Loop algebra

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

Definition

For a Lie algebra \mathfrak{g} over a field K, if K[t,t^{-1}] is the space of Laurent polynomials, then

L\mathfrak{g} := \mathfrak{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 \mathfrak{g} is a Lie algebra, the tensor product of \mathfrak{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),

\mathfrak{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 \mathfrak{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 \mathfrak{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 \mathfrak{g}; a smooth parametrized loop in \mathfrak{g}, in other words. This is why it is called the loop algebra.

Gradation

Defining \mathfrak{g}_i to be the linear subspace \mathfrak{g}_i = \mathfrak{g}\otimes t^i < L\mathfrak{g}, the bracket restricts to a product

[\cdot\, , \, \cdot]: \mathfrak{g}_i \times \mathfrak{g}_j \rightarrow \mathfrak{g}_{i+j},

hence giving the loop algebra a \mathbb{Z}-graded Lie algebra structure.

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

Derivation

There is a natural derivation on the loop algebra, conventionally denoted d acting as

d: L\mathfrak{g} \rightarrow L\mathfrak{g}

d(X\otimes t^n) = nX\otimes t^n

and so can be thought of formally as d = t\frac{d}{dt}.

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 \mathfrak{g} is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra L\mathfrak g gives rise to an affine Lie algebra. Furthermore, this central extension is unique.^[1]

The central extension is given by adjoining a central element \hat k, that is, for all X\otimes t^n \in L\mathfrak{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} + mB(X,Y) \delta_{m+n,0} \hat k,

where B(\cdot, \cdot) is the Killing form.

The central extension is, as a vector space, L\mathfrak{g} \oplus \mathbb{C}\hat k (in its usual definition, as more generally, \mathbb{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

\varphi: L\mathfrak g \times L\mathfrak g \rightarrow \mathbb{C}

satisfying

\varphi(X\otimes t^m, Y\otimes t^n) = mB(X,Y)\delta_{m+n,0}.

Then the extra term added to the bracket is \varphi(X\otimes t^m, Y\otimes t^n)\hat k.

Affine Lie algebra

In physics, the central extension L\mathfrak g \oplus \mathbb 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^[2]

\hat \mathfrak{g} = L\mathfrak{g} \oplus \mathbb C \hat k \oplus \mathbb 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.

References

1. ^ Kac, V.G. (1990). Infinite-dimensional Lie algebras. 3rd ed. Cambridge University Press. Exercise 7.8. ISBN 978-0-521-37215-2.
2. ^ P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X

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