Verma module

Definition

A Verma module is a specific type of module constructed from a highest weight representation of a Lie algebra, defined using the universal enveloping algebra. It is generated by the action of the algebra on a highest weight vector, which makes it essential in understanding representations of semisimple Lie algebras. These modules are crucial for studying irreducible representations, as they provide a systematic way to build them and analyze their properties through highest weight theory.

5 Must Know Facts For Your Next Test

1. Verma modules are not always irreducible; they can contain submodules that are important in studying the structure of representations.
2. Each Verma module is indexed by a highest weight, and the structure of these modules can often be described using weights associated with the underlying Lie algebra.
3. The process of taking a Verma module and factoring out certain submodules leads to irreducible representations, highlighting the connection between Verma modules and higher weight theory.
4. The characters of Verma modules can be computed using formulas involving Weyl groups, connecting them to the broader framework of representation theory.
5. Verma modules can sometimes be realized as spaces of functions on certain geometric objects, providing a bridge between algebra and geometry.

Review Questions

• How do Verma modules relate to the construction of irreducible representations?
  • Verma modules serve as foundational building blocks in the study of irreducible representations. By starting with a highest weight vector, one can construct a Verma module, which may not be irreducible. Factoring out certain submodules from a Verma module often yields an irreducible representation, demonstrating how Verma modules help identify and analyze these fundamental components in representation theory.
• Discuss the significance of highest weight vectors in the context of Verma modules and how they affect their structure.
  • Highest weight vectors are central to the structure of Verma modules as they define the representation's starting point. The action of the Lie algebra on these vectors generates the entire module. The nature of the highest weight vector determines the properties of the module, including its composition series and potential reducibility. This highlights how key features like weight play critical roles in determining both the structure and classification of representations.
• Evaluate the role of Verma modules in connecting representation theory with geometric concepts, particularly through their realizations as function spaces.
  • Verma modules bridge representation theory and geometry by being realizable as spaces of functions on geometric objects like flag varieties or symmetric spaces. This connection provides insights into both fields, allowing mathematicians to leverage techniques from geometry to understand representations better. By interpreting these modules in a geometric context, one can explore deeper structural properties and relationships within representation theory, showcasing their significance beyond pure algebraic definitions.

Related terms

Highest weight representation: A representation of a Lie algebra where there exists a vector such that all other vectors can be generated from it by the action of the algebra, and it is annihilated by certain elements.

Universal enveloping algebra: An associative algebra that allows for the representation of Lie algebras, capturing their structure in a way that makes them easier to work with.

Irreducible representation: A representation that has no nontrivial invariant subspaces, meaning it cannot be decomposed into simpler components.