Alexey Ovchinnikov
We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.
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We determine internal characterisations for when a tensor category is (super) tannakian, for fields of positive characteristic. This generalises the corresponding characterisations in characteristic zero by P. Deligne. We also explore notions of Frobenius twists in tensor categories in positive characteristic.
We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our treatment of the problem is via differential Hopf algebras and Deligne's fibre functor construction.
Some fast-growing tensor categories
本文利用Kuhn关于有限域一般表示理论的结果，证明有限域上半单预Tannakian范畴在特征0域上的线性化仍保持半单预Tannakian性质，并构造出增长速度显著快于已知例子的新预Tannakian范畴。
We consider semisimple super Tannakian categories generated by an object whose symmetric or alternating tensor square is simple up to trivial summands. Using representation theory, we provide a criterion to identify the corresponding Tannaka super groups that applies in many situations. As an example we discuss the tensor category generated by the convolution powers of an algebraic curve inside its Jacobian variety.
Discrete pre-Tannakian categories
本文提出离散 pre-Tannakian 范畴的概念，证明其与寡态群之间的对应关系，并在具有足够多投射对象等情形下给出完全分类。
Pre-Galois categories and Fraïssé's theorem
本文引入预伽罗瓦范畴作为预坦纳卡范畴的组合类比，证明寡态群的光滑G-集范畴构成预伽罗瓦范畴且此类例子穷尽，从而重新表述弗拉伊塞定理，并借助置换类给出非预伽罗瓦的B-范畴例子。
The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to the category Rep_k G of finite dimensional representations of some affine group scheme G and field k, and conversely. Originally motivated by an attempt to find a first-order explanation for generic cohomology of algebraic groups, we study neutral tannakian categories as abstract first-order structures and, in particular, ultraproducts of them. One of the main theorems of this dissertation is that certain naturally definable subcategories of these ultraproducts are themselves neutral tann…
On endomorphisms of extensions in Tannakian categories
本文证明了Tannakian范畴中扩张自同态的Schur引理类似结论，研究了稳定子对象的特定自同态诱导标量映射的条件，并在一般与过滤设定下给出了相应结果及精确性讨论。
We develop a theory of tensor categories over a field endowed with abstract operators. Our notion of a "field with operators", coming from work of Moosa and Scanlon, includes the familiar cases of differential and difference fields, Hasse-Schmidt derivations, and their combinations. We develop a corresponding Tannakian formalism, describing the category of representations of linear groups defined over such fields. The paper extends the previously know (classical) algebraic and differential algebraic Tannakian formalisms.
Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hol…
The Tannakian radical and the mantle of a braided fusion category
本文定义了辫子融合范畴的Tannakian根及其对应的局部化mantle，证明了mantle具有平凡Tannakian根并允许一个作为完备不变量的典范中心扩张，同时研究了此类reductive范畴的性质、结构与分类结果。
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde…
We prove an analog of the Künneth formula for the groups of minimal non-degenerate extensions arXiv:1602.05936 of symmetric fusion categories. We describe in detail the structure of the group of minimal extensions of a pointed super-Tannakian fusion category. This description resembles that of the third cohomology group of a finite abelian group. We explicitly compute this group in several concrete examples.
We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R …
Tannakian QFT: from spark algebras to quantum groups
本文提出一种基于spark代数的非微扰构造，实现拓扑量子场论中线算子范畴的Hopf代数表示，并应用于3d规范理论，获得新结果。
The Drinfeld center of an oligomorphic tensor category
本文为分析寡态张量范畴的Drinfeld中心开发工具，并显式计算若干实例，发现多个与自身中心通过典范函子等同的有限张量生成预Tannakian范畴。
A characterization of the Delannoy category by Adams operations
本文通过 Adams 运算刻画 Delannoy 范畴，证明了它是唯一具有被第二个 Adams 运算固定的生成元且外幂为单的半单 pre-Tannakian 范畴，并可由其 Grothendieck 半环唯一确定。
Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard-Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially…
We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for w…
Remarks on symmetric fusion categories of low rank in positive characteristic
本文给出正特征下对称融合范畴秩的下界，证明第二 Adams 运算非恒等且满足特定条件的范畴为超 Tannakian，并分类了秩 3 及部分秩 4 的范畴。
On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$
本文研究一般线性超群 $GL(n|n)$ 的表示范畴的商结构，确定其半单 tannakian 商对应的代数群及其导出子群，并揭示不可约表示张量积的分解法则，部分结论依赖 $2$-挠假设。
Let $G$ (resp. $H$) be the group of orientation preserving self-homeomorphisms of the unit circle (resp. real line). In previous work, the first two authors constructed pre-Tannakian categories $\underline{\mathrm{Rep}}(G)$ and $\underline{\mathrm{Rep}}(H)$ associated to these groups. In the predecessor to this paper, we analyzed the category $\underline{\mathrm{Rep}}(H)$ (which we named the ``Delannoy category'') in great detail, and found it to have many special properties. In this paper, we study $\underline{\mathrm{Rep}}(G)$. The primary difference between these two categories is that $\un…
Oligomorphic groups and tensor categories
本文为寡态群引入测度概念，构造刚性张量范畴及其阿贝尔包络，推广了Deligne插值范畴，并首次得到正特征下超指数增长的半单pre-Tannakian范畴，核心贡献是建立寡态群上的积分理论。
The first author constructed a $q$-parameterized spherical category $\sC$ over $\mathbb{C}(q)$ in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.
A tensor-triangular property for categories of representations of restricted Lie algebras
本文基于上同调支撑与张量三角几何为限制李代数定义了一种新性质，探讨了群概结构变化下极小根式厚张量理想的 Green 环结构，并猜想该性质等价于具有有限或驯顺表示型。
Nilpotence, radicaux et structures mono\"ıdales
本文研究 Wedderburn K-线性范畴中投影的截面存在性与唯一性，并将其推广到具有幺半结构的情形，进而得到任意仿射群概形的 pro-reductive 包络，推广了 Jacobson-Morozov 定理。
Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories
该研究针对Khovanov-Sazdanovic提出的Deligne插值范畴的推广形式，通过引入带滤过Krull-Schmidt范畴的相伴分次范畴等工具，完成了不可分解对象的分类并确定了相伴分次Grothendieck环。
Representations of $\mathbb{N}^{\infty}$-type combinatorial categories
本文研究四类$\n\mathbb{N}^{\infty}$型组合范畴的表示，给出同调度显式上界，证明域上有限表现表示构成阿贝尔子范畴，还刻画了环化原子场上的层范畴并证明不可约层由本原单位根参数化。
By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the …
First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to construct adjoint and coadjoint representations as usual. For a finite group G of nilpotency class 2 of odd order, I construct a basis in its group algebra C[G], parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G]. That construction gives us a one-to-one correspondence between G-orbits in g* and class…
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