相关论文: Finite symmetric tensor categories with the Cheval…

Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$,…

In a companion work on the combinatorial quantization of 4d 2-Chern-Simons theory, the author has constructed the Hopf category of quantum 2-gauge transformations $\tilde{C}=\mathbb{U}_q\mathfrak{G}$ acting on the discrete surface-holonomy…

In this paper, we study the Chevalley property of Cayley-Hamilton Hopf algebras in the sense of De Concini-Procesi-Reshetikhin-Rosso using discriminant ideals. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ whose identity…

We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, \'etale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type…

We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal…

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a…

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks…

It is well known that if two finite groups have the same symmetric tensor categories of representations over C, then they are isomorphic. We study the following question: when do two finite groups G1,G2 have the same tensor categories of…

We construct and study a nested sequence of finite symmetric tensor categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that $\mathcal{C}_{2n}$ are…

We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde…

In this paper we classify triangular semisimple and cosemisimple Hopf algebras over any algebraically closed field k. Namely, we construct, for each positive integer N, relatively prime to the characteristic of k if it is positive, a…

In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive characteristic. We prove the conjecture in…

Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected…

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to construct generalizations ${\rm…

We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf…

Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k-representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over…

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular,…

We establish a connection between two settings of representation stability for the symmetric groups $S_n$ over $\mathbb{C}$. One is the symmetric monoidal category ${\rm Rep}(S_{\infty})$ of algebraic representations of the infinite…

We show how to compute a certain group of equivalence classes of invariant Drinfeld twists on the algebra of a finite group G over a field k of characteristic zero. This group is naturally isomorphic to the second lazy cohomology group of…

For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable…