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''Weyl–Kac formula, Kac–Weyl character formula, Kac–Weyl formula, Weyl–Kac–Borcherds character formula''

A formula describing the character of an irreducible highest weight module (with dominant integral highest weight) of a [[Kac–Moody algebra|Kac–Moody algebra]]. The formula is a generalization of Weyl's classical formula for the character of an irreducible finite-dimensional representation of a semi-simple [[Lie algebra|Lie algebra]] (cf. [[Character formula|Character formula]]). The formula is very robust and has been steadily applied (with increasing technical complications) to the representations of ever wider classes of algebras, see [[#References|[a3]]] for representations of Kac–Moody algebras and [[#References|[a2]]] for generalized Kac–Moody (or Borcherds) algebras.

Let $\frak g$ be a Borcherds (colour) [[Superalgebra|superalgebra]] (cf. also [[Borcherds Lie algebra|Borcherds Lie algebra]]) with charge $\underline{m}$ and integral Borcherds–Cartan matrix $A = ( a _ { ij} )$, restricted with respect to the colouring matrix $C$. (The charge counts the multiplicities of the simple roots.) Let $\mathfrak h $ denote the [[Cartan subalgebra|Cartan subalgebra]] of $\frak g$ and let $V$ be a weight $\frak g$-module with all weight spaces finite-dimensional. The formal character of $V$ is

\begin{equation*} \operatorname { ch } V = \sum _ { \mu \in \mathfrak{h} ^ { * } } ( \operatorname { dim } V _ { \mu } ) e ^ { \mu }. \end{equation*}

For $V ( \lambda )$ an irreducible highest weight module with dominant integral highest weight $\lambda$, U. Ray [[#References|[a6]]] and M. Miyamoto [[#References|[a5]]] have established the following generalization of the Weyl–Kac–Borcherds character formula.

Let $W$ be the [[Weyl group|Weyl group]], $\Delta^{-}$ the negative roots and $R$ the set of simple roots counted with multiplicities. Let $\rho \in \mathfrak { h } ^ { * }$ be such that

\begin{equation*} \rho ( h _ { i } ) = \frac { 1 } { 2 } a _ { i i } \end{equation*}

for all $i$. Define $S _ { \lambda } = e ^ { \lambda + \rho } \sum _ { \gamma } ( - 1 ) ^ { | \gamma | } e ^ { - \gamma }$, where the sum runs over all elements of the weight lattice of the form $\gamma = \sum _ { i = 1 } ^ { r } \alpha _ { i } + \sum _ { j = 1 } ^ { s } p _ { j } \beta _ { j }$ such that the $\alpha _ { k }$ are distinct even imaginary roots in $R$, the $\beta_l$ are distinct odd imaginary roots in $R$,

\begin{equation*} ( \alpha _ { k } | \alpha _ { l } ) = ( \beta _ { k } | \beta _ { l } ) = 0 \end{equation*}

if $k \neq l$,

\begin{equation*} ( \alpha _ { k } | \beta _ { l } ) = 0 \end{equation*}

for all $k$, $l$,

\begin{equation*} ( \beta _ { k } \mid \beta _ { k } ) = 0 \end{equation*}

if $p _ { k } &gt; 1$, and

\begin{equation*} ( \lambda | \alpha _ { k } ) = ( \lambda | \beta _ { l } ) = 0 \end{equation*}

for all $k$, $l$. Set $r = s = 0$ if $\beta = 0$, and define $| \gamma | = r + \sum _ { j = 1 } ^ { s } p _ { j }$. Then

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007038.png"/></td> </tr></table>

where $\theta$ is the colouring map induced by $C$ and $\mathfrak { g } _ { \alpha }$ is the $\alpha$ root space of $\frak g$.

In the case of Kac–Moody algebras, there are no imaginary simple roots and $\theta ( \alpha , \alpha ) = 1$ for all $\alpha$, so one recovers the Weyl–Kac formula

\begin{equation*} \operatorname { ch } V ( \lambda ) = \frac { \sum _ { w \in W } ( - 1 ) ^ { l ( w ) } e ^ { w ( \lambda + \rho ) - \rho } } { \prod _ { \alpha \in \Delta ^ { - }} ( 1 - e ^ { \alpha } ) ^ { \text{dim} \mathfrak{g} _ { \alpha }  } }. \end{equation*}

These character formulas may also be applied to representations of associated [[Quantum groups|quantum groups]] where quantum deformation theorems are known (see [[#References|[a4]]] and [[#References|[a1]]], for example).

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  G. Benkart,   S.-J. Kang,   D.J. Melville,   "Quantized enveloping algebras for Borcherds superalgebras"  ''Trans. Amer. Math. Soc.'' , '''350'''  (1998)  pp. 3297–3319</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.E. Borcherds,   "Generalized Kac–Moody algebras"  ''J. Algebra'' , '''115'''  (1988)  pp. 501–512</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  V.G. Kac,   "Infinite-dimensional Lie algebras and Dedekind's $ \eta $ function"  ''Funct. Anal. Appl.'' , '''8'''  (1974)  pp. 68–70</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  S.-J. Kang,   "Quantum deformations of generalized Kac–Moody algebras and their modules"  ''J. Algebra'' , '''175'''  (1995)  pp. 1041–1066</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Miyamoto,   "A generalization of Borcherds algebras and denominator formula"  ''J. Algebra'' , '''180'''  (1996)  pp. 631–651</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  U. Ray,   "A character formula for generalized Kac–Moody superalgebras"  ''J. Algebra'' , '''177'''  (1995)  pp. 154–163</td></tr></table>