In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.
Definition edit
For a Lie algebra over a field , if is the space of Laurent polynomials, then
Geometric definition edit
If is a Lie algebra, the tensor product of with C∞(S1), the algebra of (complex) smooth functions over the circle manifold S1 (equivalently, smooth complex-valued periodic functions of a given period),
is an infinite-dimensional Lie algebra with the Lie bracket given by
Here g1 and g2 are elements of 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 , 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 ; a smooth parametrized loop in , in other words. This is why it is called the loop algebra.
Gradation edit
Defining to be the linear subspace the bracket restricts to a product
In particular, the bracket restricts to the 'zero-mode' subalgebra .
Derivation edit
There is a natural derivation on the loop algebra, conventionally denoted acting as
It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.
Loop group edit
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.
Affine Lie algebras as central extension of loop algebras edit
If is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Lie algebra. Furthermore this central extension is unique.[1]
The central extension is given by adjoining a central element , that is, for all ,
The central extension is, as a vector space, (in its usual definition, as more generally, can be taken to be an arbitrary field).
Cocycle edit
Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra. This is the map
Affine Lie algebra edit
In physics, the central extension is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space[2]
On this space, the Killing form can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra.
References edit
- ^ Kac, V.G. (1990). Infinite-dimensional Lie algebras (3rd ed.). Cambridge University Press. Exercise 7.8. ISBN 978-0-521-37215-2.
- ^ P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X
- Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X