Loop algebra

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

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.

Here g1 and g2 are elements of \mathfrak{g} and f1 and f2 are elements of C^\infty(S^1).

This isn't precisely what would correspond to 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 in terms of 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 on 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 central extension of its loop algebra gives rise to an affine Kac-Moody 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.

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