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
Key Terms to Review (16)
Affine Lie Algebra: An affine Lie algebra is a type of infinite-dimensional Lie algebra that extends a finite-dimensional simple Lie algebra by introducing a new central element and incorporating functions on the circle. This concept is crucial in understanding representations of these algebras and their connections to loop groups, where the central extension leads to interesting topological and algebraic properties.
Based Loop Group: 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.
Cartan Subalgebra: A Cartan subalgebra is a maximal abelian subalgebra of a Lie algebra, which plays a crucial role in the structure and classification of Lie algebras. It provides a way to decompose the algebra into simpler components, making it easier to study the representations and their properties. Cartan subalgebras are intimately connected with roots and weights, helping to understand the underlying symmetry of the algebra and its representations.
Central Extension: A central extension is a type of group extension where a given group is extended by an abelian group in such a way that the abelian group lies in the center of the resulting group. This concept is crucial in understanding the structures of loop groups and affine Lie algebras, as it helps characterize representations and their behaviors. Central extensions provide insights into how these mathematical structures can be built upon simpler groups, revealing deeper symmetries and properties.
Cocycle: A cocycle is a function that assigns a value to each pair of elements in a group or algebra, satisfying certain properties that make it useful in the study of cohomology and central extensions. Cocycles are key in describing how certain algebraic structures, such as loop groups, can have additional layers of complexity through their central extensions. This concept helps in understanding how different mathematical entities relate to each other and contribute to the broader framework of topology and algebra.
Current algebra: Current algebra is a mathematical framework used to study the symmetries and conservation laws in quantum field theory, primarily through the use of operators associated with conserved currents. It connects algebraic structures, such as Lie algebras, to the physical concepts of symmetries and charges. This framework plays a significant role in understanding loop groups and their central extensions, as well as providing a foundation for exploring representations and cohomological properties within these areas.
Free Loop Group: 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.
Gerbe: A gerbe is a mathematical structure that generalizes the concept of a bundle, particularly in the context of higher category theory and topology. It is used to encode data in a way that allows for the study of more complex geometrical and topological properties, especially in relation to loop groups and their central extensions, where they help in understanding cohomology and descent theory.
Homotopy: Homotopy is a concept in topology that describes when two continuous functions can be continuously transformed into each other. This idea is crucial for understanding how spaces can be connected and how shapes can be deformed without tearing or gluing, linking to essential concepts in group theory and geometry.
Kac-Moody Algebra: A Kac-Moody algebra is a type of infinite-dimensional Lie algebra that generalizes finite-dimensional semisimple Lie algebras and is characterized by a generalized Cartan matrix. These algebras have applications in various fields such as representation theory, string theory, and conformal field theory, and their structure is closely related to root systems and their properties.
Kazhdan-Lusztig Theory: Kazhdan-Lusztig theory is a branch of representation theory focused on the study of certain algebraic structures associated with semisimple Lie algebras and their representations. It primarily explores the relationship between highest weight modules, especially Verma modules, and the representation theory of loop groups, providing deep insights into the structure of these algebraic entities through the construction of a basis for the homology of the associated varieties.
Mikhail Kac: Mikhail Kac is a mathematician known for his significant contributions to the study of loop groups and their central extensions. His work, particularly in the realm of Lie algebras and algebraic structures associated with these groups, has helped to deepen the understanding of how central extensions can arise and be classified within the context of loop groups. Kac's insights have been foundational in connecting representation theory with geometric concepts in mathematics.
Root System: A root system is a set of vectors in a Euclidean space that encodes the structure of a semisimple Lie algebra, reflecting the symmetries and relationships between its elements. These vectors, known as roots, help define the algebra's representation theory and play a crucial role in classifying the algebra's structure and properties.
Unitary: In mathematics, particularly in the context of linear algebra and group theory, a unitary transformation is a linear transformation that preserves the inner product. This means that the transformation maintains the length of vectors and the angles between them, which is crucial in quantum mechanics and various mathematical frameworks involving symmetry.
Vladimir Drinfeld: Vladimir Drinfeld is a prominent mathematician known for his groundbreaking work in algebra, particularly in the fields of quantum groups and the theory of loop groups. His contributions have significantly shaped modern mathematics, leading to advancements in areas such as representation theory and mathematical physics, particularly through the development of new algebraic structures and frameworks that connect different mathematical concepts.
Wakimoto Module: A Wakimoto module is a type of representation associated with affine Lie algebras, constructed from the underlying structure of the affine algebra and a certain kind of loop algebra. These modules play a crucial role in the study of representations of affine Lie algebras, particularly in understanding their characters and the theory of vertex operator algebras. Wakimoto modules provide a bridge between finite-dimensional representations and infinite-dimensional representations, revealing deep connections between geometry and representation theory.