Loop AlgebraEdit

Loop algebra is a classical construction in the theory of Lie algebras that takes a finite-dimensional object and enlarges it to an infinite-dimensional setting by allowing maps from the circle into the original algebra. In its most common form, one starts with a finite-dimensional Lie algebra g over a field (often the complex numbers) and forms the space of smooth maps from the circle S^1 into g, equipped with a pointwise Lie bracket. This simple idea yields a rich algebraic structure that sits at the crossroads of algebra, geometry, and mathematical physics.

The loop algebra is not merely a curiosity of abstraction. It provides the algebraic backbone for a family of infinite-dimensional objects that arise in representation theory and in the study of two-dimensional quantum field theories. It also has a close relationship to loop groups, current algebras, and the broader framework of Kac–Moody algebras. The standard notation Lg is used for the loop algebra of g, and many texts note its isomorphism with the tensor product g ⊗ Laurent polynomial ring C[t,t^{-1}], where the bracket is extended by bilinearity and the rule [x ⊗ t^m, y ⊗ t^n] = [x,y] ⊗ t^{m+n} for x,y ∈ g and integers m,n.

Definition and construction

Untwisted loop algebra. Given a finite-dimensional Lie algebra Lie algebra g over a field, the untwisted loop algebra Lg consists of all smooth maps f: S^1 → g with the Lie bracket defined pointwise: f1, f2 = [f1(θ), f2(θ)]. Since S^1 is a compact one-dimensional manifold, this construction yields an infinite-dimensional Lie algebra that encodes the original algebra g in a time-parametrized way. In many algebraic treatments, one uses the algebra of Laurent polynomial loops, identifying Lg with g ⊗ C[t,t^{-1}], where t is a formal parameter on the circle and the bracket is induced from the bracket on g: [x ⊗ t^m, y ⊗ t^n] = [x,y] ⊗ t^{m+n}. This algebraic realization emphasizes the role of the circle as the parameter space and connects to Laurent polynomial ring theory.
Twisted loop algebras. If g has an automorphism σ of finite order N, one can form a twisted loop algebra by restricting attention to loops f: S^1 → g that satisfy f(e^{2π i} θ) = σ(f(θ)). This yields a variant often denoted Lg(σ). Twisted loop algebras are important in the full landscape of infinite-dimensional Lie algebras and lead to certain families of Kac-Moody algebras that are not obtained from the untwisted construction.
Relations to current algebras. The loop algebra Lg is sometimes described as the current algebra associated with the Lie group corresponding to g, because its elements can be viewed as g-valued currents varying along the circle. In the language of representation theory, one studies modules over Lg and its central extensions, which leads naturally to the affine world.

Extensions and affine algebras

A central theme is that loop algebras admit natural central extensions. By adding a one-dimensional center and a derivation, one obtains the untwisted affine Lie algebras, often written as ĝ or ĝ, which play a central role in both mathematics and theoretical physics. The central extension is defined by a 2-cocycle that is typically built from the Killing form on g, and the derivation D encodes the degree or level, effectively measuring how many times we wind around the circle in a loop.

Central extension. The central element K commutes with all loops and encodes the level in representation theory. The extended bracket takes the form [x ⊗ t^m, y ⊗ t^n] = [x,y] ⊗ t^{m+n} + m δ_{m+n,0} ⟨x,y⟩ K, where ⟨·,·⟩ is an invariant form on g and δ is the Kronecker delta. This central term reflects the nontrivial topology of the circle and is essential for many representation-theoretic phenomena.
Derivation. The derivation D acts as t d/dt on the Laurent polynomial model, serving as a grading operator. It satisfies [D, x ⊗ t^n] = n x ⊗ t^n and helps organize representations into weight spaces of integer level or grade.

Affine Lie algebras (often called affine Kac–Moody algebras in older literature) arise as these central extensions of loop algebras together with the derivation. They provide a robust framework for studying infinite-dimensional representations and have applications ranging from number theory to string theory.

Twisted affine algebras. The twisted loop algebras lead to twisted affine Lie algebras, which are the central extensions of Lg(σ). These objects complete the classification of affine algebras and are indexed by the pair (g, σ). See Affine Lie algebra and Kac–Moody algebra for broader context.

Representations and structure

Representations of loop algebras and their affine extensions exhibit rich structures. Key themes include:

Evaluation representations. For a fixed point a ∈ C^×, the evaluation map ev_a: Lg → g sends f(θ) to f(a). If V is a finite-dimensional representation of g, composing with ev_a yields a finite-dimensional representation of Lg. These evaluation representations form a fundamental class of representations for loop algebras and their untwisted affine extensions.
Integrable representations. When passing to the affine algebra ĝ, one studies integrable highest-weight representations, characterized by finite-dimensional weight spaces and well-behaved actions of the central element and the derivation. The highest-weight theory parallels that for finite-dimensional semisimple Lie algebras but with new features arising from the infinite-dimensional setting.
Connection to vertex algebras and conformal field theory. Representations of affine Lie algebras interface with vertex algebras and play a central role in the algebraic formulation of two-dimensional conformal field theories, such as the Wess–Zumino–Witten model and related constructions. The Sugawara construction provides a way to obtain a Virasoro algebra action from a given affine algebra representation, linking symmetry algebras to the energy-mpectrum structure of models.

Examples and intuition

The sl2 example. Take g = sl2. The loop algebra consists of elements E ⊗ t^n, F ⊗ t^n, H ⊗ t^n for n ∈ Z, with the standard sl2 bracket extended linearly. The corresponding untwisted affine algebra sl2̂ adds a central element K and a derivation D. This family serves as a concrete testing ground for representation theory, including level k modules and their characters, which connect to modular forms and partition functions in physics.
Physical intuition. In two-dimensional models, the loop algebra encodes conserved currents along a spatial circle. The central extension corresponds to a quantum anomaly that shifts how charges and energy are counted, a feature that is essential in many exactly solvable systems. The mathematical structure thus reflects fundamental symmetry patterns that appear in diverse physical settings.

Historical notes

The study of loop algebras and their central extensions matured in the late 20th century as part of the development of infinite-dimensional Lie theory. The fundamental insight that loop algebras admit meaningful central extensions leading to affine Lie algebras is attributed to the combined work of researchers including Victor G. Kac and Robert Moody and their collaborators. This line of work connected finite-dimensional Lie theory to broader geometric and physical frameworks and laid the groundwork for extensive applications in representation theory, geometry, and mathematical physics.