Modified quantum dimensions and re-normalized link invariants

In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, using modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly, we give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants axising from Lie superalgebras previously defined by the first two authors. These link invariants are multivaxiable and generalize the multivariable Alexander polynomial. The second example is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots.