Loop algebra explained

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

Definition

For a Lie algebra

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

Geometric definition

If

is a Lie algebra, the tensor product of with, the algebra of (complex) smooth functions over the circle manifold (equivalently, smooth complex-valued periodic functions of a given period),

$$\backslash mathfrak\backslash otimes\; C^\backslash infty(S^1),$$

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

$$[g\_1\backslash otimes\; f\_1,g\_2\; \backslash otimes\; f\_2]=[g\_1,g\_2]\backslash otimes\; f\_1\; f\_2.$$

Here and are elements of

and and are elements of .

This isn't precisely what would correspond to the direct product of infinitely many copies of

, one for each point in, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from to ; a smooth parametrized loop in , in other words. This is why it is called the loop algebra.

Gradation

Defining

to be the linear subspace the bracket restricts to a product$$[\backslash cdot\backslash ,,\; \backslash ,\; \backslash cdot]:\; \backslash mathfrak\_i\; \backslash times\; \backslash mathfrak\_j\; \backslash rightarrow\; \backslash mathfrak\_,$$hence giving the loop algebra a -graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra

.

Derivation

See also: Derivation (differential algebra). There is a natural derivation on the loop algebra, conventionally denoted

acting as$$d:\; L\backslash mathfrak\; \backslash rightarrow\; L\backslash mathfrak$$$$d(X\backslash otimes\; t^n)\; =\; nX\backslash otimes\; t^n$$and so can be thought of formally as .

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 to a Lie group 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

See also: Affine Lie algebra. If

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

The central extension is given by adjoining a central element

, that is, for all ,$$[\backslash hat\; k,\; X\backslash otimes\; t^n]\; =\; 0,$$and modifying the bracket on the loop algebra to$$[X\backslash otimes\; t^m,\; Y\backslash otimes\; t^n]\; =\; [X,Y]\; \backslash otimes\; t^\; +\; mB(X,Y)\; \backslash delta\_\; \backslash hat\; k,$$where is the Killing form.

The central extension is, as a vector space,

(in its usual definition, as more generally, 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$$\backslash varphi:\; L\backslash mathfrak\; g\; \backslash times\; L\backslash mathfrak\; g\; \backslash rightarrow\; \backslash mathbb$$satisfying$$\backslash varphi(X\backslash otimes\; t^m,\; Y\backslash otimes\; t^n)\; =\; mB(X,Y)\backslash delta\_.$$Then the extra term added to the bracket is

Affine Lie algebra

In physics, the central extension

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] $$\backslash hat\; \backslash mathfrak\; =\; L\backslash mathfrak\; \backslash oplus\; \backslash mathbb\; C\; \backslash hat\; k\; \backslash oplus\; \backslash mathbb\; C\; d$$where 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.

Notes and References

1. Book: Kac, V.G. . Infinite-dimensional Lie algebras. 3rd. Cambridge University Press. 1990. Victor Kac. 978-0-521-37215-2 . Exercise 7.8..
2. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997,