Monoidal categories with additional structure such as a braiding or some form of dualityabound in quantum topology. They often appear in tandem with Frobenius algebras inside them. Motivations for this range from the theory of module categories to the construction of correlators in conformal field theory. We gen eralize the Baez–Dolan microcosm principle to consistently describe all these types of algebras by extending it to cyclic and modular algebras in the sense of Getzler–Kapranov. Our main result links the microcosm principle for cyclic algebras to the one for modular algebras via Costello’s modular envelope. The result can be understood as a local-to-global construction or an integration procedure for various flavors of Frobenius algebras that substantially generalizes and unifies the available (and often intrinsically semisimple) methods using for example triangulations or classical skein theory. As the main application of this rather abstract result, we solve the problem of classifying consistent systems of correlators for open conformal field theories and show that the genus zero correlators for logarithmic conformal field theories constructed by Fuchs–Schweigert can be uniquely extended to handlebodies. This establishes a very general correspondence between full genus zero conformal field theory in dimension two and skein theory in dimension three.

The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if $F:{\mathcal B}\to {\mathcal C}$ is an exact faithful monoidal functor of tensor categories, one would like to realize ${\mathcal B}$ as category of representations of a braided Hopf algebra $H(F)$ in ${\mathcal C}$. We prove that this is the case iff ${\mathcal B}$ has the additional structure of a monoidal ${\mathcal C}$-module category compatible with F, which equivalently means that F admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.

We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new proofs for the existence of several universal tensor categories as conjectured by Deligne. Another example constructs interesting tensor categories in positive characteristic via tilting modules for ${\rm SL}_2$.

Let be a field and let Q be a minimal Hopf quiver, i.e. a cyclic quiver or the infinite linear quiver, and let repln(Q) denote the category of locally nilpotent finite-dimensional -representations of Q. The category repln(Q) has natural tensor structures induced from graded Hopf structures on the path coalgebra . Tensor categories of the form repln(Q) are an interesting class of tame hereditary pointed tensor categories that are not finite. The aim of this paper is to compute the Clebsch–Gordan formulae and Green rings of such tensor categories.

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 at a root of unity. These invariants contain Kashaev’s quantum dilogarithm invariants of knots.