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''Kac–Moody Lie algebra''
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{{MSC|17B67}}
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{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550501.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550502.png" />-matrix satisfying conditions (see [[Cartan matrix|Cartan matrix]])
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A ''Kac-Moody algebra'' (also ''Kac–Moody Lie algebra'') is defined as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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Let $A=(a_{ij})_{i,j=1}^n$ be an $(n\times n)$-matrix satisfying conditions (see
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[[Cartan matrix|Cartan matrix]])
  
The associated Kac–Moody algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550504.png" /> is a Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550505.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550506.png" /> generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k0550509.png" /> (called the Chevalley generators) and the following defining relations:
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$$\begin{equation}  \left.\begin{aligned}  a_{ii}=2;\quad a_{ij}\le 0 \ &\textrm{and}\ a_{ij} \in \Z\textrm{ for }i\ne j,\\
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        a_{ij}=0\ &\;\;\Rightarrow\  a_{ji}=0.       \end{aligned}
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  \right\}\qquad\tag{a1}  \end{equation}$$
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The associated Kac–Moody algebra $\def\fg{\mathfrak{g}}\fg(A)$ is a Lie algebra over $\C$ on $3n$ generators $e_i$, $f_i$, $h_i$ (called the Chevalley generators) and the following defining relations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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$$ \def\ad{\textrm{ad}\;}
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\begin{equation}  \left.\begin{aligned}   
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[h_i,h_j] = 0,\quad [e_i,f_i] = h_i, \quad [e_i,f_j] = 0\quad &\textrm{ if }i\ne j,\\
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[h_i,e_j] = a_{ij}e_j,\quad [h_i,f_j]=a_{ij}f_j,\quad &\\
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(\ad e_i)^{1-a_{ij}} e_j = 0, \quad (\ad f_i)^{1-a_{ij}} f_j = 0, &\textrm{ if } i\ne j.\end{aligned}\quad\right\}\quad\tag{a2}\end{equation}$$
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The Lie algebra $\fg(A)$ is finite dimensional if and only if the matrix $A$ is positive definite (i.e. all principal minors of $A$ are positive, cf. also
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[[Minor|Minor]]). One gets in this way all finite-dimensional semi-simple Lie algebras over $\C$ (see
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[[Lie algebra, semi-simple|Lie algebra, semi-simple]]). Thus, Kac–Moody algebras are infinite-dimensional analogues of the finite-dimensional semi-simple Lie algebras.
  
The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505011.png" /> is finite dimensional if and only if the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505012.png" /> is positive definite (i.e. all principal minors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505013.png" /> are positive, cf. also [[Minor|Minor]]). One gets in this way all finite-dimensional semi-simple Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505014.png" /> (see [[Lie algebra, semi-simple|Lie algebra, semi-simple]]). Thus, Kac–Moody algebras are infinite-dimensional analogues of the finite-dimensional semi-simple Lie algebras.
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A systematic study of Kac–Moody algebras was started independently by V.G. Kac
 +
{{Cite|Ka}} and R.V. Moody
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{{Cite|Mo}}, and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac–Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf.
 +
[[Casimir element|Casimir element]]), which can be constructed provided that the matrix $A$ is symmetrizable, i.e. $A=DB$ for some invertible diagonal matrix $D$ and symmetric matrix $B$
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{{Cite|Ka2}}. In the non-symmetrizable case more sophisticated geometric methods are required
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{{Cite|Ku}},
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{{Cite|Ma}}.
  
A systematic study of Kac–Moody algebras was started independently by V.G. Kac [[#References|[a1]]] and R.V. Moody [[#References|[a2]]], and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac–Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf. [[Casimir element|Casimir element]]), which can be constructed provided that the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505015.png" /> is symmetrizable, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505016.png" /> for some invertible diagonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505017.png" /> and symmetric matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505018.png" /> [[#References|[a3]]]. In the non-symmetrizable case more sophisticated geometric methods are required [[#References|[a4]]], [[#References|[a5]]].
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One of the most important ingredients of the theory of Kac–Moody algebras are integrable highest-weight representations (cf. also
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[[Representation with a highest weight vector|Representation with a highest weight vector]]). Given an $n$-tuple of non-negative integers $\def\L{\Lambda}\def\l{\lambda}\L=(\l_1,\dots,\l_n)$, the integrable highest-weight representation $\pi_\L$ of a Kac–Moody algebra $\fg(A)$ is its irreducible representation on a complex vector space $L(\L)$, which is determined by the property that there exists a non-zero vector $v_\L\in L(\L)$ such that
  
One of the most important ingredients of the theory of Kac–Moody algebras are integrable highest-weight representations (cf. also [[Representation with a highest weight vector|Representation with a highest weight vector]]). Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505019.png" />-tuple of non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505020.png" />, the integrable highest-weight representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505021.png" /> of a Kac–Moody algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505022.png" /> is its irreducible representation on a complex vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505023.png" />, which is determined by the property that there exists a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505024.png" /> such that
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$$\pi_\L(e_i)v_\L = 0\quad\textrm{and}\quad \pi_\L(h_i)v_\L = \l_i v_\L,\quad i=1,\dots,n.$$
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Note that $\pi_\L$ are precisely all irreducible finite-dimensional representations of a finite-dimensional Kac–Moody algebra $\fg(A)$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505025.png" /></td> </tr></table>
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The basic result of the theory of integrable highest-weight representations is the Weyl–Kac character formula
 
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{{Cite|Ka2}}, which gives an explicit expression for the formal power series $\def\tr{\textrm{tr}\;}\tr_{L(\L)} e^{\sum_i x_i\pi_\L(h_i)}$ in the $x_i$ in terms of $\L$ (see also
Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505026.png" /> are precisely all irreducible finite-dimensional representations of a finite-dimensional Kac–Moody algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505027.png" />.
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[[Character formula|Character formula]]).
 
 
The basic result of the theory of integrable highest-weight representations is the Weyl–Kac character formula [[#References|[a3]]], which gives an explicit expression for the formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505028.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505029.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505030.png" /> (see also [[Character formula|Character formula]]).
 
  
 
The numerous applications of Kac–Moody algebras are mainly related to the fact that the Kac–Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.) These Kac–Moody algebras are called affine algebras.
 
The numerous applications of Kac–Moody algebras are mainly related to the fact that the Kac–Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.) These Kac–Moody algebras are called affine algebras.
  
Below a construction of  "non-twisted"  affine algebras is given. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505031.png" /> be a positive-definite indecomposable Cartan matrix and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505032.png" /> be the associated simple finite-dimensional Lie algebra with Chevalley generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505034.png" />. There exists a unique (up to a constant multiple) non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505035.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505036.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505038.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505039.png" />) vanishes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505040.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505041.png" />, a linear combination of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505042.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505043.png" />), and one normalizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505045.png" /> by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505047.png" />. Then: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505050.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505051.png" /> are certain non-positive integers, and one puts
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Below a construction of  "non-twisted"  affine algebras is given. Let $A$ be a positive-definite indecomposable Cartan matrix and let $\fg=\fg(A)$ be the associated simple finite-dimensional Lie algebra with Chevalley generators $E_i,\;F_i\;H_i\;$, $i=1,\dots,r$. There exists a unique (up to a constant multiple) non-zero element $E_0$ (respectively, $F_0$) in $\fg$ such that $[E_0,F_i]$ (respectively, $[F_0,E_i]$) vanishes for $i=1,\dots,r$. Then $[E_0,F_0] = H_0$, a linear combination of the $H_i$ ($i=1,\dots,r$), and one normalizes $E_0$ and $F_0$ by the conditions $[H_0,E_0]=2E_0$, $[H_0,F_0]=-2F_0$. Then: $[H_0,E_i] = a_{0i} E_i$, $[H_i,E_0] = a_{i0} E_0$ for $i=1,\dots,r$, where the $a_{0i}, a_{i0}$ are certain non-positive integers, and one puts
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505052.png" /></td> </tr></table>
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$$A^{(1)} = \begin{pmatrix} 2 & a_{01} & \dots & a_{0r}\\
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a_{10} & &  & \\
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\vdots & & A & \\
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a_{r0} & &  & \end{pmatrix}.$$
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This is a positive semi-definite $(r+1,r+1)$ matrix satisfying (a1) (called the extended Cartan matrix of $\fg$). These matrices, along with all affine matrices, are listed in
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[[Lie algebra, graded|Lie algebra, graded]]. The associated affine algebra
  
This is a positive semi-definite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505053.png" /> matrix satisfying (a1) (called the extended Cartan matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505054.png" />). These matrices, along with all affine matrices, are listed in [[Lie algebra, graded|Lie algebra, graded]]. The associated affine algebra
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$$\fg(A^{(1)}) = \Big(\bigoplus_{n\in\Z} \fg^{(n)}\Big) \oplus \C k,$$
 +
where $\fg^{(n)}$ is the $n$-th copy of $\fg$ and $k$ is a central element (i.e. $[k,\fg(A)] = 0$), with the following commutation relations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505055.png" /></td> </tr></table>
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$$[x^{(m)},y^{(n)}] = [x,y]^{(m+n)} + m\delta_{m,-n}(x|y)k.$$
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Here $x,y\in\fg$, $x^{(n)}$ denotes the element $x$ taken from $\fg^{(n)}$ and $(\cdot|\cdot)$ is the
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[[Killing form|Killing form]] on $\fg$ normalized by the condition $(H_0|H_0) = 2$. (Note that for $\fg = \def\fsl{\mathfrak{sl}}\fsl_n(\C)$, one has $(x|y) = \tr xy$.) The canonical generators of $\fg(A^{(1)})$ are:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505056.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505057.png" />-th copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505059.png" /> is a central element (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505060.png" />), with the following commutation relations:
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$$e_0 = E_0^{(1)},\quad f_0 = F_0^{(-1)},\quad h_0 = c - H_0^{(0)},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505061.png" /></td> </tr></table>
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$$e_i=E_i^{(0)},\quad f_i=F_i^{(0)},$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505063.png" /> denotes the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505064.png" /> taken from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505066.png" /> is the [[Killing form|Killing form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505067.png" /> normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505068.png" />. (Note that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505069.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505070.png" />.) The canonical generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505071.png" /> are:
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$$h_i=H_i^{(0)},\quad \textrm{ for } i=1,\dots,r.$$
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In more geometric terms, $\fg(A^{(1)})$ is a central extension (by $\C k$) of the loop algebra, i.e. the Lie algebra of regular mappings of $\C^*$ to $\fg$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505072.png" /></td> </tr></table>
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$$\fg(A^{(1)}) = (\C[z,z^{-1}]\otimes_\C \fg)+\C k.$$
 +
This observation leads to geometric applications of affine algebras and the corresponding groups, called the [[loop group]]s (see
 +
{{Cite|PrSe}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505073.png" /></td> </tr></table>
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In an integrable highest-weight representation $\pi_\L$, the central element $k$ acts as a non-negative integral scalar, also denoted by $k$, which is called the level of $\pi_\L$. The only $\pi_\L$ of level $0$ is the trivial representation. A remarkable feature of the representation theory of the affine algebras is the existence of explicit canonical constructions for the $\pi_\L$ of level $1$. Below the simplest  "vertex operator" construction for the basic representation $\pi=\pi_{\L_0}$, where $\L_0 = (1,0,\dots,0)$, is explained. It uses the vertex operators, which are defined as follows. Let $\def\fh{\mathfrak{h}}$ be an $r$-dimensional complex vector space with a symmetric bilinear form $(\cdot|\cdot)$ and let $Q$ be a lattice in $\fh$ of rank $r$. For each $n\in\Z$, take a copy $\fh^{(n)}$ of $\fh$ and let $\fh^- = \otimes_{n<0} h^{(n)}$. Let $S(\fh^-)$ be the symmetric algebra over $\fh^-$ and let $\C[Q]$ be the group algebra of $Q$ with the inclusion $Q\to\C[Q]$ denoted by $\def\a{\alpha}\a \mapsto e^\a$. Consider the complex commutative associative algebra
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505074.png" /></td> </tr></table>
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$$V=S(\fh^-)\otimes_\C\C[A].$$
 +
For $u\in \fh$ and $n\in\Z$ define an operator $u(n)$ on $V$ as follows. For $n>0$, $u(-n)$ is the operator of multiplication by $u{(-n)}\in\fh^{(-n)}$; for $n\ge 0$, $u(n)$ is the derivation of $V$ defined by:
  
In more geometric terms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505075.png" /> is a central extension (by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505076.png" />) of the loop algebra, i.e. the Lie algebra of regular mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505077.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505078.png" />:
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$$u(n)(u_1^{(-n_1)}) = n(u|u_1)\delta_{n,n_1},\quad u(n)(e^\a) = \delta_{n,0}(u|\a)e^\a.$$
 +
For $\def\g{\gamma}\g\in Q$ such that $(\g|\g) = 2$, define the vertex operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505079.png" /></td> </tr></table>
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$$X(\g,z) = \Big(\exp\sum_{j\ge 1}\frac{z^j}{j}\g(-j)\Big)
 +
\Big(\exp -\sum_{j\ge 1}\frac{z^{-j}}{j}\g(j)\Big)e^\g z^{\g(0)},$$
 +
where $z\in\C^*$. Expanding in powers of $z$: $X(\a,z) = \sum_{n\in\Z}X_n(\g)z^{-n-1}$, one obtains a sequence of operators $X_n(\g)$ on $V$. Let now $\fg$ be a simple Lie algebra with the Cartan matrix $A=A_r$, $D_r$ or $E_r$, choose a
 +
[[Cartan subalgebra|Cartan subalgebra]] $\fh$ of $\fg$, let $Q\subset \fh$ be the root lattice (identifying $\fh$ with $\fh^*$ using the form $(\cdot|\cdot)$), and let $\def\D{\Delta}\D=\{\a\in Q\;|\; (\a|\a)=2\}$ be the
 +
[[Root system|root system]] of $\fg$. Choose a bimultiplicative function $\def\b{\beta}\def\e{\epsilon}\e(\a,\b)$ on $Q$ with values $\pm 1$ such that $\e(\a,\a) = (-1)^{(\a|\a)/2}$. For $\g\in Q$, define an operator $c_\g$ on $V$ by $c_\g(f\otimes e^\b) = \e(\g,\b)f\otimes e^\b$. Then $\fg =\fh \oplus \sum_{\a\in\D}\C E_\a$ with commutation relations:
  
This observation leads to geometric applications of affine algebras and the corresponding groups, called the loop groups (see [[#References|[a6]]]).
+
$$[h,h]=0;\quad [h,E_\a] = (\a|h)E_\a \quad \textrm{ for } h\in\fh;$$
  
In an integrable highest-weight representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505080.png" />, the central element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505081.png" /> acts as a non-negative integral scalar, also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505082.png" />, which is called the level of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505083.png" />. The only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505084.png" /> of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505085.png" /> is the trivial representation. A remarkable feature of the representation theory of the affine algebras is the existence of explicit canonical constructions for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505086.png" /> of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505087.png" />. Below the simplest  "vertex operator"  construction for the basic representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505089.png" />, is explained. It uses the vertex operators, which are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505090.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505091.png" />-dimensional complex vector space with a symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505092.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505093.png" /> be a lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505094.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505095.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505096.png" />, take a copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505097.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505098.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k05505099.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050100.png" /> be the symmetric algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050101.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050102.png" /> be the group algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050103.png" /> with the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050104.png" /> denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050105.png" />. Consider the complex commutative associative algebra
+
$$[E_\a,E_\b] = 0 \quad \textrm{ if } \a+\b\not\in\D\cup\{0\};$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050106.png" /></td> </tr></table>
+
$$[E_\a,E_{-\a}] = -\a; \quad [E_\a,E_\b] = \e(\a,\b) \quad \textrm{  if } \a+\b\in\D.$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050108.png" /> define an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050109.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050110.png" /> as follows. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050112.png" /> is the operator of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050113.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050115.png" /> is the derivation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050116.png" /> defined by:
+
The basic representation of $\fg(A^{(1)})$ is then defined on $V$ by the following formulas
 +
{{Cite|FrKa}}:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050117.png" /></td> </tr></table>
+
$$\pi(u^{(n)}) = u(n),\quad u\in\fh$$
 
 
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050118.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050119.png" />, define the vertex operator
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050120.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050121.png" />. Expanding in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050122.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050123.png" />, one obtains a sequence of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050124.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050125.png" />. Let now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050126.png" /> be a simple Lie algebra with the Cartan matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050128.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050129.png" />, choose a [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050130.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050131.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050132.png" /> be the root lattice (identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050133.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050134.png" /> using the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050135.png" />), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050136.png" /> be the [[Root system|root system]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050137.png" />. Choose a bimultiplicative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050138.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050139.png" /> with values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050140.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050141.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050142.png" />, define an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050143.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050144.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050145.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050146.png" /> with commutation relations:
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050147.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050148.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050149.png" />;
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050150.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050151.png" />;
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050152.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050153.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050154.png" />.
 
 
 
The basic representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050155.png" /> is then defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050156.png" /> by the following formulas [[#References|[a11]]]:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050157.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050158.png" /></td> </tr></table>
 
  
 +
$$\pi(E_\a^{(n)}) = X_n(\a)c_\a, \quad \pi(k) = 1;$$
 
This is called the homogeneous vertex operator construction of the basic representation.
 
This is called the homogeneous vertex operator construction of the basic representation.
  
The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [[#References|[a12]]]).
+
The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see
 
+
{{Cite|GrScWi}}).
The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the [[Leech lattice|Leech lattice]] produced the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050159.png" />-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see [[Sporadic simple group|Sporadic simple group]]) [[#References|[a13]]].
 
 
 
The vertex operator constructions were, quite unexpectedly, applied to the theory of [[Soliton|soliton]] equations. This was based on the observation (see [[#References|[a14]]]) that the orbit of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050160.png" /> of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the [[Korteweg–de Vries equation|Korteweg–de Vries equation]].
 
  
The link of the representation theory of affine algebras to the conformal field theory is given by the Sugawara construction. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050161.png" /> be a simple finite-dimensional algebra and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050162.png" /> be the corresponding affine algebra. Choose bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050164.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050165.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050166.png" />. Let
+
The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the
 +
[[Leech lattice|Leech lattice]] produced the $196883$-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see
 +
[[Sporadic simple group|Sporadic simple group]])
 +
{{Cite|FrLeMe}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050167.png" /></td> </tr></table>
+
The vertex operator constructions were, quite unexpectedly, applied to the theory of
 +
[[Soliton|soliton]] equations. This was based on the observation (see
 +
{{Cite|DaJiKaMi}}) that the orbit of the vector $v_{\L_0}$ of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the
 +
[[Korteweg–de Vries equation|Korteweg–de Vries equation]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050168.png" /></td> </tr></table>
+
The link of the representation theory of affine algebras to the conformal field theory is given by the Sugawara construction. Let $\fg(A)$ be a simple finite-dimensional algebra and let $\fg(A^{(1)})$ be the corresponding affine algebra. Choose bases $u_i$ and $v_i$ of $\fg(A)$ such that $\def\d{\delta}(u_i|v_j) =\d_{ij}$. Let
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050169.png" /> is the dual Coxeter number (defined by: Killing form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050170.png" />). Then one has
+
$$L_0 = \frac{1}{2(k+h^\nu)}\Big(\sum_i(u_iv_i + 2\sum_{n>0} u_i^{(n)}v_i^{(n)})\Big),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050171.png" /></td> </tr></table>
+
$$L_m = \frac{1}{2(k+h^\nu)}\sum_i\sum_{n\in\Z} u_i^{(n)}v_i^{(m+n)}\quad\textrm{if } m\ne 0.$$
 +
Here $h^\nu$ is the dual Coxeter number (defined by: Killing form $=2h^\nu(\cdot|\cdot)$). Then one has
  
 +
$$[L_m,L_n] = (m-n)L_{m+n}\d_{m,-n}\frac{m^3-m}{12}c(k),$$
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050172.png" /></td> </tr></table>
+
$$c(k) = \frac{k\dim\fg(A)}{k+h^\nu},$$
 
+
producing thereby a representation of the
producing thereby a representation of the [[Virasoro algebra|Virasoro algebra]].
+
[[Virasoro algebra|Virasoro algebra]].
 
 
The character of an integrable highest-weight representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050173.png" /> of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050174.png" /> of an affine algebra, multiplied by a suitable power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050175.png" />, can be written in the following form:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050176.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050178.png" />. This is a series which converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050179.png" /> to a [[Modular function|modular function]]. Moreover, the linear span of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050180.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050181.png" /> of fixed level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050182.png" /> is invariant under the modular transformations
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050183.png" /></td> </tr></table>
+
The character of an integrable highest-weight representation $L(\L)$ of level $k$ of an affine algebra, multiplied by a suitable power of $\exp 2\pi i\tau$, can be written in the following form:
  
and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050184.png" /> of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050185.png" /> is known explicitly [[#References|[a7]]]. For example, in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050186.png" />,
+
$$\chi_\L(\tau,z) = \tr_{L(\L)} e^{2\pi i \tau(L_0 - c(k)/24)+2\pi iz},$$
 +
where $\tau\in\C$ and $z\in\fh$. This is a series which converges for $\def\Im{\textrm{Im}\;}\Im \tau > 0$ to a
 +
[[Modular function|modular function]]. Moreover, the linear span of the functions $\chi_\Lambda(\tau,0)$ for $\Lambda$ of fixed level $k$ is invariant under the modular transformations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050187.png" /></td> </tr></table>
+
$$\tau\mapsto \frac{a\tau + b}{c\tau + d},\quad \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\textrm{SL}_2(\Z),$$
 +
and the matrix $S$ of the transformation $\tau\mapsto -1/\tau$ is known explicitly
 +
{{Cite|KaPe}}. For example, in the case $\fg = \fsl_2(\C)$,
  
This turned out to be a key fact in the representation theory of affine algebras, as well as its applications to conformal field theory (see [[#References|[a8]]]), to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050188.png" />-dimensional lattice models [[#References|[a9]]], and even to [[Knot theory|knot theory]] [[#References|[a10]]].
+
$$S=\Bigg(\sqrt{\frac{2}{k+2}}\sin\frac{\pi(r+1)(s+1)}{k+2}\Bigg)_{r,s=0}^k$$
 +
This turned out to be a key fact in the representation theory of affine algebras, as well as its applications to conformal field theory (see
 +
{{Cite|Ve}}), to $2$-dimensional lattice models
 +
{{Cite|DaJiKuMiOk}}, and even to
 +
[[Knot theory|knot theory]]{{Cite|YaGe}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.G. Kac,  "Simple irreducible graded Lie algebras of finite growth"  ''Math. USSR Izv.'' , '''2''' (1968)  pp. 1271–1311 ''Izv. Akad. Nauk USSR Ser. Mat.'' , '''32'''  (1968)  pp. 1923–1967</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.V. Moody,  "A new class of Lie algebras"  ''J. of Algebra'' , '''10'''  (1968)  pp. 211–230</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.G. Kac,  "Infinite-dimensional Lie algebras and Dedekind's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050189.png" />-function"  ''Funct. Anal. Appl.'' , '''8'''  (1974)  pp. 68–70 ''Funkts. Anal. i Prilozhen.'' , '''8''' :  1 (1974)  pp. 77–78</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Kumar,  "Demazure character formula in arbitrary Kac–Moody setting"  ''Invent. Math.'' , '''89'''  (1987)  pp. 395–423</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  O. Mathieu,  "Formules de caractères pour les algèbres de Kac–Moody générales" ''Astérisque'' , '''159–160'''  (1988)  pp. 1–266</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Pressley,  G. Segal,  "Loop groups" , Oxford Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> V.G. Kac,  D.H. Peterson,  "Infinite-dimensional Lie algebras, theta functions and modular forms"  ''Adv. in Math.'' , '''53'''  (1984)  pp. 125–264</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> E. Verlinde"Fusion rules and modular transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055050/k055050190.png" /> conformal field theory''Nucl. Phys.'' , '''B300''' pp. 360–375</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> E. Date,  M. Jimbo,  A. Kuniba,  T. Miwa,  M. Okado,  "Exactly solvable SOS models"  ''Nucl. Phys.'' , '''B290'''  (1987)  pp. 231–273</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> C.N. Yang (ed.) M.L. Ge (ed.) , ''Braid group, knot theory and statistical mechanics'' , World Sci. (1989)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> I.B. Frenkel,  V.G. Kac,  "Basic representations of affine Lie algebras and dual resonance models"  ''Invent. Math.'' , '''62'''  (1980)  pp. 23–66</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> M.B. GreenJ.H. Schwarz,  E. Witten,  "Superstring theory" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> I. Frenkel,  J. Lepowsky,  A. Meurman,  "Vertex operator algebras and the Monster" , Acad. Press  (1989)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> E. Date,  M. Jimbo,  M. Kashiwara,  T. Miwa,  "Transformation groups for soliton equations"  M. Jimbo (ed.)  T. Miwa (ed.) , ''Proc. RIMS Symp.'' , World Sci.  (1983pp. 39–120</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> V.G. Kac,  "Infinite-dimensional Lie algebras" , Cambridge Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  V.G. Kac,  A.K. Raina,  "Bombay lectures on highest weight representations" , World Sci.  (1987)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|DaJiKaMi}}||valign="top"| E. Date,  M. Jimbo,  M. Kashiwara,  T. Miwa,  "Transformation groups for soliton equationsM. Jimbo (ed.)  T. Miwa (ed.), ''Proc. RIMS Symp.'', World Sci. (1983)  pp. 39–120 
 +
|-
 +
|valign="top"|{{Ref|DaJiKuMiOk}}||valign="top"|  E. Date,  M. Jimbo,  A. Kuniba,  T. Miwa,  M. Okado,  "Exactly solvable SOS models" ''Nucl. Phys.'', '''B290'''  (1987)  pp. 231–273  {{MR|0910849}}  {{ZBL|0679.17010}}
 +
|-
 +
|valign="top"|{{Ref|FrKa}}||valign="top"| I.B. Frenkel,  V.G. Kac,  "Basic representations of affine Lie algebras and dual resonance models"  ''Invent. Math.'', '''62'''  (1980)  pp. 23–66  {{MR|0595581}}  {{ZBL|0493.17010}}
 +
|-
 +
|valign="top"|{{Ref|FrLeMe}}||valign="top"| I. Frenkel,  J. Lepowsky,  A. Meurman,  "Vertex operator algebras and the Monster", Acad. Press  (1989)  {{MR|1167718}} {{MR|0996026}}  {{ZBL|0674.17001}}
 +
|-
 +
|valign="top"|{{Ref|GrScWi}}||valign="top"|  M.B. Green,  J.H. Schwarz,  E. Witten,  "Superstring theory", Cambridge Univ. Press  (1987)  {{MR|0922731}} {{MR|0915347}} {{MR|0878144}} {{MR|0878143}}  {{ZBL|0637.53111}} {{ZBL|0619.53002}}
 +
|-
 +
|valign="top"|{{Ref|Ka}}||valign="top"|  V.G. Kac,  "Simple irreducible graded Lie algebras of finite growth"  ''Math. USSR Izv.'', '''2'''  (1968)  pp. 1271–1311 ''Izv. Akad. Nauk USSR Ser. Mat.'', '''32'''  (1968)  pp. 1923–1967  {{MR|0259961}}  {{ZBL|0222.17007}}
 +
|-
 +
|valign="top"|{{Ref|Ka2}}||valign="top"| V.G. Kac,  "Infinite-dimensional Lie algebras and Dedekind's $\eta$-function"  ''Funct. Anal. Appl.'', '''8'''  (1974)  pp. 68–70 ''Funkts. Anal. i Prilozhen.'', '''8''' :  1 (1974)  pp. 77–78  {{MR|0374210}}  {{ZBL|0299.17005}}
 +
|-
 +
|valign="top"|{{Ref|Ka3}}||valign="top"| V.G. Kac,  "Infinite-dimensional Lie algebras", Cambridge Univ. Press  (1985) {{MR|0823672}} {{MR|0823320}} {{MR|0802097}}  {{ZBL|0614.22006}} {{ZBL|0581.17009}} {{ZBL|0574.17010}}
 +
|-
 +
|valign="top"|{{Ref|KaPe}}||valign="top"| V.G. Kac,  D.H. Peterson,  "Infinite-dimensional Lie algebras, theta functions and modular forms"  ''Adv. in Math.'', '''53'''  (1984)  pp. 125–264 {{MR|0750341}}  {{ZBL|0584.17007}}
 +
|-
 +
|valign="top"|{{Ref|KaRa}}||valign="top"| V.G. KacA.K. Raina,  "Bombay lectures on highest weight representations", World Sci. (1987)  {{MR|1021978}} {{ZBL|0668.17012}}
 +
|-
 +
|valign="top"|{{Ref|Ku}}||valign="top"| S. Kumar,  "Demazure character formula in arbitrary Kac–Moody setting"  ''Invent. Math.'', '''89'''  (1987)  pp. 395–423  {{MR|0894387}}  {{ZBL|0635.14023}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"| O. Mathieu,  "Formules de caractères pour les algèbres de Kac–Moody générales" ''Astérisque'', '''159–160'''  (1988) pp. 1–266  {{ZBL|0683.17010}}
 +
|-
 +
|valign="top"|{{Ref|Mo}}||valign="top"| R.V. Moody,  "A new class of Lie algebras"  ''J. of Algebra'', '''10'''  (1968)  pp. 211–230  {{MR|0229687}}  {{ZBL|0191.03005}}
 +
|-
 +
|valign="top"|{{Ref|PrSe}}||valign="top"| A. PressleyG. Segal,  "Loop groups", Oxford Univ. Press  (1986) {{MR|0900587}} {{MR|0849057}}  {{ZBL|0618.22011}} {{ZBL|0603.17012}}
 +
|-
 +
|valign="top"|{{Ref|Ve}}||valign="top"| E. Verlinde,  "Fusion rules and modular transformations in $2D$ conformal field theory" ''Nucl. Phys.'', '''B300'''  pp. 360–375
 +
|-
 +
|valign="top"|{{Ref|YaGe}}||valign="top"| C.N. Yang (ed.)  M.L. Ge (ed.), ''Braid group, knot theory and statistical mechanics'', World Sci.  (1989{{MR|1062420}} {{ZBL|0716.00010}}
 +
|-
 +
|}

Latest revision as of 20:27, 15 November 2017

2020 Mathematics Subject Classification: Primary: 17B67 [MSN][ZBL]

A Kac-Moody algebra (also Kac–Moody Lie algebra) is defined as follows:

Let $A=(a_{ij})_{i,j=1}^n$ be an $(n\times n)$-matrix satisfying conditions (see Cartan matrix)

$$\begin{equation} \left.\begin{aligned} a_{ii}=2;\quad a_{ij}\le 0 \ &\textrm{and}\ a_{ij} \in \Z\textrm{ for }i\ne j,\\ a_{ij}=0\ &\;\;\Rightarrow\ a_{ji}=0. \end{aligned} \right\}\qquad\tag{a1} \end{equation}$$ The associated Kac–Moody algebra $\def\fg{\mathfrak{g}}\fg(A)$ is a Lie algebra over $\C$ on $3n$ generators $e_i$, $f_i$, $h_i$ (called the Chevalley generators) and the following defining relations:

$$ \def\ad{\textrm{ad}\;} \begin{equation} \left.\begin{aligned} [h_i,h_j] = 0,\quad [e_i,f_i] = h_i, \quad [e_i,f_j] = 0\quad &\textrm{ if }i\ne j,\\ [h_i,e_j] = a_{ij}e_j,\quad [h_i,f_j]=a_{ij}f_j,\quad &\\ (\ad e_i)^{1-a_{ij}} e_j = 0, \quad (\ad f_i)^{1-a_{ij}} f_j = 0, &\textrm{ if } i\ne j.\end{aligned}\quad\right\}\quad\tag{a2}\end{equation}$$ The Lie algebra $\fg(A)$ is finite dimensional if and only if the matrix $A$ is positive definite (i.e. all principal minors of $A$ are positive, cf. also Minor). One gets in this way all finite-dimensional semi-simple Lie algebras over $\C$ (see Lie algebra, semi-simple). Thus, Kac–Moody algebras are infinite-dimensional analogues of the finite-dimensional semi-simple Lie algebras.

A systematic study of Kac–Moody algebras was started independently by V.G. Kac [Ka] and R.V. Moody [Mo], and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac–Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf. Casimir element), which can be constructed provided that the matrix $A$ is symmetrizable, i.e. $A=DB$ for some invertible diagonal matrix $D$ and symmetric matrix $B$ [Ka2]. In the non-symmetrizable case more sophisticated geometric methods are required [Ku], [Ma].

One of the most important ingredients of the theory of Kac–Moody algebras are integrable highest-weight representations (cf. also Representation with a highest weight vector). Given an $n$-tuple of non-negative integers $\def\L{\Lambda}\def\l{\lambda}\L=(\l_1,\dots,\l_n)$, the integrable highest-weight representation $\pi_\L$ of a Kac–Moody algebra $\fg(A)$ is its irreducible representation on a complex vector space $L(\L)$, which is determined by the property that there exists a non-zero vector $v_\L\in L(\L)$ such that

$$\pi_\L(e_i)v_\L = 0\quad\textrm{and}\quad \pi_\L(h_i)v_\L = \l_i v_\L,\quad i=1,\dots,n.$$ Note that $\pi_\L$ are precisely all irreducible finite-dimensional representations of a finite-dimensional Kac–Moody algebra $\fg(A)$.

The basic result of the theory of integrable highest-weight representations is the Weyl–Kac character formula [Ka2], which gives an explicit expression for the formal power series $\def\tr{\textrm{tr}\;}\tr_{L(\L)} e^{\sum_i x_i\pi_\L(h_i)}$ in the $x_i$ in terms of $\L$ (see also Character formula).

The numerous applications of Kac–Moody algebras are mainly related to the fact that the Kac–Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.) These Kac–Moody algebras are called affine algebras.

Below a construction of "non-twisted" affine algebras is given. Let $A$ be a positive-definite indecomposable Cartan matrix and let $\fg=\fg(A)$ be the associated simple finite-dimensional Lie algebra with Chevalley generators $E_i,\;F_i\;H_i\;$, $i=1,\dots,r$. There exists a unique (up to a constant multiple) non-zero element $E_0$ (respectively, $F_0$) in $\fg$ such that $[E_0,F_i]$ (respectively, $[F_0,E_i]$) vanishes for $i=1,\dots,r$. Then $[E_0,F_0] = H_0$, a linear combination of the $H_i$ ($i=1,\dots,r$), and one normalizes $E_0$ and $F_0$ by the conditions $[H_0,E_0]=2E_0$, $[H_0,F_0]=-2F_0$. Then: $[H_0,E_i] = a_{0i} E_i$, $[H_i,E_0] = a_{i0} E_0$ for $i=1,\dots,r$, where the $a_{0i}, a_{i0}$ are certain non-positive integers, and one puts

$$A^{(1)} = \begin{pmatrix} 2 & a_{01} & \dots & a_{0r}\\ a_{10} & & & \\ \vdots & & A & \\ a_{r0} & & & \end{pmatrix}.$$ This is a positive semi-definite $(r+1,r+1)$ matrix satisfying (a1) (called the extended Cartan matrix of $\fg$). These matrices, along with all affine matrices, are listed in Lie algebra, graded. The associated affine algebra

$$\fg(A^{(1)}) = \Big(\bigoplus_{n\in\Z} \fg^{(n)}\Big) \oplus \C k,$$ where $\fg^{(n)}$ is the $n$-th copy of $\fg$ and $k$ is a central element (i.e. $[k,\fg(A)] = 0$), with the following commutation relations:

$$[x^{(m)},y^{(n)}] = [x,y]^{(m+n)} + m\delta_{m,-n}(x|y)k.$$ Here $x,y\in\fg$, $x^{(n)}$ denotes the element $x$ taken from $\fg^{(n)}$ and $(\cdot|\cdot)$ is the Killing form on $\fg$ normalized by the condition $(H_0|H_0) = 2$. (Note that for $\fg = \def\fsl{\mathfrak{sl}}\fsl_n(\C)$, one has $(x|y) = \tr xy$.) The canonical generators of $\fg(A^{(1)})$ are:

$$e_0 = E_0^{(1)},\quad f_0 = F_0^{(-1)},\quad h_0 = c - H_0^{(0)},$$

$$e_i=E_i^{(0)},\quad f_i=F_i^{(0)},$$

$$h_i=H_i^{(0)},\quad \textrm{ for } i=1,\dots,r.$$ In more geometric terms, $\fg(A^{(1)})$ is a central extension (by $\C k$) of the loop algebra, i.e. the Lie algebra of regular mappings of $\C^*$ to $\fg$:

$$\fg(A^{(1)}) = (\C[z,z^{-1}]\otimes_\C \fg)+\C k.$$ This observation leads to geometric applications of affine algebras and the corresponding groups, called the loop groups (see [PrSe]).

In an integrable highest-weight representation $\pi_\L$, the central element $k$ acts as a non-negative integral scalar, also denoted by $k$, which is called the level of $\pi_\L$. The only $\pi_\L$ of level $0$ is the trivial representation. A remarkable feature of the representation theory of the affine algebras is the existence of explicit canonical constructions for the $\pi_\L$ of level $1$. Below the simplest "vertex operator" construction for the basic representation $\pi=\pi_{\L_0}$, where $\L_0 = (1,0,\dots,0)$, is explained. It uses the vertex operators, which are defined as follows. Let $\def\fh{\mathfrak{h}}$ be an $r$-dimensional complex vector space with a symmetric bilinear form $(\cdot|\cdot)$ and let $Q$ be a lattice in $\fh$ of rank $r$. For each $n\in\Z$, take a copy $\fh^{(n)}$ of $\fh$ and let $\fh^- = \otimes_{n<0} h^{(n)}$. Let $S(\fh^-)$ be the symmetric algebra over $\fh^-$ and let $\C[Q]$ be the group algebra of $Q$ with the inclusion $Q\to\C[Q]$ denoted by $\def\a{\alpha}\a \mapsto e^\a$. Consider the complex commutative associative algebra

$$V=S(\fh^-)\otimes_\C\C[A].$$ For $u\in \fh$ and $n\in\Z$ define an operator $u(n)$ on $V$ as follows. For $n>0$, $u(-n)$ is the operator of multiplication by $u{(-n)}\in\fh^{(-n)}$; for $n\ge 0$, $u(n)$ is the derivation of $V$ defined by:

$$u(n)(u_1^{(-n_1)}) = n(u|u_1)\delta_{n,n_1},\quad u(n)(e^\a) = \delta_{n,0}(u|\a)e^\a.$$ For $\def\g{\gamma}\g\in Q$ such that $(\g|\g) = 2$, define the vertex operator

$$X(\g,z) = \Big(\exp\sum_{j\ge 1}\frac{z^j}{j}\g(-j)\Big) \Big(\exp -\sum_{j\ge 1}\frac{z^{-j}}{j}\g(j)\Big)e^\g z^{\g(0)},$$ where $z\in\C^*$. Expanding in powers of $z$: $X(\a,z) = \sum_{n\in\Z}X_n(\g)z^{-n-1}$, one obtains a sequence of operators $X_n(\g)$ on $V$. Let now $\fg$ be a simple Lie algebra with the Cartan matrix $A=A_r$, $D_r$ or $E_r$, choose a Cartan subalgebra $\fh$ of $\fg$, let $Q\subset \fh$ be the root lattice (identifying $\fh$ with $\fh^*$ using the form $(\cdot|\cdot)$), and let $\def\D{\Delta}\D=\{\a\in Q\;|\; (\a|\a)=2\}$ be the root system of $\fg$. Choose a bimultiplicative function $\def\b{\beta}\def\e{\epsilon}\e(\a,\b)$ on $Q$ with values $\pm 1$ such that $\e(\a,\a) = (-1)^{(\a|\a)/2}$. For $\g\in Q$, define an operator $c_\g$ on $V$ by $c_\g(f\otimes e^\b) = \e(\g,\b)f\otimes e^\b$. Then $\fg =\fh \oplus \sum_{\a\in\D}\C E_\a$ with commutation relations:

$$[h,h]=0;\quad [h,E_\a] = (\a|h)E_\a \quad \textrm{ for } h\in\fh;$$

$$[E_\a,E_\b] = 0 \quad \textrm{ if } \a+\b\not\in\D\cup\{0\};$$

$$[E_\a,E_{-\a}] = -\a; \quad [E_\a,E_\b] = \e(\a,\b) \quad \textrm{ if } \a+\b\in\D.$$

The basic representation of $\fg(A^{(1)})$ is then defined on $V$ by the following formulas [FrKa]:

$$\pi(u^{(n)}) = u(n),\quad u\in\fh$$

$$\pi(E_\a^{(n)}) = X_n(\a)c_\a, \quad \pi(k) = 1;$$ This is called the homogeneous vertex operator construction of the basic representation.

The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).

The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice produced the $196883$-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].

The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) that the orbit of the vector $v_{\L_0}$ of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the Korteweg–de Vries equation.

The link of the representation theory of affine algebras to the conformal field theory is given by the Sugawara construction. Let $\fg(A)$ be a simple finite-dimensional algebra and let $\fg(A^{(1)})$ be the corresponding affine algebra. Choose bases $u_i$ and $v_i$ of $\fg(A)$ such that $\def\d{\delta}(u_i|v_j) =\d_{ij}$. Let

$$L_0 = \frac{1}{2(k+h^\nu)}\Big(\sum_i(u_iv_i + 2\sum_{n>0} u_i^{(n)}v_i^{(n)})\Big),$$

$$L_m = \frac{1}{2(k+h^\nu)}\sum_i\sum_{n\in\Z} u_i^{(n)}v_i^{(m+n)}\quad\textrm{if } m\ne 0.$$ Here $h^\nu$ is the dual Coxeter number (defined by: Killing form $=2h^\nu(\cdot|\cdot)$). Then one has

$$[L_m,L_n] = (m-n)L_{m+n}\d_{m,-n}\frac{m^3-m}{12}c(k),$$ where

$$c(k) = \frac{k\dim\fg(A)}{k+h^\nu},$$ producing thereby a representation of the Virasoro algebra.

The character of an integrable highest-weight representation $L(\L)$ of level $k$ of an affine algebra, multiplied by a suitable power of $\exp 2\pi i\tau$, can be written in the following form:

$$\chi_\L(\tau,z) = \tr_{L(\L)} e^{2\pi i \tau(L_0 - c(k)/24)+2\pi iz},$$ where $\tau\in\C$ and $z\in\fh$. This is a series which converges for $\def\Im{\textrm{Im}\;}\Im \tau > 0$ to a modular function. Moreover, the linear span of the functions $\chi_\Lambda(\tau,0)$ for $\Lambda$ of fixed level $k$ is invariant under the modular transformations

$$\tau\mapsto \frac{a\tau + b}{c\tau + d},\quad \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\textrm{SL}_2(\Z),$$ and the matrix $S$ of the transformation $\tau\mapsto -1/\tau$ is known explicitly [KaPe]. For example, in the case $\fg = \fsl_2(\C)$,

$$S=\Bigg(\sqrt{\frac{2}{k+2}}\sin\frac{\pi(r+1)(s+1)}{k+2}\Bigg)_{r,s=0}^k$$ This turned out to be a key fact in the representation theory of affine algebras, as well as its applications to conformal field theory (see [Ve]), to $2$-dimensional lattice models [DaJiKuMiOk], and even to knot theory[YaGe].

References

[DaJiKaMi] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, "Transformation groups for soliton equations" M. Jimbo (ed.) T. Miwa (ed.), Proc. RIMS Symp., World Sci. (1983) pp. 39–120
[DaJiKuMiOk] E. Date, M. Jimbo, A. Kuniba, T. Miwa, M. Okado, "Exactly solvable SOS models" Nucl. Phys., B290 (1987) pp. 231–273 MR0910849 Zbl 0679.17010
[FrKa] I.B. Frenkel, V.G. Kac, "Basic representations of affine Lie algebras and dual resonance models" Invent. Math., 62 (1980) pp. 23–66 MR0595581 Zbl 0493.17010
[FrLeMe] I. Frenkel, J. Lepowsky, A. Meurman, "Vertex operator algebras and the Monster", Acad. Press (1989) MR1167718 MR0996026 Zbl 0674.17001
[GrScWi] M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Cambridge Univ. Press (1987) MR0922731 MR0915347 MR0878144 MR0878143 Zbl 0637.53111 Zbl 0619.53002
[Ka] V.G. Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv., 2 (1968) pp. 1271–1311 Izv. Akad. Nauk USSR Ser. Mat., 32 (1968) pp. 1923–1967 MR0259961 Zbl 0222.17007
[Ka2] V.G. Kac, "Infinite-dimensional Lie algebras and Dedekind's $\eta$-function" Funct. Anal. Appl., 8 (1974) pp. 68–70 Funkts. Anal. i Prilozhen., 8 : 1 (1974) pp. 77–78 MR0374210 Zbl 0299.17005
[Ka3] V.G. Kac, "Infinite-dimensional Lie algebras", Cambridge Univ. Press (1985) MR0823672 MR0823320 MR0802097 Zbl 0614.22006 Zbl 0581.17009 Zbl 0574.17010
[KaPe] V.G. Kac, D.H. Peterson, "Infinite-dimensional Lie algebras, theta functions and modular forms" Adv. in Math., 53 (1984) pp. 125–264 MR0750341 Zbl 0584.17007
[KaRa] V.G. Kac, A.K. Raina, "Bombay lectures on highest weight representations", World Sci. (1987) MR1021978 Zbl 0668.17012
[Ku] S. Kumar, "Demazure character formula in arbitrary Kac–Moody setting" Invent. Math., 89 (1987) pp. 395–423 MR0894387 Zbl 0635.14023
[Ma] O. Mathieu, "Formules de caractères pour les algèbres de Kac–Moody générales" Astérisque, 159–160 (1988) pp. 1–266 Zbl 0683.17010
[Mo] R.V. Moody, "A new class of Lie algebras" J. of Algebra, 10 (1968) pp. 211–230 MR0229687 Zbl 0191.03005
[PrSe] A. Pressley, G. Segal, "Loop groups", Oxford Univ. Press (1986) MR0900587 MR0849057 Zbl 0618.22011 Zbl 0603.17012
[Ve] E. Verlinde, "Fusion rules and modular transformations in $2D$ conformal field theory" Nucl. Phys., B300 pp. 360–375
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How to Cite This Entry:
Kac-Moody algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kac-Moody_algebra&oldid=22623