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.

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.

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in [6], lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects. Dedicated to Jose Maria Montesinos on the occasion of his 65th birthday

In this paper we consider the representation theory of a non-standard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this non-standard quantization of sl(2). The results of this paper are used extensively in [4] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT).

15 pagesWe provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. As a consequence we prove that they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the (1,p) minimal model in conformal field theory, and quantum groups at a root of unity

Abstract. We construct invariants of C -colored spherical graphs from traces on ideals in a pivotal category C. Then we provide a systematic approach to defining such traces from ambidextrous and spherical traces on a class of objects of C. This extends the notion of an ambidextrous object of a braided category given by the first two authors in a previous work.Mathematics Subject Classification (2010). Primary 18D10; Secondary 16T05.