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\begin{document}

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\section*{loop group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{lie_algebra}{Lie algebra}\dotfill \pageref*{lie_algebra} \linebreak
\noindent\hyperlink{complexification}{Complexification}\dotfill \pageref*{complexification} \linebreak
\noindent\hyperlink{central_extensions}{Central extensions}\dotfill \pageref*{central_extensions} \linebreak
\noindent\hyperlink{Representations}{Representations}\dotfill \pageref*{Representations} \linebreak
\noindent\hyperlink{positive_energy}{Positive energy}\dotfill \pageref*{positive_energy} \linebreak
\noindent\hyperlink{ByGeometricQuantization}{By geometric quantization (looped orbit method)}\dotfill \pageref*{ByGeometricQuantization} \linebreak
\noindent\hyperlink{RelationToEquivariantEllipticCohomology}{Relation to equivariant elliptic cohomology}\dotfill \pageref*{RelationToEquivariantEllipticCohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[loop space]] of a [[topological group]] $G$ inherits the structure of a [[group]] under pointwise group multiplication of [[loops]]. This is called a \emph{loop group} of $G$.

(Notice that this is a group structure in addition to the [[infinity-group]]-structure of any [[loop space]] under \emph{composition} of loops.)

If $G$ is a [[Lie group]], then there is a smooth version of the loop group consisting of [[smooth functions]] $S^1 \to G$. By the discussion at [[manifold structure of mapping spaces]] the collection of such smooth maps is itself an [[infinite-dimensional smooth manifold]] and so the smooth loop group of a Lie group is an [[infinite-dimensional Lie group]].

Among all [[infinite-dimensional Lie groups]], loop groups are a most well behaved class. In particular their [[representation theory]] is similar to that of [[compact Lie groups]].

Some of these nice properties are solely due to the [[circle]] $S^1$ being a [[compact manifold]]. For $X$ any other compact manifold there is similarly an infinite-dimensional Lie group $[X,G]$ of smooth functions $X \to G$ under pointwise multiplication in $G$.

Such mapping groups appear in [[physics]] notably as groups of [[gauge transformations]] over a [[spacetime]]/[[worldvolume]] $X$. Accordingly, loop groups play a prominint role in 1- and 2-dimensional [[quantum field theory]], notably the [[WZW model]] describing the propagation of a [[string]] on $G$. The \emph{[[current algebras]]} ([[affine algebras]]) which arise as [[Lie algebras]] of ([[central extension|centrally extended]]) loop groups derive their name from this relation to physics. Accordingly, as for [[compact Lie groups]], the [[representation theory]] of loop groups is naturally understood in terms of their [[geometric quantization]] (by a loop variant of the [[orbit method]]).

On the other hand, for $X$ of [[dimension]] greater that 1 there are very few known results about the properties of the mapping group $[X,G]$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{lie_algebra}{}\subsubsection*{{Lie algebra}}\label{lie_algebra}

Let $G$ be a [[compact Lie group]]. Write $\mathfrak{g}$ for its [[Lie algebra]].

\begin{prop}
\label{}\hypertarget{}{}
The [[Lie algebra]] of $L G$ is the [[loop Lie algebra]]

\begin{displaymath}
Lie(L G) \simeq L Lie(G) = L \mathfrak{g}
  \,.
\end{displaymath}
\end{prop}
\hypertarget{complexification}{}\subsubsection*{{Complexification}}\label{complexification}

Let $G$ be a [[compact Lie group]].

\begin{prop}
\label{}\hypertarget{}{}
The [[complexification of a Lie group|complexification]] of $L G$ is the loop group of the complexification of $G$

\begin{displaymath}
(L G)_{\mathbb{C}}
  \simeq
  L (G_\mathbb{C})
  \,.
\end{displaymath}
\end{prop}
\hypertarget{central_extensions}{}\subsubsection*{{Central extensions}}\label{central_extensions}

Loop groups of [[compact space|compact]] [[Lie groups]] have canonical [[central extensions]], often called \emph{Kac-Moody central extensions} . A detailed discussion is in (\hyperlink{PressleySegal}{PressleySegal}). A review is in (\hyperlink{BCSS}{BCSS})

\hypertarget{Representations}{}\subsubsection*{{Representations}}\label{Representations}

\hypertarget{positive_energy}{}\paragraph*{{Positive energy}}\label{positive_energy}

Write

\begin{displaymath}
t_\theta \colon L G \to L G
\end{displaymath}
for the [[automorphism]] which rotates loops by an [[angle]] $\theta$.

The corresponding [[semidirect product group]] we write $L G \rtimes S^1$

\begin{defn}
\label{}\hypertarget{}{}
Let $V$ be a [[topological vector space]]. A linear representation

\begin{displaymath}
S^1 \to Aut(V)
\end{displaymath}
of the [[circle group]] is called \textbf{positive} if $\exp(i \theta)$ acts by $\exp(i A \theta)$ where $A \in End(V)$ is a [[linear operator]] with positive [[spectrum of an operator|spectrum]].

A linear [[representation]]

\begin{displaymath}
\rho : L G \to Aut(V)
\end{displaymath}
is said to have \textbf{positive energy} or to be a \textbf{[[positive energy representation]]} if it extends to a representation of the [[semidirect product group]] $L G \rtimes S^1$ such that the restriction to $S^1$ is positive.

\end{defn}
\hypertarget{ByGeometricQuantization}{}\paragraph*{{By geometric quantization (looped orbit method)}}\label{ByGeometricQuantization}

We discuss the [[quantization of loop groups]] in the sense of [[geometric quantization]] of their canonical [[prequantum bundle]].

Let $G$ be a [[compact Lie group]]. Let $T \hookrightarrow G$ be the inclusion of a [[maximal torus]]. There is a [[fiber sequence]]

\begin{displaymath}
\itexarray{
    G/T &\to& L G / T
    \\
    && \downarrow
    \\
    && L G / G & \simeq \Omega G
  }
  \,.
\end{displaymath}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[orbit method]]}, if $G$ is a [[semisimple Lie group]], then $G/T$ is isomorphic to the [[coadjoint orbit]] of an element $\langle \lambda , -\rangle \in \mathfrak{g}^*$ for which $T \simeq G_\lambda$ is the [[stabilizer subgroup]].

If moreover $G$ is [[simply connected topological space|simply connected]], then the [[weight lattice]] $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is [[isomorphism|isomorphic]] to the group of [[group characters]]

\begin{displaymath}
\Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1))
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{RepsFromGeometricQuantization}\hypertarget{RepsFromGeometricQuantization}{}
The [[irreducible representation|irreducible]] [[projective representation|projective]] [[positive energy representations]] of $L G$ correspond precisley to the possible [[geometric quantizations]] of $L G / T$ (as in the [[orbit method]]).

More in detail:

The degree-2 [[integral cohomology]] of $L G / T$ is

\begin{displaymath}
H^2(L G / T) \simeq \mathbb{Z} \oplus H^2(G / T, \mathbb{Z}) \simeq \mathbb{Z} \oplus \hat T
  \,.
\end{displaymath}
Writing $L_{n,\lambda}$ for the corresponding [[complex line bundle]] with level $n \in \mathbb{Z}$ and weight $\lambda \in \hat T$ we have that

\begin{enumerate}%
\item the space of [[holomorphic sections]] of $L_{n,\lambda}$ is either zero or is an irreducible positive energy representation;


\item every such arises this way;


\item and is non-zero precisely if $(n,\lambda)$ is positive in the sense that for each positive [[coroot]] $h_\alpha$ of $G$

\begin{displaymath}
0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle
 \,.
\end{displaymath}


\end{enumerate}
\end{prop}
This appears for instance as (\hyperlink{Segal}{Segal, prop. 4.2}).

\hypertarget{RelationToEquivariantEllipticCohomology}{}\subsubsection*{{Relation to equivariant elliptic cohomology}}\label{RelationToEquivariantEllipticCohomology}

Under mild conditions (but over the complex numbers) the representation ring of a loop group $L G$ is equivalent to the $G$-[[equivariant elliptic cohomology]] (see there for more) of the point (\hyperlink{Ando00}{Ando 00, theorem 10.10}).

This is a higher analog of how $G$-[[equivariant K-theory]] of the point gives the [[representation ring]] of $G$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Kac-Moody group]]


\item [[Verlinde ring]]


\item [[string group]]


\item [[caloron correspondence]]


\item [[equivariant elliptic cohomology]]


\item [[Kac character formula]]


\item [[Kac-Weyl character]]


\item [[double loop group]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The standard textbook on loop groups is

\begin{itemize}%
\item Andrew Pressley, [[Graeme Segal]], \emph{Loop groups} Oxford University Press (1988)

\end{itemize}
A review talk is

\begin{itemize}%
\item [[Graeme Segal]], \emph{Loop groups} (\href{http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/chapter08.pdf}{pdf})

\end{itemize}
A review of some aspects with an eye towards loop groups as part of the [[crossed module]] of groups representing a [[string 2-group]] is in

\begin{itemize}%
\item [[John Baez]], [[Alissa Crans]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{From loop groups to 2-groups}, Homology Homotopy Appl. Volume 9, Number 2 (2007), 101-135. (\href{https://arxiv.org/abs/math/0504123}{arXiv:math/0504123})

\end{itemize}
The relation between [[representations]] of loop groups and [[twisted K-theory]] over the group is the topic of

\begin{itemize}%
\item [[Dan Freed]], [[Mike Hopkins]], [[Constantin Teleman]], \emph{[[Loop Groups and Twisted K-Theory]]}

\end{itemize}
The relation between representations of loop groups an [[equivariant elliptic cohomology]] of the point is discussed in

\begin{itemize}%
\item [[Matthew Ando]], \emph{Power operations in elliptic cohomology and representations of loop groups} Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (\href{http://www.jstor.org/stable/221905}{JSTOR}, \href{http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf}{pdf})

\end{itemize}
Discussion with respect to [[flag varieties]] is in

\begin{itemize}%
\item [[Shrawan Kumar]], \emph{Kac-Moody Groups, their Flag Varieties and Representation Theory}, Birkh\"a{}user 2002

\end{itemize}
[[!redirects loop group]] [[!redirects loop groups]]

[[!redirects loop group representation]] [[!redirects loop group representations]]

[[!redirects Kac-Moody central extension]] [[!redirects Kac-Moody central extensions]]

[[!redirects Kac-Moody loop group]] [[!redirects Kac-Moody loop groups]]



\end{document}