Loop groups are infinite-dimensional Lie groups of smooth maps from a circle to a finite-dimensional Lie group. They inherit properties from their underlying group and have applications in physics and mathematics.
Central extensions of loop groups create larger groups with additional structure. The most common is the Kac-Moody group, whose Lie algebra is an . These extensions are crucial in conformal field theory and integrable systems.
Loop Groups and Their Properties
Definition and Basic Structure
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Loop groups are infinite-dimensional Lie groups consisting of smooth maps from the circle S1 to a finite-dimensional Lie group G
The group operation in a loop group is point-wise multiplication of the maps
(fg)(θ)=f(θ)g(θ) for all θ∈S1, where f,g are elements of the loop group
Loop groups inherit many properties from their underlying finite-dimensional Lie group
Connectedness
Compactness
Semisimplicity
Loop Algebras and Constructions
The Lie algebra of a loop group, called a loop algebra, consists of smooth maps from S1 to the Lie algebra of G
Loop groups can be constructed from various finite-dimensional Lie groups
group U(n)
Special unitary group SU(n)
Orthogonal group O(n)
Examples of loop groups include the loop group of SU(2), denoted as LSU(2), and the loop group of U(1), denoted as LU(1)
Central Extensions of Loop Groups
Motivation and Definition
Central extensions of loop groups are new loop groups that contain the original loop group as a quotient by a central subgroup
Central extensions allow for the construction of new, larger loop groups with additional structure and properties
The most common of a loop group is the Kac-Moody group
Kac-Moody group is a central extension by the circle group S1
Affine Lie Algebras and Cocycles
The Lie algebra of a Kac-Moody group is an affine Lie algebra
Affine Lie algebra is a central extension of the loop algebra by a one-dimensional center
Central extensions can be classified by the second cohomology group H2(G,S1), where G is the original loop group
The construction of central extensions involves the use of cocycles
Cocycles are maps satisfying certain properties that define the extension
An example of a is the Kac-Moody cocycle, which defines the central extension of a loop algebra to an affine Lie algebra
Loop Groups vs Affine Lie Algebras
Relationship and Correspondence
Affine Lie algebras are infinite-dimensional Lie algebras that are central extensions of loop algebras
The Kac-Moody group, which is a central extension of a loop group, has an affine Lie algebra as its Lie algebra
The highest weight representations of affine Lie algebras correspond to the positive energy representations of loop groups
Representation Theory and Applications
Affine Lie algebras have a rich representation theory that is closely related to the representation theory of loop groups
Affine Lie algebras have important applications in various areas
Conformal field theory
Study of critical phenomena in statistical mechanics
Examples of affine Lie algebras include the affine Kac-Moody algebras su^(2) and su^(3), which are central extensions of the loop algebras of SU(2) and SU(3), respectively
Applications of Loop Groups in Physics
Conformal Field Theory and String Theory
Loop groups and their central extensions have numerous applications in mathematical physics
In conformal field theory, loop groups and affine Lie algebras are used to construct the Wess-Zumino-Witten (WZW) model
WZW model describes the propagation of strings on group manifolds
The representation theory of loop groups and affine Lie algebras plays a crucial role in the classification and study of conformal field theories
In string theory, loop groups arise naturally in the description of closed strings propagating on group manifolds
Central extensions of loop groups are related to the anomalies that appear in the quantization of these strings
Integrable Systems
Loop groups and affine Lie algebras also appear in the study of integrable systems
Examples of integrable systems include
Korteweg-de Vries (KdV) equation
Sine-Gordon equation
Loop groups and affine Lie algebras provide a framework for constructing and classifying solutions to these integrable systems
The representation theory of loop groups and affine Lie algebras is used to construct soliton solutions and study the integrability of these systems