Semisimple

Definition

In the context of Lie algebras, a semisimple Lie algebra is one that can be expressed as a direct sum of simple Lie algebras, where a simple Lie algebra is non-abelian and has no nontrivial ideals. Semisimple Lie algebras play a crucial role in the classification of classical Lie algebras and are characterized by their ability to represent finite-dimensional representations that are completely reducible. This property leads to a rich structure and provides insight into the symmetries found in various mathematical and physical contexts.

5 Must Know Facts For Your Next Test

1. Every finite-dimensional semisimple Lie algebra over a field of characteristic zero is a direct sum of simple Lie algebras.
2. Semisimple Lie algebras are characterized by having finite-dimensional representations that are completely reducible, meaning every representation can be decomposed into irreducible components.
3. The classification of semisimple Lie algebras is closely linked to the theory of root systems, which helps in understanding their structure and representations.
4. The Killing form is a bilinear form that plays an essential role in determining whether a Lie algebra is semisimple; a Lie algebra is semisimple if and only if its Killing form is non-degenerate.
5. Semisimple Lie algebras can be classified into types such as A_n, B_n, C_n, and D_n based on their root systems and corresponding Dynkin diagrams.

Review Questions

• How does the property of being semisimple impact the representation theory of Lie algebras?
  • The property of being semisimple significantly impacts representation theory because it ensures that every finite-dimensional representation of a semisimple Lie algebra is completely reducible. This means that any representation can be broken down into irreducible components without any loss. This reduction simplifies the analysis of representations and helps in understanding the underlying structure of the algebra through its representations.
• Discuss how Cartan subalgebras relate to the classification of semisimple Lie algebras.
  • Cartan subalgebras are critical in the classification of semisimple Lie algebras as they provide a maximal abelian structure within the algebra. Each semisimple Lie algebra has a unique (up to conjugacy) Cartan subalgebra, which allows for the identification of root systems associated with the algebra. The roots, together with the Cartan subalgebra, help construct Dynkin diagrams that visually represent the relationships between different simple components, aiding in their classification.
• Evaluate the significance of the Killing form in determining whether a Lie algebra is semisimple and its implications for representation theory.
  • The Killing form serves as a powerful tool in determining whether a Lie algebra is semisimple; specifically, a Lie algebra is considered semisimple if its Killing form is non-degenerate. This non-degeneracy condition implies that there are no nontrivial ideals within the algebra, which directly influences representation theory. If the Killing form is non-degenerate, every finite-dimensional representation will exhibit complete reducibility. This enhances our ability to analyze and categorize representations, making it easier to connect abstract mathematical concepts to practical applications across physics and geometry.

Related terms

Simple Lie Algebra: A simple Lie algebra is a non-abelian Lie algebra that has no nontrivial ideals, meaning it cannot be decomposed into smaller subalgebras.

Reducible Representation: A representation of a Lie algebra is called reducible if it can be decomposed into smaller representations, allowing for the possibility of nontrivial invariant subspaces.

Cartan Subalgebra: A Cartan subalgebra is a maximal abelian subalgebra of a Lie algebra, important in the classification and structure theory of semisimple Lie algebras.