Loop algebra

Not to be confused with loop (algebra).

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

Definition

If $g$ is a Lie algebra, the tensor product of $g$ with $C \infty (S 1)$ , the algebra of (complex) smooth functions over the circle manifold $S 1$ ,

{\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 $g 1$ and $g 2$ are elements of $g$ and $f 1$ and $f 2$ are elements of $C \infty (S 1)$ .

This isn't precisely what would correspond to the direct product of infinitely many copies of $g$ , one for each point in $S 1$ , because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from $S 1$ to $g$ ; a smooth parametrized loop in $g$ , in other words. This is why it is called the loop algebra.

Loop group

Similarly, a set of all smooth maps from $S 1$ 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.

Fourier transform

We can take the Fourier transform on this loop algebra by defining

g\otimes t^{n}

g\otimes e^{{-in\sigma }}

where

0 ≤ σ <2π

is a coordinatization of $S 1$ .

Applications

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

Notes

References

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

This article is issued from Wikipedia - version of the 9/2/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.