Loop Algebra

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

If <math>mathfrak</math> is a Lie algebra, the tensor product of <math>mathfrak</math> with <math>C^infty(S^1)</math>,

the algebra of (complex) smooth functions over the circle manifold S<sup>1</sup> is an infinite-dimensional Lie algebra with the Lie bracket given by

Here g<sub>1</sub> and g<sub>2</sub> are elements of <math>mathfrak</math> and f<sub>1</sub> and f<sub>2</sub> are elements of <math>C^infty(S^1)</math>.

This isn't precisely what would correspond to the direct product of infinitely many copies of <math>mathfrak</math>, one for each point in S<sup>1</sup>, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from S<sup>1</sup> to <math>mathfrak</math>; a smooth parameterized loop in <math>mathfrak</math>, in other words. This is why it is called the**loop algebra**.

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

as

where

is a coordinatization of S<sup>1</sup>.

If...

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If <math>mathfrak</math> is a Lie algebra, the tensor product of <math>mathfrak</math> with <math>C^infty(S^1)</math>,

- <math>mathfrakotimes C^infty(S^1)</math>,

the algebra of (complex) smooth functions over the circle manifold S<sup>1</sup> is an infinite-dimensional Lie algebra with the Lie bracket given by

- <math>=otimes f_1 f_2</math>.

Here g<sub>1</sub> and g<sub>2</sub> are elements of <math>mathfrak</math> and f<sub>1</sub> and f<sub>2</sub> are elements of <math>C^infty(S^1)</math>.

This isn't precisely what would correspond to the direct product of infinitely many copies of <math>mathfrak</math>, one for each point in S<sup>1</sup>, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from S<sup>1</sup> to <math>mathfrak</math>; a smooth parameterized loop in <math>mathfrak</math>, in other words. This is why it is called the

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

- <math>gotimes t^n</math>

as

- <math>gotimes e^</math>

where

- 0 ≤ σ <2π

is a coordinatization of S<sup>1</sup>.

If...

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