Loop algebra

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,

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

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 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

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.

Fourier transform

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.

Applications

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

Notes
References

Mathematics Encyclopedia