Maschke functors, semisimple functors and separable functors of the second kind. Applications

On normal Hopf subalgebras of semisimple Hopf algebras

A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka's criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is isomorphic to the image of the restriction functor.

Shifted functors of linear representations are semisimple

We prove that, for any fields $k$ and $\mathbb{F}$ of characteristic $0$ and any finite group $T$, the category of modules over the shifted Green biset functor $(kR_{\mathbb{F}})_T$ is semisimple.

New examples of Green functors arising from representation theory of semisimple Hopf algebras

A general Mackey type decomposition for representations of semisimple Hopf algebras is investigated. We show that such a decomposition occurs in the case that the module is induced from an arbitrary Hopf subalgebra and it is restricted back to a group subalgebra. Some other examples when such a decomposition occurs are also constructed. They arise from gradings on the category of corepresentations of a semisimple Hopf algebra and provide new examples of Green functors in the literature.

Algebra depth in tensor categories

Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It i…

Semiseparable functors

In this paper we introduce and investigate the notion of semiseparable functor. One of its first features is that it allows a novel description of separable and naturally full functors in terms of faithful and full functors, respectively. To any semiseparable functor we attach an invariant, given by an idempotent natural transformation, which controls when the functor is separable and yields a characterization of separable functors in terms of (dual) Maschke and conservative functors. We prove that any semiseparable functor admits a canonical factorization as a naturally full functor followed …

群体学分级半单环

Groupoid Graded Semisimple Rings

本文研究群体学分级半单环的理论，提出Γ₀-artinian概念，并证明了Wedderburn-Artin定理的群体学分级版本，同时探讨了群体学分级环的自由性及与半单范畴的关系。

论（自然）半满与（半）可分半函子

On (naturally) semifull and (semi)separable semifunctors

本文将满性、忠实性、可分性与自然满性推广到半函子，引入半满及自然半满概念，并用半伴随的单位、余单位半分裂条件加以刻画，证明 Rafael 型与 Maschke 型定理。

A generalization of dual symmetry and reciprocity for symmetric algebras

Slicing a module into semisimple ones is useful to study modules. Loewy structures provide a means of doing so. To establish the Loewy structures of projective modules over a finite dimensional symmetric algebra over a field $F$, the Landrock lemma is a primary tool. The lemma and its corollary relate radical layers of projective indecomposable modules to radical layers of the $F$-duals of those modules ("dual symmetry") and to socle layers of those modules ("reciprocity"). We generalize these results to an arbitrary finite dimensional algebra $A$. Our main theorem, which is the same as the La…

双代数的积分

Integrals for Bialgebras

本文提出一种基于范畴方法的双代数积分新概念，证明其在存在antipode时退化为经典定义，并通过对多种特殊双代数的测试，建立了对偶Maschke型定理及有限维情况下的积分存在唯一性定理。

Derived $H$-module endomorphism rings

Let $H$ be a Hopf algebra, $A/B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^\cdot\in D(A)$ may be regarded as an object in $D(B)$ via the restriction functor. We discuss the relations of the derived endomorphism rings $E_A(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(A)}(M^\cdot,M^\cdot[i])$ and $E_B(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(B)}(M^\cdot,M^\cdot[i])$. If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^\cdot)$ is a graded subalgebra of $E_B(M^\cdot)$. In particular, if $…