Character

Definition

In the context of Lie algebras and Lie groups, a character is a homomorphism from a Lie group to the multiplicative group of non-zero complex numbers, which can also be viewed as a trace of a representation. Characters are essential in understanding the representations of compact Lie groups and play a crucial role in the analysis of their structures and properties.

5 Must Know Facts For Your Next Test

1. Characters provide a way to study representations through their traces, simplifying complex group actions into scalar functions.
2. The characters of irreducible representations of compact Lie groups are orthogonal with respect to the inner product defined on the space of functions.
3. For compact Lie groups, characters can be used to obtain information about representation theory without explicitly computing the representations themselves.
4. The Weyl integral formula relates characters to integrals over the group, helping to compute characters in practical situations.
5. In representations of SU(2) and SO(3), characters can be explicitly calculated, illustrating their fundamental nature in linking geometry and algebra.

Review Questions

• How do characters help in understanding representations of compact Lie groups?
  • Characters help simplify the study of representations by providing scalar functions derived from the trace of these representations. This means instead of dealing with complex matrices or operators, we can work with functions that capture important structural information about the representations. This reduction makes it easier to analyze and categorize different representations and understand how they relate to each other.
• Discuss how characters are utilized in the Peter-Weyl theorem and its implications for harmonic analysis on compact Lie groups.
  • Characters play a significant role in the Peter-Weyl theorem by allowing us to express continuous functions on compact Lie groups in terms of matrix coefficients from irreducible representations. The theorem states that these functions can be approximated using finite sums involving these matrix coefficients, thus linking harmonic analysis with representation theory. This connection enables deeper insights into the structure and behavior of functions defined on such groups.
• Evaluate the importance of characters in relation to irreducible representations, particularly in the context of SU(2) and SO(3).
  • Characters are critically important for understanding irreducible representations because they encapsulate essential features such as dimensions and how different representations relate to one another. In the context of SU(2) and SO(3), characters provide explicit calculations that show how these groups' geometrical aspects correspond with algebraic structures. This evaluation helps bridge abstract algebraic concepts with practical applications in physics and geometry, reinforcing why characters are a central focus in representation theory.

Related terms

Representation: A representation of a Lie group is a way to express group elements as linear transformations on a vector space.

Irreducible Representation: An irreducible representation is one that has no non-trivial invariant subspaces under the action of the group.

Peter-Weyl Theorem: A theorem that states every continuous function on a compact Lie group can be approximated in an L² sense by finite sums of matrix coefficients of the irreducible representations of the group.