10.3 Kac-Moody algebras and their structure

Kac-Moody algebras are infinite-dimensional Lie algebras that extend finite-dimensional semisimple Lie algebras. They're defined by generators and relations encoded in a generalized Cartan matrix, and come in three types: finite-dimensional, affine, and indefinite.

These algebras have a triangular decomposition and a Weyl group crucial to their structure. Their representation theory, especially for affine types, connects to conformal field theory and has applications in physics and mathematics.

Kac-Moody Algebras

Definition and Generalization

Top images from around the web for Definition and Generalization

• 

  File:DynkinC5Affine.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  Cartan matrix - Wikipedia View original

  Is this image relevant?

• 

  File:DynkinC4Affine.svg - Wikipedia View original

  Is this image relevant?

• 

  File:DynkinC5Affine.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  Cartan matrix - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Generalization

• 

  File:DynkinC5Affine.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  Cartan matrix - Wikipedia View original

  Is this image relevant?

• 

  File:DynkinC4Affine.svg - Wikipedia View original

  Is this image relevant?

• 

  File:DynkinC5Affine.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  Cartan matrix - Wikipedia View original

  Is this image relevant?

1 of 3

• Kac-Moody algebras are infinite-dimensional Lie algebras that generalize the finite-dimensional semisimple Lie algebras ($\mathfrak{sl}_n, \mathfrak{so}_n, \mathfrak{sp}_n$)
• Defined by generators and relations encoded in a generalized Cartan matrix
• Three main types: finite-dimensional, affine, and indefinite
  • Finite-dimensional Kac-Moody algebras are the classical semisimple Lie algebras
  • Affine Kac-Moody algebras are associated with affine Dynkin diagrams ($\tilde{A}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$)
  • Indefinite Kac-Moody algebras are associated with indefinite Cartan matrices and can be further classified as hyperbolic or Lorentzian
• Borcherds algebras, also known as generalized Kac-Moody algebras, allow for imaginary simple roots and have applications in mathematical physics (string theory, moonshine)

Structure and Properties

• Kac-Moody algebras have a triangular decomposition $\mathfrak{g} = \mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$
  • $\mathfrak{h}$ is the Cartan subalgebra, and $\mathfrak{n}_\pm$ are the positive and negative root spaces
• The Weyl group of a Kac-Moody algebra is generated by reflections associated with the simple roots and plays a crucial role in the structure and representation theory
• The Weyl-Kac denominator formula relates the denominator of the Weyl character formula to the root system and the Weyl group
• The Virasoro algebra is a central extension of the Witt algebra and plays a fundamental role in the representation theory of affine Kac-Moody algebras and conformal field theory

Classifying Kac-Moody Algebras

Dynkin Diagrams and Cartan Matrices

• Dynkin diagrams are graphical representations of the generalized Cartan matrix associated with a Kac-Moody algebra
  • Nodes represent simple roots, and edges represent inner products between simple roots
• Cartan matrices encode the inner products between the simple roots of a Kac-Moody algebra
  • A generalized Cartan matrix $A = (a_{ij})$ satisfies $a_{ii} = 2$, $a_{ij} \leq 0$ for $i \neq j$, and $a_{ij} = 0$ if and only if $a_{ji} = 0$
  • The type of Kac-Moody algebra (finite, affine, or indefinite) is determined by the determinant and principal minors of the Cartan matrix
• Affine Dynkin diagrams are obtained from finite Dynkin diagrams by adding a node and correspond to affine Kac-Moody algebras ($\tilde{A}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$)
• Hyperbolic and Lorentzian Kac-Moody algebras are associated with specific types of indefinite Cartan matrices and Dynkin diagrams

Examples and Classification

• The affine Kac-Moody algebra $\tilde{\mathfrak{sl}}_2$ has a Dynkin diagram with two nodes connected by a double edge
• The hyperbolic Kac-Moody algebra $\mathfrak{e}_{10}$ is associated with the Dynkin diagram obtained by extending the $E_8$ Dynkin diagram with two additional nodes
• The monster Lie algebra, a Borcherds algebra, is related to the monster group and has applications in moonshine and string theory
• Kac-Moody algebras can be classified using the Cartan matrix and the associated Dynkin diagram, which determine the type (finite, affine, hyperbolic, or Lorentzian) and the structure of the algebra

Representation Theory of Kac-Moody Algebras

Highest Weight Representations and Characters

• Representations of Kac-Moody algebras are vector spaces on which the algebra acts as linear transformations
• Highest weight representations are generated by a highest weight vector, which is annihilated by the positive root generators of the algebra
  • The highest weight determines the structure and properties of the representation
• The character of a representation encodes important information, such as its dimension and weight space decomposition
• The Weyl-Kac character formula expresses the characters of irreducible highest weight representations in terms of the Weyl group and the highest weight

Integrable Representations and Applications

• Integrable representations are a special class of representations with desirable properties, such as complete reducibility
• The representation theory of affine Kac-Moody algebras has connections to conformal field theory and the theory of vertex operator algebras
  • Integrable highest weight representations of affine Kac-Moody algebras correspond to primary fields in conformal field theory
  • The characters of these representations are modular forms and satisfy certain differential equations (Knizhnik-Zamolodchikov equations)
• Representations of Kac-Moody algebras have applications in various areas of mathematics and physics, such as:
  • Conformal field theory and string theory
  • Quantum groups and knot invariants
  • Modular forms and moonshine

Kac-Moody vs Affine Lie Algebras

Relationship between Kac-Moody and Affine Lie Algebras

• Affine Lie algebras are a special class of infinite-dimensional Lie algebras closely related to affine Kac-Moody algebras
• An affine Lie algebra is the central extension of a loop algebra, which is the algebra of polynomial maps from the circle to a finite-dimensional simple Lie algebra
• Untwisted affine Lie algebras correspond to affine Kac-Moody algebras associated with the extended Dynkin diagrams of the finite-dimensional simple Lie algebras
• Twisted affine Lie algebras are obtained by considering twisted loop algebras, which involve automorphisms of the underlying finite-dimensional simple Lie algebra

Representation Theory and Applications

• The representation theory of affine Lie algebras is closely related to that of affine Kac-Moody algebras
  • Highest weight representations and characters play a central role in both theories
• Affine Lie algebras have applications in various areas, such as:
  • Conformal field theory and vertex operator algebras
  • Integrable systems in mathematical physics (Korteweg-de Vries equation, Toda lattice)
  • Representation theory of quantum groups and knot invariants (Reshetikhin-Turaev invariants)
• The study of affine Lie algebras and their representations has led to important developments in mathematics and physics, such as:
  • The Sugawara construction, which relates the energy-momentum tensor in conformal field theory to the currents of an affine Lie algebra
  • The Knizhnik-Zamolodchikov equations, which are differential equations satisfied by the correlation functions in conformal field theory
  • The Wess-Zumino-Witten model, a conformal field theory based on affine Lie algebras with applications in string theory and condensed matter physics