Loop algebra

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

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

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

is an infinite-dimensional Lie algebra with the Lie bracket given by

$${\displaystyle [g_{1}\otimes f_{1},g_{2}\otimes f_{2}]=[g_{1},g_{2}]\otimes f_{1}f_{2}.}$$

Here g₁ and g₂ are elements of ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ , 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 ${\displaystyle {\mathfrak {g}}}$ ; a smooth parametrized loop in ${\displaystyle {\mathfrak {g}}}$ , in other words. This is why it is called the loop algebra.

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

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

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

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

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.

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

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

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

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

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