Loop algebra explained

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

Definition

For a Lie algebra

ak{g} over a field K , if K[t,t ^{-1} ] 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

ak{g} is a Lie algebra, the tensor product of ak{g} 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

ak{g} and and are elements of .

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

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

Gradation

Defining

ak{g} _{i} to be the linear subspace ak{g} _{i} = ak{g} ⊗ t ^{i} < L ak{g}, 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 Z -graded Lie algebra structure.

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

ak{g} _{0} \cong ak{g} .

Derivation

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

d 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 d = t (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 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

ak{g} is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra L ak 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 ⊗ t ^{n} \in L ak{g} ,$$[\backslash hat\; k,\; X\backslash otimes\; t^n]\; =\; 0,$$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 B( ⋅ , ⋅ ) is the Killing form.

The central extension is, as a vector space,

L ak{g} ⊕ 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$$\backslash varphi:\; L\backslash mathfrak\; g\; \backslash times\; L\backslash mathfrak\; g\; \backslash rightarrow\; \backslash mathbb$$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

\varphi(X ⊗ t ^{m,} Y ⊗ t ^{n)\hat} k.

Affine Lie algebra

In physics, the central extension

L ak g ⊕ 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] $$\backslash hat\; \backslash mathfrak\; =\; L\backslash mathfrak\; \backslash oplus\; \backslash mathbb\; C\; \backslash hat\; k\; \backslash oplus\; \backslash mathbb\; C\; d$$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.

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,