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Loop algebra

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

Definition

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For a Lie algebra over a field , if is the space of Laurent polynomials, then with the inherited bracket

Geometric definition

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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

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Defining to be the linear subspace the bracket restricts to a product hence giving the loop algebra a -graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra .

Derivation

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There is a natural derivation on the loop algebra, conventionally denoted acting as and so can be thought of formally as .

It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.

Loop group

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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

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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 , and modifying the bracket on the loop algebra to where is the Killing form.

The central extension is, as a vector space, (in its usual definition, as more generally, can be taken to be an arbitrary field).

Cocycle

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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 satisfying Then the extra term added to the bracket is

Affine Lie algebra

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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] where is the derivation defined above.

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

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  1. ^ Kac, V.G. (1990). Infinite-dimensional Lie algebras (3rd ed.). Cambridge University Press. Exercise 7.8. ISBN 978-0-521-37215-2.
  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