A denominator identity for affine Lie superalgebras with zero dual Coxeter number (Q1932500)

In [\textit{M. Gorelik}, J. Algebra 337, No. 1, 50--62 (2011; Zbl 1266.17008)], it was shown that the affine denominator identity takes the form \(\hat{R}e^{\hat{\rho}}=\sum_{w\in T'} w(Re^{\hat{\rho}})\) where \(T'\) is the affine translation group corresponding to the largest root subsystem of \(\Delta_0\). In the case of zero Killing form the authors of the present paper prove tha the formula has an extra factor depending on the algebra being considered: either \(\mathfrak{gl}(n|n)\) or \(D(n+1,n), D(2,1,a)\) and \(T'\) now corresponds to the smallest root subsystem of \(\Delta_0\).