Realisation of groups as automorphism groups in categories

Isotropy Groups of Free Racks and Quandles

In this article, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that are "coherently extendible". We then use this result to compute the global isotropy groups of the categories of racks and quandles, i.e. the automorphism groups of the identity functors of these categories.

双侧移位的自同构与Higman--Thompson群 III：扩张

Automorphisms of the two-sided shift and the Higman--Thompson groups III: extensions

本文用Higman--Thompson群的外自同构子群解释移位动力系统自同构理论，实现惰性子群并证明自同构群包含多种重要群结构，同时扩展了相关自同构到外自同构群。

Clifford's theorem for orbit categories

Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this result to the situation of a Krull-Schmidt category on which a finite group acts as automorphisms. This then provides the orbit category introduced by Cibils and Marcos, and studied intensively by Keller in the context of cluster algebras, and by Asashiba in the context of Galois covering functors. We formulate and prove Clifford's theorem for Krull-Schmidt orbit categories with respect to a finite…

Composition Algebras and Outer Automorphisms of Algebraic Groups

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form $n$ over a field $k$ of characteristic not two, and a category arising from an action of the projective similarity group of $n$ on certain pairs of automorphisms of the group scheme $\mathbf{PGO}^+(n)$ defined over $k$. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.

关于某些代数结构自同构数的一些评论

A remark on the number of automorphisms of some algebraic structures

研究在不同代数结构类别中是否存在元素，其自同构群大小与元素大小比为有理数，证明在演化代数、图、单体、部分群和偏序集等类别中成立。

Systems and Categories

The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we define a group of automorphisms; we are defining an ordered group of functors with a natural transformation between any two. We give an isomorphism from the group of integers into the group of automorphisms, as guaranteed by Cayley's Theorem. The ultimate aim is to use these definitions and concepts, of system and category, to give a general description of math…

保持粗中位数的自同构

Coarse-median preserving automorphisms

本文引入保持粗中位数的自同构，刻画 RAAG/RACG 不动子群的几何性质，并将无限阶外自同构实现为有限秩中位数空间上的同位似。

The rational stable homology of mapping class groups of universal nil-manifolds

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.

利用 Jones 技术构造的群的刚性与自同构

Rigidity and automorphisms of groups constructed using Jones' technology

本文推广 Brothier 的结果，给出 Jones 技术产生的 Thompson 群 V 分裂扩张同构的结构定理，并据此分类无限制扭曲置换式圈积、分解其自同构群，揭示协变与反变函子构造之间的对偶性。

寡态群、部分双射范畴与有限域上超齐次三次空间

Oligomorphic groups, categories of partial bijections, and ultrahomogeneous cubic spaces over finite fields

本文为某类寡态群构造了 Ismagilov–Olshanski 意义下的双陪集乘法，并将约化双陪集范畴实现为部分双射范畴，进而以有限域上带三次型的超齐次空间为例描述其开子群并分类其酉表示。

On tensor factorizations of Hopf algebras

We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automo…