Loop Groups

Definition

Loop groups are mathematical structures that arise when considering the group of loops in a topological space, particularly in the context of Lie groups. They can be thought of as the group of smooth mappings from a circle into a Lie group, which are equipped with a natural group structure. This concept is essential for understanding various geometric realizations and representations, particularly in relation to the Borel-Weil theorem.

5 Must Know Facts For Your Next Test

1. Loop groups can be denoted as $LG$, where $G$ is a connected Lie group, capturing the structure of paths in the group space.
2. They play a key role in the theory of integrable systems and conformal field theory, linking geometry and physics.
3. The representation theory of loop groups reveals deep connections with modular forms and algebraic geometry.
4. Loop groups can be constructed from based loops, which have a distinguished base point in the circle, leading to a richer structure.
5. The Borel-Weil theorem applies directly to representations of loop groups by providing a geometric framework for understanding line bundles over complex projective spaces.

Review Questions

• How do loop groups relate to the structure of Lie groups and their representations?
  • Loop groups serve as an extension of Lie groups by allowing for the study of smooth mappings from circles into these groups. They provide a framework to analyze the representations of Lie groups through their infinite-dimensional counterparts. By exploring loop groups, one can gain insights into both the algebraic properties and geometric realizations of these structures.
• Discuss how the Borel-Weil theorem utilizes loop groups in understanding line bundles over projective varieties.
  • The Borel-Weil theorem leverages the properties of loop groups to establish a correspondence between line bundles on projective varieties and representations of algebraic groups. By considering loop groups, one can examine how these geometric objects interact with algebraic structures. This connection enhances the understanding of how line bundles can be constructed and characterized through their associated representation theories.
• Evaluate the implications of loop groups in integrable systems and conformal field theory, particularly regarding their applications.
  • Loop groups have significant implications in both integrable systems and conformal field theory by providing a means to study complex geometries and symmetries. In integrable systems, they help classify solutions and understand underlying structures that govern dynamics. In conformal field theory, they facilitate the exploration of vertex operator algebras and modular forms, showcasing how geometry informs physical theories and contributes to advancements in mathematical physics.

Related terms

Lie Group:

A Lie group is a group that is also a differentiable manifold, allowing for the study of algebraic properties in a geometric context.

Borel-Weil Theorem:

The Borel-Weil theorem provides a powerful correspondence between line bundles on projective varieties and representations of algebraic groups.

Kac-Moody Algebra:

Kac-Moody algebras are infinite-dimensional Lie algebras that generalize finite-dimensional semisimple Lie algebras and are associated with loop groups.