Loop algebra

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 $C \infty (S 1)$ , the algebra of (complex) smooth functions over the circle manifold $S 1$ (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 $g 1$ and $g 2$ are elements of ${\mathfrak {g}}$ and $f 1$ and $f 2$ 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 $S 1$ , because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from $S 1$ 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 ${\mathfrak {g}}_{0}\cong {\mathfrak {g}}$ .

Derivation

There is a natural derivation on the loop algebra, conventionally denoted $d$ acting as $d:L{\mathfrak {g}}\rightarrow L{\mathfrak {g}}$ $d(X\otimes t^{n})=nX\otimes t^{n}$ and so can be thought of formally as $d=t{\frac {d}{dt}}$ .

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

Loop group

Similarly, a set of all smooth maps from $S 1$ 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

If ${\mathfrak {g}}$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra $L{\mathfrak {g}}$ gives rise to an affine Lie algebra. Furthermore this central extension is unique.^[1]

The central extension is given by adjoining a central element ${\hat {k}}$ , that is, for all $X\otimes t^{n}\in L{\mathfrak {g}}$ , $[{\hat {k}},X\otimes t^{n}]=0,$ and modifying the bracket on the loop algebra to $[X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}+mB(X,Y)\delta _{m+n,0}{\hat {k}},$ where $B(\cdot ,\cdot )$ is the Killing form.

The central extension is, as a vector space, $L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}$ (in its usual definition, as more generally, $\mathbb {C}$ can be taken to be an arbitrary field).

Cocycle

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 $\varphi :L{\mathfrak {g}}\times L{\mathfrak {g}}\rightarrow \mathbb {C}$ satisfying $\varphi (X\otimes t^{m},Y\otimes t^{n})=mB(X,Y)\delta _{m+n,0}.$ Then the extra term added to the bracket is $\varphi (X\otimes t^{m},Y\otimes t^{n}){\hat {k}}.$

Affine Lie algebra

In physics, the central extension $L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}$ 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] ${\hat {\mathfrak {g}}}=L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}\oplus \mathbb {C} d$ where $d$ 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

^ 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

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

[BYB-2] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X

[1]

[2]

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