10.1 Affine Lie algebras and their representations

Affine Lie algebras expand finite-dimensional simple Lie algebras into infinite dimensions. They add a central extension and derivation, following Chevalley-Serre relations. These algebras are classified by extended Dynkin diagrams and have real and imaginary roots.

Highest weight modules are key representations of affine Lie algebras. They're built from a highest weight vector and have interesting properties like modular invariance of characters. This connects them to conformal field theory and string theory.

Affine Lie algebras

Definition and basic properties

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• Affine Lie algebras are infinite-dimensional Lie algebras that extend finite-dimensional simple Lie algebras by adding a central extension and a derivation
• The generators of an affine Lie algebra satisfy the Chevalley-Serre relations, which generalize the relations for finite-dimensional simple Lie algebras
• Affine Lie algebras are classified by the extended Dynkin diagrams, which are obtained from the Dynkin diagrams of finite-dimensional simple Lie algebras by adding an extra node (affine node)
• The Cartan subalgebra of an affine Lie algebra includes the Cartan subalgebra of the underlying finite-dimensional Lie algebra, the central element, and the derivation

Root system and Weyl group

• The root system of an affine Lie algebra consists of real roots, which are the roots of the underlying finite-dimensional Lie algebra, and imaginary roots, which are integer multiples of the null root $\delta$
• The imaginary roots form a lattice, called the root lattice, which is the integer span of the simple roots
• The Weyl group of an affine Lie algebra is generated by the reflections corresponding to the real roots and acts on the root lattice by affine transformations
• The affine Weyl group is the semidirect product of the finite Weyl group and the translation group of the root lattice ($\tilde{W} = W \ltimes Q^\vee$)

Highest weight modules

Construction and properties

• Highest weight modules are representations of affine Lie algebras that are generated by a highest weight vector, which is annihilated by the positive root spaces and is an eigenvector for the Cartan subalgebra
• The highest weight of a module determines its structure and is specified by a dominant integral weight of the underlying finite-dimensional Lie algebra and a level, which is the eigenvalue of the central element
• Verma modules are universal highest weight modules that are obtained by inducing from one-dimensional representations of the Borel subalgebra (positive root spaces and Cartan subalgebra)
• Irreducible highest weight modules are the unique simple quotients of Verma modules and are characterized by their highest weights

Characters and modular invariance

• The character of a highest weight module is a formal power series that encodes the multiplicities of the weight spaces and is a generating function for the dimensions of the graded components
• Characters of irreducible highest weight modules satisfy certain modular invariance properties, which relate them to the characters of the irreducible representations of the underlying finite-dimensional Lie algebra
• The modular invariance of characters is a consequence of the Weyl-Kac character formula, which expresses the characters in terms of the Weyl group and the Dedekind eta function
• The modular invariance of characters plays a crucial role in the study of affine Lie algebras and their connections to conformal field theory and string theory

Central extensions in affine Lie algebras

Construction and properties

• Central extensions of loop algebras are necessary to obtain non-trivial representations of affine Lie algebras, as the loop algebras themselves have no finite-dimensional representations
• The central extension is determined by a two-cocycle, which is a bilinear form on the loop algebra satisfying certain cohomological conditions (cocycle condition and invariance under the adjoint action)
• The central element acts as a scalar multiple of the identity on any representation of the affine Lie algebra, and the scalar is called the level of the representation
• The level of a representation determines the central charge of the Virasoro algebra, which acts on the representation space via the Sugawara construction

Importance in representation theory

• The central extension is crucial for the existence of non-trivial representations of affine Lie algebras, as it allows for the construction of highest weight modules
• The level of a representation plays a key role in the classification of irreducible highest weight modules, as it determines the possible highest weights and the structure of the module
• The central charge of the Virasoro algebra, which is related to the level by the Sugawara construction, determines the conformal properties of the representation and its connection to conformal field theory
• The study of representations of affine Lie algebras has important applications in various areas of mathematics and physics, such as conformal field theory, integrable systems, and quantum groups

Affine Lie algebras vs loop algebras

Relation between the two

• Affine Lie algebras can be constructed as central extensions of loop algebras, which are the algebras of smooth maps from the circle to a finite-dimensional simple Lie algebra
• The loop algebra has a natural basis consisting of the Fourier modes of the generators of the finite-dimensional Lie algebra, and its commutation relations are determined by the commutation relations of the finite-dimensional Lie algebra and the Lie bracket of Laurent polynomials
• The central extension of the loop algebra is obtained by adding a central element and modifying the commutation relations by a term proportional to the two-cocycle, which is a bilinear form on the loop algebra satisfying certain properties
• The resulting affine Lie algebra is a non-trivial extension of the loop algebra, and its representations are closely related to the representations of the Virasoro algebra and the conformal properties of the underlying finite-dimensional Lie algebra

Untwisted and twisted cases

• Untwisted affine Lie algebras correspond to the case where the loop algebra is the tensor product of a finite-dimensional simple Lie algebra with the algebra of Laurent polynomials ($\mathfrak{g} \otimes \mathbb{C}[t, t^{-1}]$)
• In the untwisted case, the affine Lie algebra is simply-laced and has a symmetric generalized Cartan matrix, which is obtained from the Cartan matrix of the finite-dimensional Lie algebra by adding an extra row and column corresponding to the affine root
• Twisted affine Lie algebras arise from loop algebras with a non-trivial automorphism of the underlying finite-dimensional Lie algebra, which is used to twist the multiplication in the loop algebra
• The twisted affine Lie algebras are non-simply-laced and have a non-symmetric generalized Cartan matrix, which is obtained from the Cartan matrix of the orbit Lie algebra (the fixed-point subalgebra under the automorphism) by a folding procedure
• The classification of affine Lie algebras includes both the untwisted and twisted cases, and the latter are related to the outer automorphisms of the finite-dimensional simple Lie algebras, which are classified by the symmetries of the Dynkin diagrams