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

$[g_{1}\otimes f_{1},g_{2}\otimes f_{2}]=[g_{1},g_{2}]\otimes f_{1}f_{2}$.

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

This isn't precisely what would correspond to the group 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

$g\otimes t^{n}$

as

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

where

0 ≤ σ <2π

is a coordinatization of S¹.

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