Admissible module

Definition

An admissible module is a type of representation of a Lie algebra that satisfies certain finiteness conditions, making it suitable for studying the structure and representation theory of the algebra. This concept plays a crucial role in the theory of highest weight modules, as admissible modules can be thought of as the building blocks from which more complex representations are constructed. They typically arise in the context of Verma modules, which are specific types of highest weight modules associated with a particular weight and are essential for understanding the broader classification of representations.

5 Must Know Facts For Your Next Test

1. Admissible modules are often defined in terms of their relation to finitely generated representations, ensuring they have a well-defined structure and properties.
2. These modules are closely linked to the concept of integrability, meaning they exhibit behaviors similar to finite-dimensional representations.
3. Admissible modules can be constructed from Verma modules by considering their submodules and quotient structures.
4. The category of admissible modules is important for understanding the representation theory of both semisimple and solvable Lie algebras.
5. In many cases, admissible modules help determine the classification and decomposition of representations, leading to deeper insights into the algebra's structure.

Review Questions

• How do admissible modules relate to highest weight modules and Verma modules within representation theory?
  • Admissible modules serve as an essential foundation for highest weight modules, as they satisfy specific finiteness conditions that make them well-suited for representation theory. Verma modules are a particular instance of highest weight modules and can often be transformed into admissible modules by exploring their submodules. This relationship allows researchers to understand how different representations can be constructed and classified based on their underlying structures.
• Discuss the significance of integrability in the context of admissible modules and how it affects their representation properties.
  • Integrability is crucial for admissible modules because it ensures that these modules behave similarly to finite-dimensional representations. When an admissible module is integrable, it indicates that it has a well-defined action under the Lie algebra that allows for effective manipulation and analysis within the broader representation theory framework. This property plays an important role in classifying representations and understanding their relationships with one another.
• Evaluate the impact of admissible modules on the classification and decomposition of representations in Lie algebra theory.
  • Admissible modules significantly influence the classification and decomposition of representations by providing a structured approach to understanding how these representations relate to one another. They help researchers analyze how higher-dimensional or more complex representations can be constructed from simpler building blocks, such as Verma modules. This process not only facilitates a clearer classification scheme but also uncovers deep connections within the representation theory, enhancing our overall understanding of Lie algebras and their applications.

Related terms

Highest Weight Module: A representation of a Lie algebra characterized by a highest weight vector, which serves as a foundational element in constructing other representations.

Verma Module: A specific type of highest weight module that is induced from a one-dimensional representation of a Borel subalgebra, often used to study the category of representations.

Weight Space: A subspace of a representation that corresponds to a specific eigenvalue (weight) under the action of a Cartan subalgebra, crucial for analyzing the structure of representations.