Killing Form

Definition

The Killing form is a symmetric bilinear form associated with a Lie algebra that provides important insights into its structure and classification. It is defined using the Lie bracket, and its properties help in identifying whether the algebra is semisimple or not. The Killing form plays a significant role in understanding the representation theory of Lie algebras and how they relate to their corresponding Lie groups.

5 Must Know Facts For Your Next Test

1. The Killing form is defined as $K(X,Y) = ext{Tr}( ext{ad}(X) \circ \text{ad}(Y))$, where $ ext{ad}(X)$ is the adjoint representation of the Lie algebra.
2. A Lie algebra is semisimple if and only if its Killing form is non-degenerate, meaning it has an inverse.
3. The Killing form is invariant under automorphisms of the Lie algebra, making it a valuable tool in representation theory.
4. For non-semisimple Lie algebras, the Killing form can be degenerate, leading to complications in their classification.
5. In the classification of classical Lie algebras, the properties of the Killing form help distinguish between different types of algebras and their representations.

Review Questions

• How does the Killing form help determine whether a Lie algebra is semisimple?
  • The Killing form helps determine if a Lie algebra is semisimple by being non-degenerate when evaluated on the algebra's elements. If the Killing form has an inverse, it indicates that the algebra does not possess any nontrivial ideals, which defines its semisimplicity. This relationship is critical for classifying and understanding the structure of various Lie algebras.
• Discuss the significance of the Killing form's invariance under automorphisms for representation theory.
  • The invariance of the Killing form under automorphisms of a Lie algebra means that if you apply an automorphism to elements of the algebra, the value of the Killing form remains unchanged. This property is significant for representation theory because it ensures that the relations defined by the Killing form are preserved across different representations. As a result, it provides a consistent framework for studying how these representations behave under changes in basis or other transformations.
• Evaluate the implications of having a degenerate Killing form on a Lie algebra's classification.
  • A degenerate Killing form indicates that a Lie algebra cannot be classified as semisimple, which has broader implications for its structure and representation theory. This degeneration suggests the existence of nontrivial ideals within the algebra, complicating its classification and limiting the types of representations that can exist. Understanding these implications allows mathematicians to better navigate the landscape of more complex algebras and their relationships to classical ones.

Related terms

Lie Algebra: A Lie algebra is a mathematical structure formed by a vector space equipped with a binary operation called the Lie bracket that satisfies bilinearity, antisymmetry, and the Jacobi identity.

Semisimple Lie Algebra: A semisimple Lie algebra is one that can be decomposed into a direct sum of simple Lie algebras, indicating that it has no nontrivial ideals.

Root System: A root system is a combinatorial structure associated with semisimple Lie algebras that helps classify the algebraic structure and identify its representations.