Loop algebra
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, loop algebras are certain types of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, of particular interest in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
.


Definition

For a Lie algebra \mathfrak over a field K, if K ,t^/math> is the space of Laurent polynomials, then L\mathfrak := \mathfrak\otimes K ,t^ with the inherited bracket \otimes t^m, Y\otimes t^n= ,Yotimes t^.


Geometric definition

If \mathfrak is a Lie algebra, the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of \mathfrak with , the
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
of (complex)
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s over the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(equivalently, smooth complex-valued
periodic functions A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
of a given period), \mathfrak\otimes C^\infty(S^1), is an infinite-dimensional Lie algebra with the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
given by _1\otimes f_1,g_2 \otimes f_2 _1,g_2otimes f_1 f_2. Here and are elements of \mathfrak and and are elements of . This isn't precisely what would correspond to the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of infinitely many copies of \mathfrak, one for each point in , because of the smoothness restriction. Instead, it can be thought of in terms of
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
from to \mathfrak; a smooth parametrized loop in \mathfrak, in other words. This is why it is called the loop algebra.


Gradation

Defining \mathfrak_i to be the
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
\mathfrak_i = \mathfrak\otimes t^i < L\mathfrak, the bracket restricts to a product cdot\, , \, \cdot \mathfrak_i \times \mathfrak_j \rightarrow \mathfrak_, hence giving the loop algebra a \mathbb-
graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket oper ...
structure. In particular, the bracket restricts to the 'zero-mode' subalgebra \mathfrak_0 \cong \mathfrak.


Derivation

There is a natural derivation on the loop algebra, conventionally denoted d acting as d: L\mathfrak \rightarrow L\mathfrak d(X\otimes t^n) = nX\otimes t^n and so can be thought of formally as d = t\frac. It is required to define
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
s, which are used in physics, particularly
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
.


Loop group

Similarly, a set of all smooth maps from to a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
forms an infinite-dimensional Lie group (Lie group in the sense we can define
functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
s over it) called the
loop group In mathematics, a loop group is a Group (mathematics), group of Loop (topology), loops in a topological group ''G'' with multiplication defined pointwise. Definition In its most general form a loop group is a group of continuous mappings from a ...
. The Lie algebra of a loop group is the corresponding loop algebra.


Affine Lie algebras as central extension of loop algebras

If \mathfrak is a
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...
, then a nontrivial central extension of its loop algebra L\mathfrak g gives rise to an
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
. Furthermore this central extension is unique. Exercise 7.8. The central extension is given by adjoining a central element \hat k, that is, for all X\otimes t^n \in L\mathfrak, hat k, X\otimes t^n= 0, and modifying the bracket on the loop algebra to \otimes t^m, Y\otimes t^n= ,Y\otimes t^ + mB(X,Y) \delta_ \hat k, where B(\cdot, \cdot) is the
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
. The central extension is, as a vector space, L\mathfrak \oplus \mathbb\hat k (in its usual definition, as more generally, \mathbb can be taken to be an arbitrary field).


Cocycle

Using the language of
Lie algebra cohomology In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to p ...
, 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 satisfying \varphi(X\otimes t^m, Y\otimes t^n) = mB(X,Y)\delta_. 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 spaceP. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, \hat \mathfrak = L\mathfrak \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

* Lie algebras {{algebra-stub