Loop algebra

Not to be confused with quasigroup with an identity element, also called an algebraic loop.

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

Definition

For a Lie algebra ${\mathfrak {g}}$ over a field $K$, if $K[t,t^{-1}]$ is the space of Laurent polynomials, then $$L{\mathfrak {g}}:={\mathfrak {g}}\otimes K[t,t^{-1}],$$ with the inherited bracket $$[X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}.$$

Geometric definition

If ${\mathfrak {g}}$ is a Lie algebra, the tensor product of ${\mathfrak {g}}$ with C^∞(S¹), the algebra of (complex) smooth functions over the circle manifold S¹ (equivalently, smooth complex-valued periodic functions of a given period),

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

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 ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$, 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 parametrized loop in ${\mathfrak {g}}$, in other words. This is why it is called the loop algebra.

Gradation

Defining ${\mathfrak {g}}_{i}$ to be the linear subspace ${\mathfrak {g}}_{i}={\mathfrak {g}}\otimes t^{i}<L{\mathfrak {g}},$ the bracket restricts to a product $$[\cdot \,,\,\cdot ]:{\mathfrak {g}}_{i}\times {\mathfrak {g}}_{j}\rightarrow {\mathfrak {g}}_{i+j},$$ hence giving the loop algebra a $\mathbb {Z}$-graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra ${\mathfrak {g}}_{0}\cong {\mathfrak {g}}$.

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 mapping

$g\otimes t^{n}$

to

$g\otimes e^{-in\sigma }$

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