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''Weyl–Kac formula, Kac–Weyl character formula, Kac–Weyl formula, Weyl–Kac–Borcherds character formula''
 
''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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300701.png" /> be a Borcherds (colour) [[Superalgebra|superalgebra]] (cf. also [[Borcherds Lie algebra|Borcherds Lie algebra]]) with charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300702.png" /> and integral Borcherds–Cartan matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300703.png" />, restricted with respect to the colouring matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300704.png" />. (The charge counts the multiplicities of the simple roots.) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300705.png" /> denote the [[Cartan subalgebra|Cartan subalgebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300706.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300707.png" /> be a weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300708.png" />-module with all weight spaces finite-dimensional. The formal character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w1300709.png" /> is
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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
  
<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/w/w130/w130070/w13007010.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { ch } V = \sum _ { \mu \in \mathfrak{h} ^ { * } } ( \operatorname { dim } V _ { \mu } ) e ^ { \mu }. \end{equation*}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007011.png" /> an irreducible highest weight module with dominant integral highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007012.png" />, U. Ray [[#References|[a6]]] and M. Miyamoto [[#References|[a5]]] have established the following generalization of the Weyl–Kac–Borcherds character formula.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007013.png" /> be the [[Weyl group|Weyl group]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007014.png" /> the negative roots and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007015.png" /> the set of simple roots counted with multiplicities. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007016.png" /> be such that
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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
  
<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/w/w130/w130070/w13007017.png" /></td> </tr></table>
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\begin{equation*} \rho ( h _ { i } ) = \frac { 1 } { 2 } a _ { i i } \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007018.png" />. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007019.png" />, where the sum runs over all elements of the weight lattice of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007020.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007021.png" /> are distinct even imaginary roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007022.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007023.png" /> are distinct odd imaginary roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007024.png" />,
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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 $R$,
  
<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/w/w130/w130070/w13007025.png" /></td> </tr></table>
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\begin{equation*} ( \alpha _ { k } | \alpha _ { l } ) = ( \beta _ { k } | \beta _ { l } ) = 0 \end{equation*}
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007026.png" />,
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if $k \neq l$,
  
<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/w/w130/w130070/w13007027.png" /></td> </tr></table>
+
\begin{equation*} ( \alpha _ { k } | \beta _ { l } ) = 0 \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007029.png" />,
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for all $k$, $l$,
  
<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/w/w130/w130070/w13007030.png" /></td> </tr></table>
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\begin{equation*} ( \beta _ { k } \mid \beta _ { k } ) = 0 \end{equation*}
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007031.png" />, and
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if $p _ { k } &gt; 1$, and
  
<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/w/w130/w130070/w13007032.png" /></td> </tr></table>
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\begin{equation*} ( \lambda | \alpha _ { k } ) = ( \lambda | \beta _ { l } ) = 0 \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007034.png" />. Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007035.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007036.png" />, and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007037.png" />. Then
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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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007038.png" /></td> </tr></table>
+
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007039.png" /> is the colouring map induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007041.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007042.png" /> root space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007043.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007044.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007045.png" />, so one recovers the Weyl–Kac formula
+
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
  
<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/w/w130/w130070/w13007046.png" /></td> </tr></table>
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\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).
 
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007047.png" /> 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>
+
<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>

Latest revision as of 17:43, 1 July 2020

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. 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 (cf. 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 [a3] for representations of Kac–Moody algebras and [a2] for generalized Kac–Moody (or Borcherds) algebras.

Let $\frak g$ be a Borcherds (colour) superalgebra (cf. also 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 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 [a6] and M. Miyamoto [a5] have established the following generalization of the Weyl–Kac–Borcherds character formula.

Let $W$ be the 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 $p _ { k } > 1$, 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

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 where quantum deformation theorems are known (see [a4] and [a1], for example).

References

[a1] G. Benkart, S.-J. Kang, D.J. Melville, "Quantized enveloping algebras for Borcherds superalgebras" Trans. Amer. Math. Soc. , 350 (1998) pp. 3297–3319
[a2] R.E. Borcherds, "Generalized Kac–Moody algebras" J. Algebra , 115 (1988) pp. 501–512
[a3] V.G. Kac, "Infinite-dimensional Lie algebras and Dedekind's $ \eta $ function" Funct. Anal. Appl. , 8 (1974) pp. 68–70
[a4] S.-J. Kang, "Quantum deformations of generalized Kac–Moody algebras and their modules" J. Algebra , 175 (1995) pp. 1041–1066
[a5] M. Miyamoto, "A generalization of Borcherds algebras and denominator formula" J. Algebra , 180 (1996) pp. 631–651
[a6] U. Ray, "A character formula for generalized Kac–Moody superalgebras" J. Algebra , 177 (1995) pp. 154–163
How to Cite This Entry:
Weyl-Kac character formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl-Kac_character_formula&oldid=23135
This article was adapted from an original article by Duncan J. Melville (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article