Based Loop Group

Definition

A based loop group is a mathematical structure that consists of loops based at a chosen point in a topological space, specifically focusing on the maps from the circle $S^1$ to a Lie group that are based at a given point. This concept is significant as it connects topology and algebra through the study of loop spaces and their symmetries, enabling the examination of central extensions and cohomology related to these groups.

5 Must Know Facts For Your Next Test

1. Based loop groups can be denoted as $LG_{x_0}$, where $x_0$ is the base point in the Lie group $G$.
2. The study of based loop groups often involves their algebraic structures, leading to insights about their representations and actions on various spaces.
3. In addition to being important in pure mathematics, based loop groups have applications in theoretical physics, particularly in string theory and gauge theory.
4. Based loop groups can also be related to infinite-dimensional Lie groups, allowing for deeper analysis in contexts like differential geometry.
5. The fundamental group of a based loop group is closely tied to the homotopy type of the space, providing valuable information about its topological structure.

Review Questions

• How do based loop groups relate to homotopy theory and what implications does this have for their structure?
  • Based loop groups are closely linked to homotopy theory since they represent continuous maps from the circle $S^1$ into a Lie group while preserving base points. This relationship allows us to analyze their topological properties and understand how they behave under deformation. The study of these groups often leads to insights about their fundamental groups and homotopy classes, enriching our understanding of both algebra and topology.
• Discuss how central extensions play a role in the study of based loop groups and provide an example.
  • Central extensions are significant when studying based loop groups because they help us understand how these groups can be constructed from simpler components. For example, if we take a based loop group $LG_{x_0}$, a central extension may arise when we consider an abelian group that sits inside it, impacting both its representation theory and cohomology. This interplay illustrates how central extensions can elucidate the deeper structures within loop groups.
• Evaluate the importance of based loop groups in theoretical physics and how they connect with other mathematical concepts.
  • Based loop groups hold substantial importance in theoretical physics, particularly in areas like string theory and gauge theory where symmetries and fields are analyzed. Their mathematical properties allow physicists to model physical phenomena using concepts from algebraic topology and geometry. This connection demonstrates how abstract mathematical ideas find applications in understanding fundamental physical theories, revealing intricate relationships between mathematics and physics.

Related terms

Loop Space: The space of all continuous maps from the circle $S^1$ into a topological space, capturing the idea of 'loops' in that space.

Central Extension:

A group extension where the kernel is in the center of the larger group, allowing for the study of how groups can be 'built' from simpler components.

Homotopy Theory: A branch of algebraic topology that studies spaces up to continuous deformation, playing a crucial role in understanding the properties of loop groups.