Killing form

Definition

The Killing form is a bilinear form associated with a Lie algebra, defined using the structure constants of the algebra. It serves as a tool to determine important properties of the Lie algebra, such as whether it is semisimple, by analyzing its eigenvalues. The Killing form plays a significant role in understanding the classification and structure of simple Lie algebras.

5 Must Know Facts For Your Next Test

1. The Killing form is defined as $$K(X, Y) = ext{Tr}( ext{ad}(X) ext{ad}(Y))$$, where $$X$$ and $$Y$$ are elements of the Lie algebra and $$ ext{ad}$$ is the adjoint representation.
2. For a semisimple Lie algebra, the Killing form is non-degenerate, meaning that it can provide insights into the representation theory of the algebra.
3. The Killing form helps in distinguishing between different types of Lie algebras, particularly in classifying simple Lie algebras.
4. If the Killing form is positive definite, it indicates that the Lie algebra is compact, providing further geometric insights.
5. The properties of the Killing form are essential in determining the roots and weights associated with representations of semisimple Lie algebras.

Review Questions

• How does the Killing form help classify simple Lie algebras?
  • The Killing form provides crucial information about the structure of simple Lie algebras. By examining whether the Killing form is non-degenerate or degenerate, we can categorize algebras into different classes. Specifically, if the Killing form is non-degenerate, it indicates that the Lie algebra is semisimple, which is fundamental for further classification and understanding of its representations.
• In what ways does the non-degeneracy of the Killing form relate to other properties of a Lie algebra?
  • The non-degeneracy of the Killing form directly implies that a Lie algebra is semisimple. This non-degeneracy means that there are no non-zero vectors in the Lie algebra that are orthogonal to all other vectors under this bilinear form. This property links to various structural aspects like root systems and representation theory, as it ensures that the algebra can be decomposed into simpler components while maintaining certain desirable traits.
• Evaluate how the properties of the Killing form influence representation theory in semisimple Lie algebras.
  • The properties of the Killing form significantly impact representation theory in semisimple Lie algebras by determining the structure of representations and character theory. A non-degenerate Killing form allows for a consistent way to define weights and roots within these representations. Furthermore, it ensures that representations can be effectively analyzed through geometric means, aiding in understanding how these algebras act on various vector spaces and simplifying complex interactions between their elements.

Related terms

Lie Algebra: An algebraic structure formed by a vector space equipped with a binary operation called the Lie bracket, satisfying certain properties like bilinearity, antisymmetry, and the Jacobi identity.

Semisimple Lie Algebra: A Lie algebra that can be decomposed into a direct sum of simple Lie algebras, characterized by having a non-degenerate Killing form.

Cartan Subalgebra: A maximal abelian subalgebra of a Lie algebra where the elements can be simultaneously diagonalized with respect to the Killing form.