Free Loop Group

Definition

A free loop group is the group of all smooth maps from the circle $S^1$ into a given manifold, typically denoted as $L(G)$ when referring to a Lie group $G$. These maps are considered as loops based on the topology of the circle, and the group operation is defined pointwise. This structure is fundamental in understanding various aspects of topology and geometry, particularly when exploring central extensions and their implications in both algebraic and geometric contexts.

5 Must Know Facts For Your Next Test

1. The free loop group can be viewed as an infinite-dimensional manifold, which highlights its rich geometric structure compared to finite-dimensional groups.
2. It plays a crucial role in the study of gauge theories and string theory, where loops represent fundamental aspects of physical models.
3. Free loop groups are closely linked to the notion of symplectic geometry, especially in the context of Hamiltonian dynamics and integrable systems.
4. Central extensions of free loop groups often arise when examining their representations and quantization in mathematical physics.
5. In the context of homotopy theory, free loop groups serve as important tools for understanding string topology and related invariants.

Review Questions

• How does the structure of a free loop group relate to its interpretation as an infinite-dimensional manifold?
  • The structure of a free loop group as an infinite-dimensional manifold arises from considering smooth maps from $S^1$ into a manifold $G$. This allows for an understanding of loops as points in this manifold, where each loop can be differentiated and analyzed with respect to its properties. The infinite-dimensional aspect introduces complexities not present in finite-dimensional groups, leading to rich geometrical structures and interactions with other areas of mathematics.
• Discuss how central extensions of free loop groups can influence their representations and physical interpretations.
  • Central extensions of free loop groups are significant because they provide a framework to study representations that may not be captured by the group itself. These extensions often allow for new symmetries and gauge transformations that are crucial in physical theories like string theory or quantum mechanics. By examining these extensions, mathematicians can uncover deeper relationships between different types of symmetries and the underlying geometry of the associated manifolds.
• Evaluate the implications of using free loop groups in homotopy theory and their relevance to modern mathematical physics.
  • Using free loop groups in homotopy theory has far-reaching implications for understanding various topological properties through the lens of loops and their deformations. This evaluation is particularly relevant in modern mathematical physics, as it helps bridge concepts from algebraic topology with physical models, such as string theory. By studying these loops, researchers can derive important invariants and establish connections between different branches of mathematics, enhancing our grasp of complex systems and their underlying structures.

Related terms

Loop Space: The space of all continuous maps from a topological space into itself, often used in algebraic topology to study homotopy.

Central Extension:

An extension of a group that fits into a specific sequence where the kernel is in the center of the larger group, playing a crucial role in classifying loop groups.

Homotopy:

A concept in topology that studies the properties of spaces that are preserved under continuous deformations, helping analyze loops within free loop groups.