View source for Weyl-Kac 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.

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

<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>

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.

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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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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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007026.png" />,

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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" />,

<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>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130070/w13007031.png" />, and

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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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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" />.

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

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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 <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>