Kac-Moody algebra

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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].


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