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We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson algebraic groups, we give a classification of coconnected Hopf algebras, i.e. of unipotent categories of exponential growth with a fiber functor.
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This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of the current state of the theory of symmetric tensor categories and discuss their applications, including ones unavailable in the literature.
We classify various types of graded extensions of a finite braided tensor category $cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $cal B$ (consisting of invertible central $cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.
We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $mathfrak{u}_q(mathfrak{g})$, where $q$ is a root of unity.
We classify finite pointed braided tensor categories admitting a fiber functor in terms of bilinear forms on symmetric Yetter-Drinfeld modules over abelian groups. We describe the groupoid formed by braided equivalences of such categories in terms of certain metric data, generalizing the well-known result of Joyal and Street for fusion categories. We study symmetric centers and ribbon structures of pointed braided tensor categories and examine their Drinfeld centers.
In a previous work by the author it was shown that every finite dimensional algebraic structure over an algebraically closed field of characteristic zero K gives rise to a character $K[X]_{aug}to K$, where $K[X]_aug$ is a commutative Hopf algebra that encodes scalar invariants of structures. This enabled us to think of some characters $K[X]_{aug}to K$ as algebraic structures with closed orbit. In this paper we study structures in general symmetric monoidal categories, and not only in $Vec_K$. We show that every character $chi : K[X]_{aug}to K$ arises from such a structure, by constructing a category $C_{chi}$ that is analogous to the universal construction from TQFT. We then give necessary and sufficient conditions for a given character to arise from a structure in an abelian category with finite dimensional hom-spaces. We call such characters good characters. We show that if $chi$ is good then $C_{chi}$ is abelian and semisimple, and that the set of good characters forms a K-algebra. This gives us a way to interpolate algebraic structures, and also symmetric monoidal categories, in a way that generalizes Delignes categories $Rep(S_t)$, $Rep(GL_t(K))$, $Rep(O_t)$, and also some of the symmetric monoidal categories introduced by Knop. We also explain how one can recover the recent construction of 2 dimensional TQFT of Khovanov, Ostrik, and Kononov, by the methods presented here. We give new examples, of interpolations of the categories $Rep(Aut_{O}(M))$ where $O$ is a discrete valuation ring with a finite residue field, and M is a finite module over it. We also generalize the construction of wreath products with $S_t$, which was introduced by Knop.
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