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. 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 be a Borcherds (colour) superalgebra (cf. also Borcherds Lie algebra) with charge and integral Borcherds–Cartan matrix , restricted with respect to the colouring matrix . (The charge counts the multiplicities of the simple roots.) Let denote the Cartan subalgebra of and let be a weight -module with all weight spaces finite-dimensional. The formal character of is

For an irreducible highest weight module with dominant integral highest weight , U. Ray [a6] and M. Miyamoto [a5] have established the following generalization of the Weyl–Kac–Borcherds character formula.

Let be the Weyl group, the negative roots and the set of simple roots counted with multiplicities. Let be such that

for all . Define , where the sum runs over all elements of the weight lattice of the form such that the are distinct even imaginary roots in , the are distinct odd imaginary roots in ,

if ,

for all , ,

if , and

for all , . Set if , and define . Then

where is the colouring map induced by and is the root space of .

In the case of Kac–Moody algebras, there are no imaginary simple roots and for all , so one recovers the Weyl–Kac formula

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 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