Lie group representations

Definition

Lie group representations are a way to realize Lie groups as groups of transformations on vector spaces, allowing for the study of continuous symmetries in mathematical structures. By associating elements of a Lie group with linear transformations, these representations provide insights into the behavior of the group and its structure, especially in the context of finite groups where they help classify and understand the representations through characters and group actions.

5 Must Know Facts For Your Next Test

1. Lie group representations can be finite-dimensional or infinite-dimensional, depending on the nature of the Lie group being represented.
2. The classification of irreducible representations of a Lie group is closely linked to its root system and representation theory helps in understanding these irreducible components.
3. Characters play a crucial role in understanding Lie group representations, as they can simplify complex representation calculations through their orthogonality relations.
4. Every finite-dimensional representation of a compact Lie group is completely reducible, meaning it can be decomposed into a direct sum of irreducible representations.
5. The relationship between Lie groups and finite groups becomes significant when studying how finite groups can be seen as discrete subgroups of Lie groups.

Review Questions

• How do Lie group representations help in understanding the structure of finite groups?
  • Lie group representations allow us to view finite groups as subsets of Lie groups, providing tools to analyze their symmetries through linear transformations. By studying how elements from a finite group can be represented in vector spaces, we can gain insights into their structure, including the properties of their irreducible representations. This connection enhances our understanding of both finite groups and their potential embeddings into larger continuous symmetries.
• Discuss the role of characters in Lie group representations and their significance in representation theory.
  • Characters serve as crucial tools in representation theory by providing a way to capture essential information about a representation. They are defined as the traces of the corresponding linear transformations and enable us to identify and classify irreducible representations effectively. The orthogonality relations among characters facilitate calculations and lead to powerful results, such as the ability to decompose complex representations into simpler irreducible ones.
• Evaluate how the study of Lie group representations impacts both theoretical mathematics and practical applications in physics.
  • The study of Lie group representations bridges theoretical mathematics and practical applications in physics by enabling the understanding of symmetries in physical systems. In theoretical physics, symmetries governed by Lie groups underlie fundamental concepts such as conservation laws and particle interactions. This connection not only enriches mathematical theories but also has implications for formulating physical models, illustrating how abstract mathematical concepts manifest in real-world phenomena.

Related terms

Representation Theory: The study of how groups can be represented by linear transformations of vector spaces, providing a bridge between abstract algebra and linear algebra.

Character Theory: A subfield of representation theory that studies the traces of group representations, allowing for the classification of irreducible representations.

Finite Groups: Groups that have a finite number of elements, which can be studied using representation theory to analyze their structure and properties.