Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules

Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)

The category $_{A}\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor product $-\boxtimes_{A}-,$ is semimonoidal but not monoidal. Although not a unit in $_{A}\mathbb{S}%_{A},$ the base semialgebra $A$ has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories $(\mathcal{V}%, \bullet, I)$ as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call $\mathbb{J}$-monads ($\mathbb{J}$-% como…

A Problem of B. Plotkin for S-Acts: Automorphisms of Categories of Free S-Acts

In this paper, we prove that all automorphisms of categories of free S-acts are semi-inner, which solves a variation of a well known B. Plotkin's problem in the case of monoids. We also give a description of automorphisms of categories of finitely generated free algebras of varieties of unary algebras, and show that among varieties of unary algebras only the variety of mono-unary algebras is perfect.

Semicorings and Semicomodules

In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is \emph{not} straightforward and even subtle at some places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no non-zero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and un…

Dedualizing complexes of bicomodules and MGM duality over coalgebras

This paper is based on the material of Section 4 and Appendix C in arXiv:1503.05523v6, which was excluded from the subsequent versions of arXiv:1503.05523. We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras $\mathcal C$ and $\mathcal D$. Given such a complex $\mathcal B^\bullet$, we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over $\mathcal C$ and left contramodules over $\mathcal D$. Furthermore, we spell out the de…

Strongly Stable Automorphisms of the Categories of the Finitely Generated Free Algebras of the some Varieties of Linear Algebras

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this category we calculate the quotient group of the group of the all automorphisms of this category over the subgroup of the all inner automorphisms. This quotient group measures difference between the geometric equivalence and automorphic equivalence of algebras from this variety.

无不动点融合自同构

Fixed-point-free fusion automorphisms

本文研究仅固定平凡元的融合环自同构，将有限群无不动点自同构的经典结果推广至融合环，并分类秩小于9且具素数阶此类自同构的8个融合范畴Grothendieck等价类。

An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G --> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo …

$n$-angulated categories from self-injective algebras

Let $\mathcal{C}$ be a $k$-linear category with split idempotents, and $Σ:\mathcal{C}\rightarrow\mathcal{C}$ an automorphism. We show that there is an $n$-angulated structure on $(\mathcal{C},Σ)$ under certain conditions. As an application, we obtain a class of examples of $n$-angulated categories from self-injective algebras.

A general construction of $n$-angulated categories using periodic injective resolutions

Let $\mathcal{C}$ be an additive category equipped with an automorphism $Σ$. We show how to obtain $n$-angulations of $(\mathcal{C},Σ)$ using some particular periodic injective resolutions. We give necessary and sufficient conditions on $(\mathcal{C},Σ)$ admitting an $n$-angulation. Then we apply these characterizations to explain the standard construction of $n$-angulated categories and the $n$-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of $n$-angulated categories from quasi-periodic selfinjective algebras.

A Note on Braided $T$-categories over Monoidal Hom-Hopf Algebras

Let $ Aut_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{MHD}(H)$ and prove that the $T$-category $Rep(\mathcal{MHD}(H))$ of representation of $\mathcal{MHD}(H)$ is isomorphic to $\mathcal {MHYD}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $…

Constructing New Braided $T$-categories over Monoidal Hom-Hopf Algebras

Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal M…

Centers of Braided Tensor Categories

Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to the category of left $B$-comodules in $\mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable $\mathcal{C}$-subcoalgebras leads to a decomposition of $B$-$\operatorname*{Comod}_{\mathcal{C}}$ into a direct sum of indecomposable $\mathcal{C}$-module subcategories. As an application, we present an explicit characterization of the struc…

Hochschild-Mitchell (co)homology of skew categories and of Galois coverings

Let $\mathcal C$ be category over a commutative ring $k$, its Hochschild-Mitchell homology and cohomology are denoted respectively $HH_*(\mathcal C)$ and $HH^*(\mathcal C).$ Let $G$ be a group acting on $\mathcal C$, and $\mathcal C[G]$ be the skew category. We provide decompositions of the (co)homology of $\mathcal C[G]$ along the conjugacy classes of $G$. For Hochschild homology of a $k$-algebra, this corresponds to the decomposition obtained by M. Lorenz. If the coinvariants and invariants functors are exact, we obtain isomorphisms $\left(HH_*(C)\right)_G\simeq HH^{\{1\}}_* (\mathcal C[G])$…

无环量子丛代数：通过丛范畴相关的Hall代数实现

Acyclic quantum cluster algebras via Hall algebras associated to cluster categories

本文通过构造与丛范畴相关的Hall代数，实现了所有无环量子丛代数的Hall代数实现，并证明了量子丛代数同构于该代数的例外子代数，且Dynkin型时自然同构，同时给出了基和突变不变性等应用。