Weights

Definition

Weights are a fundamental concept in the representation theory of compact Lie groups, capturing how representations transform under the action of a Cartan subalgebra. They provide a way to classify the irreducible representations and relate to the roots of the Lie algebra, which are vital in understanding the structure and representation of the group. In this context, weights serve as an essential tool to express how various representations behave with respect to the maximal torus and its associated characters.

5 Must Know Facts For Your Next Test

1. Weights are typically expressed as linear combinations of fundamental weights, each corresponding to a simple root in the root system of the Lie algebra.
2. The set of weights forms a lattice in a vector space that is crucial for understanding the representation theory and can be visualized using weight diagrams.
3. In finite-dimensional representations, weights can be classified into dominant weights, which play a key role in determining irreducibility.
4. Weights are used in conjunction with the Weyl character formula to compute characters of representations, providing insights into their structure.
5. Each representation corresponds to a unique set of weights, which gives rise to multiple irreducible components when decomposed.

Review Questions

• How do weights relate to the representation theory of compact Lie groups and what is their significance?
  • Weights provide crucial insight into how representations transform under the action of a Cartan subalgebra in compact Lie groups. They help classify irreducible representations and link them with roots, allowing us to understand the underlying structure of both the group and its representations. Weights also help determine how different representations behave with respect to maximal tori and their associated characters.
• Discuss how weights interact with roots in a compact Lie group and their implications for understanding representations.
  • Weights and roots are intimately connected in representation theory; weights are often expressed as combinations of roots from a root system associated with a Lie algebra. The relationship between weights and roots helps us identify weight spaces and understand how these spaces decompose under group actions. This connection is essential for analyzing representations, especially when determining their irreducibility and structure.
• Evaluate the role of weights within the context of the Weyl character formula and their impact on representation analysis.
  • The Weyl character formula uses weights to express characters of finite-dimensional representations in terms of roots and weights associated with a compact Lie group. This formula provides an elegant way to compute the character for each representation by incorporating dominant weights and leveraging symmetry properties inherent in the group. Understanding weights allows for deeper analysis into how these characters behave under various transformations, facilitating a comprehensive grasp of representation theory.

Related terms

Cartan Subalgebra:

A maximal abelian subalgebra of a Lie algebra where all elements can be simultaneously diagonalized, playing a crucial role in the structure theory of Lie algebras.

Roots: Vectors in the dual space of the Cartan subalgebra that describe how the weight spaces transform, linking weights to the underlying algebraic structure of the Lie group.

Characters: Homomorphisms from a group to its multiplicative group of complex numbers, which help identify how weights interact with group representations.