Extension Problems

Definition

Extension problems refer to questions in homological algebra that investigate the conditions under which a given module can be extended by another module. In the context of Lie algebra cohomology, these problems involve examining how modules over Lie algebras can be constructed from simpler modules, emphasizing the relationships between different representations of the algebra and their cohomological properties.

5 Must Know Facts For Your Next Test

1. Extension problems in Lie algebra cohomology can be framed using derived functors, which measure how far a module is from being projective.
2. In this context, if a module can be extended by another, it implies that there exists a non-trivial short exact sequence involving those modules.
3. These problems often relate to group cohomology through the universal coefficient theorem, linking different algebraic structures.
4. The classification of extensions can provide insight into the structure of Lie algebras and their representations, revealing important properties such as solvability and semi-simplicity.
5. Computing extensions requires understanding how elements in cohomology groups can represent equivalence classes of extensions, leading to significant implications in representation theory.

Review Questions

• How do extension problems in Lie algebra cohomology relate to the concepts of projective and injective modules?
  • Extension problems often deal with whether a module can be constructed from others through extensions. Projective modules allow for splitting exact sequences, meaning they can be used to simplify extension problems. On the other hand, injective modules provide conditions under which any module can be extended, which is crucial when determining how representations behave within a Lie algebra framework.
• Discuss how derived functors are utilized in solving extension problems related to Lie algebra cohomology.
  • Derived functors play a critical role in understanding extension problems as they help measure how far a module is from being projective or injective. Specifically, when computing cohomology groups for a given Lie algebra, derived functors reveal essential information about possible extensions by capturing the complexities of module interactions. This allows for a deeper exploration of representation theory and how certain modules may interact under various conditions.
• Evaluate the significance of extension problems in understanding the structure and representation theory of Lie algebras.
  • Extension problems are vital for analyzing the structure and representation theory of Lie algebras as they help classify how different representations can be formed. By investigating these extensions, mathematicians gain insights into solvable and semi-simple Lie algebras' properties. This classification aids in understanding symmetry in various mathematical contexts and leads to deeper connections with other areas such as geometry and topology, influencing both theoretical developments and practical applications.

Related terms

Cohomology: A mathematical tool used to study and classify topological spaces and algebraic structures by associating a sequence of abelian groups or modules to them.

Exact Sequence: A sequence of algebraic objects and morphisms between them where the image of one morphism equals the kernel of the next, revealing properties about extensions and relationships between modules.

Lie Algebra: An algebraic structure that studies the properties of Lie brackets, which capture the concept of infinitesimal transformations, crucial for understanding symmetry in mathematical physics.