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In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.
DefinitionEdit
For a Lie algebra <math>\mathfrak{g}</math> over a field <math>K</math>, if <math>K[t,t^{-1}]</math> is the space of Laurent polynomials, then <math display=block>L\mathfrak{g} := \mathfrak{g}\otimes K[t,t^{-1}],</math> with the inherited bracket <math display=block>[X\otimes t^m, Y\otimes t^n] = [X,Y]\otimes t^{m+n}.</math>
Geometric definitionEdit
If <math>\mathfrak{g}</math> is a Lie algebra, the tensor product of <math>\mathfrak{g}</math> with Template:Math, the algebra of (complex) smooth functions over the circle manifold Template:Math (equivalently, smooth complex-valued periodic functions of a given period),
<math display=block>\mathfrak{g}\otimes C^\infty(S^1),</math>
is an infinite-dimensional Lie algebra with the Lie bracket given by
<math display=block>[g_1\otimes f_1,g_2 \otimes f_2]=[g_1,g_2]\otimes f_1 f_2.</math>
Here Template:Math and Template:Math are elements of <math>\mathfrak{g}</math> and Template:Math and Template:Math are elements of Template:Math.
This isn't precisely what would correspond to the direct product of infinitely many copies of <math>\mathfrak{g}</math>, one for each point in Template:Math, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from Template:Math to <math>\mathfrak{g}</math>; a smooth parametrized loop in <math>\mathfrak{g}</math>, in other words. This is why it is called the loop algebra.
GradationEdit
Defining <math>\mathfrak{g}_i</math> to be the linear subspace <math>\mathfrak{g}_i = \mathfrak{g}\otimes t^i < L\mathfrak{g},</math> the bracket restricts to a product<math display=block>[\cdot\, , \, \cdot]: \mathfrak{g}_i \times \mathfrak{g}_j \rightarrow \mathfrak{g}_{i+j},</math> hence giving the loop algebra a <math>\mathbb{Z}</math>-graded Lie algebra structure.
In particular, the bracket restricts to the 'zero-mode' subalgebra <math>\mathfrak{g}_0 \cong \mathfrak{g}</math>.
DerivationEdit
Template:See also There is a natural derivation on the loop algebra, conventionally denoted <math>d</math> acting as <math display=block>d: L\mathfrak{g} \rightarrow L\mathfrak{g}</math> <math display=block>d(X\otimes t^n) = nX\otimes t^n</math> and so can be thought of formally as <math>d = t\frac{d}{dt}</math>.
It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.
Loop groupEdit
Similarly, a set of all smooth maps from Template:Math to a Lie group Template:Math 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 algebrasEdit
Template:See also If <math>\mathfrak{g}</math> is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra <math>L\mathfrak g</math> gives rise to an affine Lie algebra. Furthermore this central extension is unique.<ref>Template:Cite book</ref>
The central extension is given by adjoining a central element <math>\hat k</math>, that is, for all <math>X\otimes t^n \in L\mathfrak{g}</math>, <math display=block>[\hat k, X\otimes t^n] = 0,</math> and modifying the bracket on the loop algebra to <math display=block>[X\otimes t^m, Y\otimes t^n] = [X,Y] \otimes t^{m + n} + mB(X,Y) \delta_{m+n,0} \hat k,</math> where <math>B(\cdot, \cdot)</math> is the Killing form.
The central extension is, as a vector space, <math>L\mathfrak{g} \oplus \mathbb{C}\hat k</math> (in its usual definition, as more generally, <math>\mathbb{C}</math> can be taken to be an arbitrary field).
CocycleEdit
Template:See also 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<math display=block>\varphi: L\mathfrak g \times L\mathfrak g \rightarrow \mathbb{C}</math> satisfying <math display=block>\varphi(X\otimes t^m, Y\otimes t^n) = mB(X,Y)\delta_{m+n,0}.</math> Then the extra term added to the bracket is <math>\varphi(X\otimes t^m, Y\otimes t^n)\hat k.</math>
Affine Lie algebraEdit
In physics, the central extension <math>L\mathfrak g \oplus \mathbb C \hat k</math> is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space<ref name="BYB">P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, Template:ISBN</ref><math display=block>\hat \mathfrak{g} = L\mathfrak{g} \oplus \mathbb C \hat k \oplus \mathbb C d</math> where <math>d</math> 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.