Loop algebra

Short description: Type of Lie algebra of interest in physics

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.

Definition

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

Geometric definition

If [math]\displaystyle{ \mathfrak{g} }[/math] is a Lie algebra, the tensor product of [math]\displaystyle{ \mathfrak{g} }[/math] with C^∞(S¹), the algebra of (complex) smooth functions over the circle manifold S¹ (equivalently, smooth complex-valued periodic functions of a given period),

[math]\displaystyle{ \mathfrak{g}\otimes C^\infty(S^1), }[/math]

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

[math]\displaystyle{ [g_1\otimes f_1,g_2 \otimes f_2]=[g_1,g_2]\otimes f_1 f_2. }[/math]

Here g₁ and g₂ are elements of [math]\displaystyle{ \mathfrak{g} }[/math] 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 [math]\displaystyle{ \mathfrak{g} }[/math], 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 [math]\displaystyle{ \mathfrak{g} }[/math]; a smooth parametrized loop in [math]\displaystyle{ \mathfrak{g} }[/math], in other words. This is why it is called the loop algebra.

Gradation

Defining [math]\displaystyle{ \mathfrak{g}_i }[/math] to be the linear subspace [math]\displaystyle{ \mathfrak{g}_i = \mathfrak{g}\otimes t^i \lt L\mathfrak{g}, }[/math] the bracket restricts to a product [math]\displaystyle{ [\cdot\, , \, \cdot]: \mathfrak{g}_i \times \mathfrak{g}_j \rightarrow \mathfrak{g}_{i+j}, }[/math] hence giving the loop algebra a [math]\displaystyle{ \mathbb{Z} }[/math]-graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra [math]\displaystyle{ \mathfrak{g}_0 \cong \mathfrak{g} }[/math].

Derivation

There is a natural derivation on the loop algebra, conventionally denoted [math]\displaystyle{ d }[/math] acting as [math]\displaystyle{ d: L\mathfrak{g} \rightarrow L\mathfrak{g} }[/math] [math]\displaystyle{ d(X\otimes t^n) = nX\otimes t^n }[/math] and so can be thought of formally as [math]\displaystyle{ d = t\frac{d}{dt} }[/math].

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 [math]\displaystyle{ \mathfrak{g} }[/math] is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra [math]\displaystyle{ L\mathfrak g }[/math] gives rise to an affine Lie algebra. Furthermore this central extension is unique.

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

The central extension is, as a vector space, [math]\displaystyle{ L\mathfrak{g} \oplus \mathbb{C}\hat k }[/math] (in its usual definition, as more generally, [math]\displaystyle{ \mathbb{C} }[/math] 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 [math]\displaystyle{ \varphi: L\mathfrak g \times L\mathfrak g \rightarrow \mathbb{C} }[/math] satisfying [math]\displaystyle{ \varphi(X\otimes t^m, Y\otimes t^n) = mB(X,Y)\delta_{m+n,0}. }[/math] Then the extra term added to the bracket is [math]\displaystyle{ \varphi(X\otimes t^m, Y\otimes t^n)\hat k. }[/math]

Affine Lie algebra

In physics, the central extension [math]\displaystyle{ L\mathfrak g \oplus \mathbb C \hat k }[/math] is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space [math]\displaystyle{ \hat \mathfrak{g} = L\mathfrak{g} \oplus \mathbb C \hat k \oplus \mathbb C d }[/math] where [math]\displaystyle{ d }[/math] 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.

References

• Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X