Discrete center

Definition

The discrete center of a group is the set of elements that commute with every element of the group, forming a subgroup that is discrete in the topology of the group. This concept is particularly relevant in understanding the structure of compact Lie groups and their representations, as it helps to identify how the group's center behaves and its implications for the group's representations.

5 Must Know Facts For Your Next Test

1. The discrete center of a compact Lie group consists of all elements that commute with every other element in the group, which can be finite in size.
2. In compact Lie groups, the discrete center is always a normal subgroup, which means it is invariant under conjugation by any element of the group.
3. The presence of a non-trivial discrete center can affect the character theory of representations of the group by influencing how characters are defined.
4. Understanding the discrete center helps in analyzing the irreducible representations of compact Lie groups, as it provides crucial information about how these representations decompose.
5. For connected compact Lie groups, the discrete center is usually related to the topology of the group and plays a significant role in identifying the possible representations.

Review Questions

• How does the discrete center influence the representation theory of compact Lie groups?
  • The discrete center plays a key role in representation theory by affecting how irreducible representations decompose. Since the discrete center consists of elements that commute with all other elements, it can impact the characters defined for these representations. Specifically, it can lead to multiplicities in representations and help determine how different representations relate to one another.
• Discuss the significance of normality in relation to the discrete center within compact Lie groups.
  • The fact that the discrete center is a normal subgroup within compact Lie groups is significant because it ensures that all elements of the group commute with those in the center when performing conjugation. This property allows for a consistent way to analyze how these elements interact and makes it easier to study quotient groups formed by factoring out the center. It also implies that all irreducible representations can be analyzed while considering this normal structure.
• Evaluate how understanding the discrete center can aid in classifying compact Lie groups and their representations.
  • Understanding the discrete center is crucial for classifying compact Lie groups and their representations because it provides essential insights into their structure. By identifying elements that commute across the group, we gain a clearer picture of how these groups behave under various operations and transformations. This knowledge facilitates categorizing different types of groups based on their centers and further allows us to classify their representations according to how they relate to these central elements, thus simplifying our analysis of their overall structure.

Related terms

Lie Group: A Lie group is a group that is also a smooth manifold, allowing for calculus to be applied to its structure and operations.

Compactness: A property of a space where every open cover has a finite subcover, which in the context of Lie groups ensures that the group can be studied using finite-dimensional representations.

Representation Theory: Representation theory studies how groups can act on vector spaces, often providing insights into their structure through linear transformations.