Loop algebra

If ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra, the tensor product of ${\displaystyle {\mathfrak {g}}}$ with C^∞(S¹), the algebra of (complex) smooth functions over the circle manifold S¹ (equivalently, smooth complex-valued periodic functions of a given period),

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

is an infinite-dimensional Lie algebra with the Lie bracket given by

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

Here g₁ and g₂ are elements of ${\displaystyle {\mathfrak {g}}}$ and f₁ and f₂ are elements of C^∞(S¹).

This isn't precisely what would correspond to the direct product of infinitely many copies of ${\displaystyle {\mathfrak {g}}}$, 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 ${\displaystyle {\mathfrak {g}}}$; a smooth parametrized loop in ${\displaystyle {\mathfrak {g}}}$, in other words. This is why it is called the loop 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.

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

${\displaystyle g\otimes t^{n}}$

as

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

where

0 ≤ σ <2π

is a coordinatization of S¹.

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

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