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There are different spaces known as  "loop groups" . One of the most studied is defined by considering, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201601.png" /> a compact semi-simple Lie group (cf. also [[Lie group, semi-simple|Lie group, semi-simple]]), the loop group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201602.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201603.png" /> is the circle of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201604.png" />.
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One can take spaces of polynomial, rational, real-analytic, smooth, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201605.png" /> loops, in decreasing order of regularity. The metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201606.png" /> is actually a [[Kähler metric|Kähler metric]]. The complex structure is defined by considering as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201607.png" />-forms on the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201608.png" />, those with positive Fourier coefficients. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l1201609.png" /> is a [[Homogeneous space|homogeneous space]] of the loop group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016010.png" />, with isotropy
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There are different spaces known as  "loop groups" . One of the most studied is defined by considering, for $G$ a compact semi-simple Lie group (cf. also [[Lie group, semi-simple|Lie group, semi-simple]]), the loop group $\Omega G = \{ \gamma : S ^ { 1 } \rightarrow G : \gamma ( 1 ) = 1 \}$; here, $S ^ { 1 }$ is the circle of complex numbers $z = e ^ { i \theta }$.
  
This is shown by considering the  "Grassmannian model"  of D. Quillen for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016012.png" />. For simplicity, take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016013.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016014.png" /> is naturally identified with the Hilbert–Schmidt Grassmannian of certain Hilbert subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016015.png" />, stable under multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016016.png" />. This also shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016017.png" /> has a holomorphic embedding into a projectivized Hilbert space, by generalized Plücker coordinates; and a determinant line bundle.
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One can take spaces of polynomial, rational, real-analytic, smooth, or $L _ { 1 / 2 } ^ { 2 }$ loops, in decreasing order of regularity. The metric $L _ { 1 / 2 } ^ { 2 }$ is actually a [[Kähler metric|Kähler metric]]. The complex structure is defined by considering as $( 1,0 )$-forms on the Lie algebra of $\Omega G$, those with positive Fourier coefficients. Equivalently, $\Omega G$ is a [[Homogeneous space|homogeneous space]] of the loop group $L G _ { \text{C} } = \{ \gamma : S ^ { 1 } \rightarrow G _ { \text{C} } \}$, with isotropy
  
While the smooth loop group is an infinite-dimensional Fréchet manifold, the rational and polynomial loop groups have natural filtrations by finite-dimensional algebraic subvarieties. Moreover, by a result of G. Segal, any holomorphic mapping from a compact manifold into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016018.png" /> actually goes into the rational loops (up to multiplication by a constant). This has application in gauge theory, because a theorem of M.F. Atiyah and S.K. Donaldson identifies, for a classical group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016019.png" />, the moduli space of charge-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016020.png" /> framed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016021.png" />-instantons with the moduli spaces of based holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016024.png" />-spheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016025.png" />, of topological degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016026.png" />. Another application to gauge theory and to the theory of completely integrable systems is given by a construction of K. Uhlenbeck, refining earlier work by V.E. Zakharov and others: this identifies, modulo basepoints, harmonic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016027.png" /> with certain holomorphic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016028.png" />. The degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016030.png" /> now gives the energy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016031.png" />, by a formula of G. Valli, generalizing earlier work of E. Calabi.
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\begin{equation*} L^+G _ { \mathbf{C} } = \left\{ \begin{array}{l}{  \\ \gamma \in L G _ { \mathbf{C} } :\\ }\end{array} \begin{array}{c}{  \gamma \text{ extends} \\ \text{ holomorphically in the disc } }\\{ \text { to a group } "\square" \text{valued mapping }}\end{array}  \right\}. \end{equation*}
  
The representation theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016032.png" /> has also been studied: the key point is the construction of a universal central extension, which makes it possible to define infinite-dimensional projective representations.
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This is shown by considering the  "Grassmannian model" of D. Quillen for $\Omega G$. For simplicity, take $G = U ( n )$; then $\Omega U ( n )$ is naturally identified with the Hilbert–Schmidt Grassmannian of certain Hilbert subspaces of $L ^ { 2 } ( S ^ { 1 } , \mathbf{C} ^ { n } )$, stable under multiplication by $z$. This also shows that $\Omega G$ has a holomorphic embedding into a projectivized Hilbert space, by generalized Plücker coordinates; and a determinant line bundle.
  
Other spaces commonly known as  "loop groups"  are the group of diffeomorphism of the circle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016033.png" />, and the loop space of a manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016034.png" />.
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While the smooth loop group is an infinite-dimensional Fréchet manifold, the rational and polynomial loop groups have natural filtrations by finite-dimensional algebraic subvarieties. Moreover, by a result of G. Segal, any holomorphic mapping from a compact manifold into $\Omega G$ actually goes into the rational loops (up to multiplication by a constant). This has application in gauge theory, because a theorem of M.F. Atiyah and S.K. Donaldson identifies, for a classical group $G$, the moduli space of charge-$k$ framed $G$-instantons with the moduli spaces of based holomorphic $2$-spheres in $\Omega G$, of topological degree $k$. Another application to gauge theory and to the theory of completely integrable systems is given by a construction of K. Uhlenbeck, refining earlier work by V.E. Zakharov and others: this identifies, modulo basepoints, harmonic mappings $f : S ^ { 2 } \rightarrow G$ with certain holomorphic mappings $F : S ^ { 2 } \rightarrow \Omega G$. The degree $k$ of $F$ now gives the energy of $f$, by a formula of G. Valli, generalizing earlier work of E. Calabi.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016035.png" /> has been studied as reparametrization groups for string theory. It has two natural homogeneous spaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016037.png" />, which are infinite-dimensional Kähler manifolds (the Kähler metric is the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016038.png" />). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016039.png" /> has been studied in the theory of universal Teichmüller spaces (cf. also [[Teichmüller space|Teichmüller space]]).
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The representation theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016032.png"/> has also been studied: the key point is the construction of a universal central extension, which makes it possible to define infinite-dimensional projective representations.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016040.png" /> is a [[Manifold|manifold]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016041.png" /> is defined as the space of loops in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016042.png" /> starting at a fixed basepoint. Here, the group operation is given by composition of loops. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120160/l12016043.png" /> has been considered in connection with the problem of characterizing parallel transport operators, as configuration space in string theory, and in probability theory.
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Other spaces commonly known as  "loop groups" are the group of diffeomorphism of the circle, $\operatorname{Diff}( S ^ { 1 } )$, and the loop space of a manifold, $LM$.
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The space $\operatorname{Diff}( S ^ { 1 } )$ has been studied as reparametrization groups for string theory. It has two natural homogeneous spaces: $\operatorname{Diff} ( S ^ { 1 } ) / \operatorname{SL} ( 2 , {\bf R} )$ and $\operatorname{Diff}( S ^ { 1 } ) / S ^ { 1 }$, which are infinite-dimensional Kähler manifolds (the Kähler metric is the metric $L _ { 3 / 2 } ^ { 2 }$). The space $\operatorname{Diff} ( S ^ { 1 } ) / \operatorname{SL} ( 2 , {\bf R} )$ has been studied in the theory of universal Teichmüller spaces (cf. also [[Teichmüller space|Teichmüller space]]).
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When $M$ is a [[Manifold|manifold]], $LM$ is defined as the space of loops in $M$ starting at a fixed basepoint. Here, the group operation is given by composition of loops. The space $LM$ has been considered in connection with the problem of characterizing parallel transport operators, as configuration space in string theory, and in probability theory.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pressley,  G. Segal,  "Loop groups" , Oxford Univ. Press  (1986)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Pressley,  G. Segal,  "Loop groups" , Oxford Univ. Press  (1986)</td></tr></table>

Revision as of 16:57, 1 July 2020

There are different spaces known as "loop groups" . One of the most studied is defined by considering, for $G$ a compact semi-simple Lie group (cf. also Lie group, semi-simple), the loop group $\Omega G = \{ \gamma : S ^ { 1 } \rightarrow G : \gamma ( 1 ) = 1 \}$; here, $S ^ { 1 }$ is the circle of complex numbers $z = e ^ { i \theta }$.

One can take spaces of polynomial, rational, real-analytic, smooth, or $L _ { 1 / 2 } ^ { 2 }$ loops, in decreasing order of regularity. The metric $L _ { 1 / 2 } ^ { 2 }$ is actually a Kähler metric. The complex structure is defined by considering as $( 1,0 )$-forms on the Lie algebra of $\Omega G$, those with positive Fourier coefficients. Equivalently, $\Omega G$ is a homogeneous space of the loop group $L G _ { \text{C} } = \{ \gamma : S ^ { 1 } \rightarrow G _ { \text{C} } \}$, with isotropy

\begin{equation*} L^+G _ { \mathbf{C} } = \left\{ \begin{array}{l}{ \\ \gamma \in L G _ { \mathbf{C} } :\\ }\end{array} \begin{array}{c}{ \gamma \text{ extends} \\ \text{ holomorphically in the disc } }\\{ \text { to a group } "\square" \text{valued mapping }}\end{array} \right\}. \end{equation*}

This is shown by considering the "Grassmannian model" of D. Quillen for $\Omega G$. For simplicity, take $G = U ( n )$; then $\Omega U ( n )$ is naturally identified with the Hilbert–Schmidt Grassmannian of certain Hilbert subspaces of $L ^ { 2 } ( S ^ { 1 } , \mathbf{C} ^ { n } )$, stable under multiplication by $z$. This also shows that $\Omega G$ has a holomorphic embedding into a projectivized Hilbert space, by generalized Plücker coordinates; and a determinant line bundle.

While the smooth loop group is an infinite-dimensional Fréchet manifold, the rational and polynomial loop groups have natural filtrations by finite-dimensional algebraic subvarieties. Moreover, by a result of G. Segal, any holomorphic mapping from a compact manifold into $\Omega G$ actually goes into the rational loops (up to multiplication by a constant). This has application in gauge theory, because a theorem of M.F. Atiyah and S.K. Donaldson identifies, for a classical group $G$, the moduli space of charge-$k$ framed $G$-instantons with the moduli spaces of based holomorphic $2$-spheres in $\Omega G$, of topological degree $k$. Another application to gauge theory and to the theory of completely integrable systems is given by a construction of K. Uhlenbeck, refining earlier work by V.E. Zakharov and others: this identifies, modulo basepoints, harmonic mappings $f : S ^ { 2 } \rightarrow G$ with certain holomorphic mappings $F : S ^ { 2 } \rightarrow \Omega G$. The degree $k$ of $F$ now gives the energy of $f$, by a formula of G. Valli, generalizing earlier work of E. Calabi.

The representation theory of has also been studied: the key point is the construction of a universal central extension, which makes it possible to define infinite-dimensional projective representations.

Other spaces commonly known as "loop groups" are the group of diffeomorphism of the circle, $\operatorname{Diff}( S ^ { 1 } )$, and the loop space of a manifold, $LM$.

The space $\operatorname{Diff}( S ^ { 1 } )$ has been studied as reparametrization groups for string theory. It has two natural homogeneous spaces: $\operatorname{Diff} ( S ^ { 1 } ) / \operatorname{SL} ( 2 , {\bf R} )$ and $\operatorname{Diff}( S ^ { 1 } ) / S ^ { 1 }$, which are infinite-dimensional Kähler manifolds (the Kähler metric is the metric $L _ { 3 / 2 } ^ { 2 }$). The space $\operatorname{Diff} ( S ^ { 1 } ) / \operatorname{SL} ( 2 , {\bf R} )$ has been studied in the theory of universal Teichmüller spaces (cf. also Teichmüller space).

When $M$ is a manifold, $LM$ is defined as the space of loops in $M$ starting at a fixed basepoint. Here, the group operation is given by composition of loops. The space $LM$ has been considered in connection with the problem of characterizing parallel transport operators, as configuration space in string theory, and in probability theory.

References

[a1] A. Pressley, G. Segal, "Loop groups" , Oxford Univ. Press (1986)
How to Cite This Entry:
Loop group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loop_group&oldid=50219
This article was adapted from an original article by Giorgio Valli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article