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Nathan Geer
Renormalized Quantum Invariants

In this talk I will discuss a renormalization of the Reshetikhin-Turaev quantum invariants, by "fake quantum dimensions." In the case of simple Lie algebras these "fake quantum dimensions" are proportional to the genuine quantum dimensions. More interestingly I will discuss two examples where the genuine quantum dimensions vanish but the "fake quantum dimensions" are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras defined by Patureau-Mirand and myself. These invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of links. This is joint work with Bertrand Patureau-Mirand and Vladimir Turaev.