The relative monoidal center and tensor products of monoidal categories

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over bialgebras inside a braided monoidal category, the relative center is shown to be equivalent to the category of Yetter–Drinfeld modules inside the braided category. If the braided category is given by modules over a quasitriangular Hopf algebra, then the relative center corresponds to modules over a braided version of the Drinfeld double (i.e. the double bosonization in the sense of Majid) which are locally finite for the action of the dual.

Fusion and monodromy in the Temperley-Lieb category

Graham and Lehrer (1998) introduced a Temperley-Lieb category \ctl whose objects are the non-negative integers and the morphisms in \Hom(n,m) are the link diagrams from nn to mm nodes. The Temperley-Lieb algebra \tl n is identified with \Hom(n,n). The category \ctl is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on \ctl. We introduce a module category \modtl whose objects are functors from \ctl to \mathsf{Vect}_{\mathbb C}𝖵𝖾𝖼𝗍ℂ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for \ctl to induce the structure of a ribbon category on \modtl(\beta=-q-q^{-1}), when qq is not a root of unity. We discuss how the braiding on \ctl and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.

The core of a weakly group-theoretical braided fusion category

We show that the core of a weakly group-theoretical braided fusion category [Formula: see text] is equivalent as a braided fusion category to a tensor product [Formula: see text], where [Formula: see text] is a pointed weakly anisotropic braided fusion category, and [Formula: see text] or [Formula: see text] is an Ising braided category. In particular, if [Formula: see text] is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius–Perron dimension at most 2 is necessarily group-theoretical.

A Frobenius homomorphism for Lusztig’s quantum groups for arbitrary roots of unity

For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius–Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

MORE EXAMPLES OF PSEUDOSYMMETRIC BRAIDED CATEGORIES

We study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.

ON 4-DIMENSIONAL 2-HANDLEBODIES AND 3-MANIFOLDS

We show that for any n ≥ 4 there exists an equivalence functor [Formula: see text] from the category [Formula: see text] of n-fold connected simple coverings of B3 × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category [Formula: see text] of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S3 branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S3 branched over embedded graphs. Then, we factor the functor above as [Formula: see text], where [Formula: see text] is an equivalence functor to a universal braided category [Formula: see text] freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category [Formula: see text]. From this we derive an analogous description of the category [Formula: see text] of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

THE DRINFEL'D DOUBLE AND TWISTING IN STRINGY ORBIFOLD THEORY

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.