Killing Form

Definition

The Killing form is a bilinear, symmetric, and non-degenerate bilinear form defined on a Lie algebra, which plays a crucial role in determining the structure of semisimple Lie algebras. It is used to measure the interaction between elements of the algebra and can provide information about the algebra's semisimplicity. The Killing form is connected to Cartan's criterion for semisimplicity and is instrumental in the classification of symmetric spaces.

5 Must Know Facts For Your Next Test

1. The Killing form is defined as $$K(X, Y) = ext{tr}( ext{ad}(X) ext{ad}(Y))$$ for elements $$X$$ and $$Y$$ in a Lie algebra, where $$ ext{ad}$$ is the adjoint representation.
2. The Killing form is used to test whether a Lie algebra is semisimple; if the Killing form is non-degenerate, the algebra is semisimple.
3. The Killing form vanishes for nilpotent Lie algebras, which means it provides a clear distinction between nilpotent and semisimple structures.
4. In the context of symmetric spaces, the Killing form helps classify these spaces by providing insight into the structure of their associated Lie groups.
5. The Killing form has applications in representation theory, particularly in understanding how representations decompose based on the structure of the underlying Lie algebra.

Review Questions

• How does the Killing form determine whether a Lie algebra is semisimple?
  • The Killing form determines if a Lie algebra is semisimple by assessing its non-degeneracy. If the Killing form is non-degenerate, it indicates that there are no non-trivial ideals within the algebra, affirming its semisimplicity. This connection is essential because it provides a concrete method for classifying Lie algebras based on their internal structure.
• Discuss how the Killing form relates to Cartan's criterion and its implications for classifying semisimple Lie algebras.
  • Cartan's criterion states that a finite-dimensional Lie algebra is semisimple if and only if its Killing form is non-degenerate. This relationship allows us to use the properties of the Killing form to classify semisimple Lie algebras systematically. By analyzing the Killing form in relation to Cartan subalgebras, we can gain insights into their structure and representation theory.
• Evaluate the significance of the Killing form in the study of symmetric spaces and their classification.
  • The significance of the Killing form in studying symmetric spaces lies in its ability to connect the geometry of these spaces with their underlying algebraic structures. The properties revealed by the Killing form help classify symmetric spaces by examining how they relate to their corresponding Lie groups. This understanding allows mathematicians to draw deeper connections between algebraic properties and geometric behavior, ultimately enriching both fields.

Related terms

Semisimple Lie Algebra: A Lie algebra that can be decomposed into a direct sum of simple Lie algebras, meaning it has no non-trivial solvable ideals.

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

Adjoint Representation: The representation of a Lie algebra on itself via the adjoint action, where each element acts by commutation with others in the algebra.