition

The Killing form is a bilinear form defined on a Lie algebra, used to measure the 'size' of the Lie algebra in terms of its structure constants. It is a crucial tool for determining properties such as semisimplicity and whether a given Lie algebra is reductive. By analyzing the Killing form, one can deduce important information about the representation theory and the roots of the algebra, linking it to concepts like the exponential map.

5 Must Know Facts For Your Next Test

1. The Killing form, denoted as $K(X, Y) = ext{Tr}( ext{ad}(X) \circ ext{ad}(Y))$, where $X$ and $Y$ are elements of the Lie algebra, is constructed using the adjoint representation.
2. For a semisimple Lie algebra, the Killing form is non-degenerate, which implies that it can be used to define an inner product structure on the algebra.
3. If the Killing form is degenerate, it indicates that the Lie algebra is not semisimple and has non-trivial solvable ideals.
4. The Killing form can also be used to identify Cartan subalgebras and determine root systems within the context of Lie algebras.
5. In the context of the exponential map, understanding the Killing form helps in studying the local structure of the corresponding Lie group.

Review Questions

• How does the Killing form help in classifying Lie algebras as semisimple or non-semisimple?
  • The Killing form serves as a crucial tool in classifying Lie algebras by determining whether they are semisimple or non-semisimple. A Lie algebra is considered semisimple if its Killing form is non-degenerate, meaning it has an inverse. In contrast, if the Killing form is degenerate, this indicates the presence of non-trivial solvable ideals within the algebra, thus classifying it as non-semisimple.
• Discuss the significance of the Killing form in relation to representations of Lie algebras.
  • The Killing form plays a significant role in representation theory by providing insights into how representations of a Lie algebra can be constructed and classified. Specifically, it allows for understanding the inner product structures on representation spaces and assists in identifying highest weight representations. The behavior of characters and weights under this bilinear form helps mathematicians analyze how various representations relate to each other within the framework of the Lie algebra's structure.
• Evaluate how the properties of the Killing form influence the connection between Lie algebras and their corresponding Lie groups through the exponential map.
  • The properties of the Killing form significantly influence how we connect Lie algebras to their corresponding Lie groups via the exponential map. Since the Killing form defines an inner product when it is non-degenerate, it provides a geometrical interpretation to elements of the Lie group derived from its algebra. This inner product helps in analyzing geodesics and curvature within the group manifold, facilitating insights into how local structures of the Lie group reflect those of its algebra through representation theory and root system analysis.

Related terms

Lie Algebra:

A mathematical structure that studies algebraic operations and their relations, defined by a binary operation known as the Lie bracket.

Semisimple Lie Algebra:

A type of Lie algebra that is a direct sum of simple Lie algebras, characterized by having a non-degenerate Killing form.

Exponential Map:

A map that relates elements of a Lie algebra to elements of a corresponding Lie group, essential for connecting algebraic structures with geometric properties.