Loop Algebra

# Loop algebra

to get instant updates about 'Loop Algebra' on your MyPage. Meet other similar minded people. Its Free!

X

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

If [itex]mathfrak[/itex] is a Lie algebra, the tensor product of [itex]mathfrak[/itex] with [itex]C^infty(S^1)[/itex],

[itex]mathfrakotimes C^infty(S^1)[/itex],

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

[itex]=otimes f_1 f_2[/itex].

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

This isn't precisely what would correspond to the direct product of infinitely many copies of [itex]mathfrak[/itex], 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 [itex]mathfrak[/itex]; a smooth parameterized loop in [itex]mathfrak[/itex], in other words. This is why it is called the loop algebra.

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

[itex]gotimes t^n[/itex]

as

[itex]gotimes e^[/itex]

where

0 &le; &sigma; <2&pi;

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

If...

No feeds found

All