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We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds. We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotoic integers for surfaces with boundary. We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element. These results are combined and applied to show that the Hennings invariant associated to quantum-$sl_2$ takes values in the cyclotomic integers. Using prior results of Chen et al we infer integrality also of the Witten-Reshetikhin-Turaev $SO(3)$ invariant for rational homology spheres. As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras., Algebraic criteria for underlying Hopf algebra and commutative ring have been corrected, including several lemmas, propositions, and theorems depending on these. (The prior Dedekind condition did not suffice due to possible non-trivial Picard groups. It has been replaced by a Frobenius condition)

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed., 22 pages, 4 figures

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for the ${\cal S}^{\pm 1}$-matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projective $SL(2, Z)$-action on the center of $U_q(sl_2)$ for $q$ an $l=2m+1$-st root of unity. It appears that the $3m+1$-dimensional representation decomposes into an $m+1$-dimensional finite representation and a $2m$-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of $SL(2, Z)$ and the finite, $m$-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category of $U_q(sl_2)\,$., 45 pages

We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find that both TQFT's are $SL(2,\R)$-equivariant functors and also as such isomorphic. The $SL(2,\R)$-action in the Hennings construction comes from the natural action on $\cal N$ and in the case of the Frohman-Nicas theory from the Hard-Lefschetz decomposition of the U(1)-moduli spaces given that they are naturally K\"ahler. The irreducible components of this TQFT, corresponding to simple representations of $SL(2,\Z)$ and $Sp(2g,\Z)$, thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg-Witten theories, Casson type theories for homology circles \'a la Donaldson, higher rank gauge theories following Frohman and Nicas, and the $\Z/r$ reductions of Reshetikhin-Turaev theories over the cyclotomic integers $\Z[\zeta_r]$. We also conjecture that the Hennings TQFT for quantum-${\mathfrak sl}_2$ is the product of the Reshetikhin-Turaev TQFT and such a homological TQFT., Comment: 50 pages 38 figures. Added chapters on Reidemeister torsion and skein theory, as well as new examples. Final revisions. To appear Canad. J. Math

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. The fully braided version of Radford's formula for the fourth power of the antipode is obtained. Connections of integration with cross-product and transmutation are studied. The results apply to topological Hopf algebras, e.g. a torus with a hole, which do not have additive structure., Comment: 55 pages, Latex2e with AMSLatex

We consider a generalization of the axioms of a TQFT, so called half-projective TQFT's, with an anomaly, $x^{\mu}$, in the composition law. $\mu$ is a coboundary on the cobordism categories with non-negative, integer values. The element $x$ of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be 0. This modification makes it possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be trivial all together). Under a few natural assumptions the notion of a half-projective TQFT is shown to be the only possible generalization. Based on separate work with Lyubashenko on connected TQFT's, we construct a large class of half-projective TQFT's with $x=0$. Their invariants vanish on $S^1\times S^2$, and they coincide with the Hennings invariant for non-semisimple Hopf algebras. Several toplogical tools that are relevant for vanishing properties of such TQFT's are developed. They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of ``interior'' homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT's with $x=0$ vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism in the ``classical limit'' can be estimated by the rank of its maximal free interior group., Comment: 55 pages, Latex

We review the definitions of braided tensor-categories and relate them in the semisimple case to the structural data given by braid- and fusion-matrices. A number of duality-relations involving semisimple, Tannakian and weakly-Tannakian categories are summarized. We introduce a GNS-type construction to define the quotient of a rigid, abelian tensor-category onto a semisimple category. This yields a consistent calculus of truncated 6-j-symbols derived from non-semisimple Hopfalgebras. We illustrate how non-Tannakian categories are obtained in the examples of U q (sl 2), q a root of unity, and SU(2)-WZW-models. We show that the two categories are equal for \( q\; = \;\exp \left( {\frac{{i\pi }}{{k + 2}}} \right) \) and that U q (sl 2) is unique as a dual Hopf algebra if the fundamental representation is required to be two-dimensional. This leads us to formulate a duality-problem for non-Tannakian categories.