In this paper, we construct invariants of 3-manifolds 'à la Reshetikhin-Turaev' in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories that lead to a family of 3-manifold invariants indexed by the integers. We prove that this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Witten-Reshetikhin-Turaev invariants do not and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links. n i=1 V i ⊕ W where V i is a simpleŪ -module with non-zero quantum dimension and W is aŪ -module with zero quantum dimension. By quotienting the category of finite-dimensionalŪ -modules byŪ -modules with zero quantum dimension, one obtains a modular category D. Loosely speaking, a modular category is a semi-simple ribbon category with a finite number of isomorphism classes of simple objects satisfying some axioms. Let M be a manifold obtained by surgery on L. If the ith component of L is labeled by a

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 arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable 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.

For every semi-simple Lie algebra g one can construct the Drinfeld-Jimbo algebra U DJ h (g). This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U DJ h (g), Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A g . He proved that particular categories of modules over the algebras U DJ h (g) and A g are tensor equivalent. Analogous constructions of the algebras U DJ h (g) and A g exist in the case when g is a Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the EtingofKazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.

Abstract. We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in [12] including the Kashaev invariant of links. Here the modular category framework does not apply and we use the "universal construction". Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a root of unity of order 2r. The functor is always symmetric monoidal but for even values of r the braiding on GrVect has to be the super-symmetric one, thus our TQFT may be considered as a super-TQFT. In the special case r = 2 our construction yields a TQFT for a canonical normalization of Reidemeister torsion and we re-prove the classification of Lens spaces via the non-semi-simple quantum invariants defined in [12]. We prove that the representations of mapping class groups and Torelli groups resulting from our constructions are potentially more sensitive than those obtained from the standard Reshetikhin-Turaev functors; in particular we prove that the action of the bounding pairs generators of the Torelli group has always infinite order.

Abstract. In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.