Loop algebra

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

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

where g₁ and g₂ are elements of ${\mathfrak {g}}$ and f₁ and f₂ are elements of $C^{\infty }(S^{1})$. This group isn't exactly 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 as a 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 of 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.