Loop algebra

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

If $mathfrak\{g\}$ is a Lie algebra, the tensor product of $mathfrak\{g\}$ with $C^infty(S^1)$,

:$mathfrak\{g\}otimes\; C^infty(S^1)$,

the algebra of (complex) smooth functions over the circle manifold S¹ is an infinite-dimensional Lie algebra with the Lie bracket given by

:$[g\_1otimes\; f\_1,g\_2\; otimes\; f\_2]\; =\; [g\_1,g\_2]\; otimes\; f\_1\; f\_2$.

Here g₁ and g₂ are elements of $mathfrak\{g\}$ and f₁ and f₂ are elements of $C^infty(S^1)$.

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 parameterized loop in $mathfrak\{g\}$, in other words. This is why it is called the loop algebra.

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

:$gotimes\; t^n$

as

:$gotimes\; e^\{-insigma\}$

where

:0 &le; &sigma; <2&pi;

is a coordinatization of S¹.

If $mathfrak\{g\}$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Kac-Moody algebra.

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.