# Loop algebra

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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(S1), the algebra of (complex) smooth functions over the circle manifold S1,

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

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 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 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 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 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.

## Fourier transform

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.

## Applications

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

## References

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