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Definition

For a Lie algebra $g$ over a field $K$ , if $K [t, t^{- 1}]$ is the space of Laurent polynomials, then $L g := g \otimes K [t, t^{- 1}],$ with the inherited bracket $[X \otimes t^{m}, Y \otimes t^{n}] = [X, Y] \otimes t^{m + n} .$

Geometric definition

If $g$ is a Lie algebra, the tensor product of $g$ with C^∞(S¹), the algebra of (complex) smooth functions over the circle manifold S¹ (equivalently, smooth complex-valued periodic functions of a given period),

$g \otimes C^{\infty} (S^{1}),$

is an infinite-dimensional Lie algebra with the Lie bracket given by

$[g_{1} \otimes f_{1}, g_{2} \otimes f_{2}] = [g_{1}, g_{2}] \otimes f_{1} f_{2} .$

Here g₁ and g₂ are elements of $g$ and f₁ and f₂ are elements of C^∞(S¹).

This isn't precisely what would correspond to the direct product of infinitely many copies of $g$ , one for each point in S¹, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from S¹ to $g$ ; a smooth parametrized loop in $g$ , in other words. This is why it is called the loop algebra.

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Gradation

Defining $g_{i}$ to be the linear subspace $g_{i} = g \otimes t^{i} < L g,$ the bracket restricts to a product $[\cdot, \cdot] : g_{i} \times g_{j} \to g_{i + j},$ hence giving the loop algebra a $Z$ -graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra $g_{0} ≅ g$ .

Derivation

There is a natural derivation on the loop algebra, conventionally denoted $d$ acting as $d : L g \to L g$ $d (X \otimes t^{n}) = n X \otimes t^{n}$ and so can be thought of formally as $d = t \frac{d}{d t}$ .

It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.

Loop group

Similarly, a set of all smooth maps from S¹ to a Lie group G forms an infinite-dimensional Lie group (Lie group in the sense we can define functional derivatives over it) called the loop group. The Lie algebra of a loop group is the corresponding loop algebra.

Affine Lie algebras as central extension of loop algebras

If $g$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra $L g$ gives rise to an affine Lie algebra. Furthermore, this central extension is unique.

The central extension is given by adjoining a central element $\hat{k}$ , that is, for all $X \otimes t^{n} \in L g$ , $[\hat{k}, X \otimes t^{n}] = 0,$ and modifying the bracket on the loop algebra to $[X \otimes t^{m}, Y \otimes t^{n}] = [X, Y] \otimes t^{m + n} + m B (X, Y) δ_{m + n, 0} \hat{k},$ where $B (\cdot, \cdot)$ is the Killing form.

The central extension is, as a vector space, $L g \oplus C \hat{k}$ (in its usual definition, as more generally, $C$ can be taken to be an arbitrary field).

Cocycle

Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra. This is the map $φ : L g \times L g \to C$ satisfying $φ (X \otimes t^{m}, Y \otimes t^{n}) = m B (X, Y) δ_{m + n, 0} .$ Then the extra term added to the bracket is $φ (X \otimes t^{m}, Y \otimes t^{n}) \hat{k} .$

Affine Lie algebra

In physics, the central extension $L g \oplus C \hat{k}$ is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space $\hat{g} = L g \oplus C \hat{k} \oplus C d$ where $d$ is the derivation defined above.

On this space, the Killing form can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra.