Killing Form

Definition

The Killing form is a bilinear form associated with a Lie algebra that helps determine the structure and properties of the algebra. It is particularly important in the study of semisimple Lie algebras, as it provides a way to understand their representations and classify them through the use of roots and weights. The Killing form can also reveal whether a Lie algebra is semisimple or not, serving as a key tool in connecting algebraic and geometric properties.

5 Must Know Facts For Your Next Test

1. The Killing form is defined for a Lie algebra \(\mathfrak{g}\) as \(K(X,Y) = Tr(\text{ad}(X) \circ \text{ad}(Y))\), where \(X\) and \(Y\) are elements of \(\mathfrak{g}\) and \(Tr\) denotes the trace operator.
2. If the Killing form is non-degenerate, it implies that the Lie algebra is semisimple.
3. The Killing form can be used to derive the Cartan subalgebra, which plays a crucial role in understanding representations of Lie algebras.
4. In addition to identifying semisimplicity, the Killing form assists in constructing invariant metrics on the corresponding Lie groups.
5. The properties of the Killing form connect deeply with the theory of roots and weights, allowing for a geometric interpretation of representation theory.

Review Questions

• How does the Killing form help in determining whether a Lie algebra is semisimple?
  • The Killing form provides a criterion for identifying semisimplicity in Lie algebras. Specifically, if the Killing form is non-degenerate, it indicates that the Lie algebra is semisimple. This connection allows mathematicians to analyze the structure and representations of the algebra using properties derived from this bilinear form.
• Discuss how the Killing form relates to the Cartan subalgebra within a Lie algebra.
  • The Killing form is instrumental in deriving the Cartan subalgebra from a given Lie algebra. By examining the eigenvalues of the adjoint representation associated with the Killing form, one can identify elements that commute with others, thus defining the Cartan subalgebra. This subalgebra then serves as a foundation for studying root systems and representation theory, showcasing the Killing form's importance in understanding structural relationships within Lie algebras.
• Evaluate the implications of using the Killing form in constructing invariant metrics on Lie groups and its impact on representation theory.
  • Using the Killing form to construct invariant metrics on Lie groups has significant implications for both geometry and representation theory. It allows for defining geometrical structures on these groups that are compatible with their algebraic properties. Furthermore, this construction provides insights into how representations of these groups behave under transformations, linking back to how roots and weights describe these representations. The interplay between geometry and algebra through the Killing form thus enriches our understanding of both areas.

Related terms

Lie Algebra:

A Lie algebra is an algebraic structure formed by a vector space equipped with a binary operation called the Lie bracket, which satisfies certain axioms like bilinearity, alternation, and the Jacobi identity.

Semisimple Lie Algebra:

A semisimple Lie algebra is a type of Lie algebra that can be expressed as a direct sum of simple Lie algebras and has no nontrivial ideals other than itself and zero.

Root System: A root system is a collection of vectors in a Euclidean space that describes the symmetries and structure of a semisimple Lie algebra, often used in conjunction with the Killing form.