Cofrobenius Coalgebra

Symmetric coalgebras

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain functors. We define an automorphism of the ring with local units constructed from a co-Frobenius coalgebra, which we call the Nakayama automorphism. This is used to give a new characterization to symmetric c…

Frobenius properties and Maschke-type theorems for entwined modules

Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems, known for Doi-Hopf modules are extended to the case of entwined modules.

On the dimension of the space of integrals on coalgebras

We study the injective envelopes of the simple right $C$-comodules, and their duals, where $C$ is a coalgebra. This is used to give a short proof and to extend a result of Iovanov on the dimension of the space of integrals on coalgebras. We show that if $C$ is right co-Frobenius, then the dimension of the space of left $M$-integrals on $C$ is $\leq {\rm dim}M$ for any left $C$-comodule $M$ of finite support, and the dimension of the space of right $N$-integrals on $C$ is $\geq {\rm dim}N$ for any right $C$-comodule $N$ of finite support. If $C$ is a coalgebra, it is discussed how far is the du…

双代数的扭曲张量积与Frobenius代数之间的二分性

A dichotomy between twisted tensor products of bialgebras and Frobenius algebras

该研究为扭曲张量积定义余单位与余乘法结构，刻画其成为余代数、双代数、Frobenius代数等的条件，方法适用于辫子幺半范畴，并得到相关代数构造应用。

Frobenius代数的Hopf范畴

The Hopf category of Frobenius algebras

本研究证明Frobenius代数间的通用测量余代数与余测量代数可分别使其范畴成为Hopf范畴与Hopf反范畴，推广了相关经典结论，还涉及有限特征下的例子及拓扑量子场论相关思考。

关于型为$(A \boxtimes H^{\text{op}}, H, \boldsymbol{\text{ψ}})$的可分与Frobenius余上余代数

On Separable and Frobenius Cowreaths of type $(A \otimes H^{\mathrm{op}}, H,ψ)$

本文针对型为$(A \boxtimes H^{\text{op}}, H, \boldsymbol{\text{ψ}})$的余上余代数，发展通用理论以简化已有结果并推广至高维，明确其Frobenius与可分性的判定条件及二者关系，还给出了可分而非Frobenius的实例。

Cyclic Homology and Quantum Orbits

A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient…

On the Hochschild homology of open Frobenius algebras

We prove that the shifted Hochschild chain complex $C\_*(A,A)[m]$ of a symmetric open Frobenius algebra $A$ of degree $m$ has a natural homotopy coBV-algebra structure. As a consequence $HH\_*(A,A)[m]$ and $HH^*(A,A^\vee)[-m]$ are respectively coBV and BV algebras. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural homotopy BV-algebra structure on $C\_*(A,A)[m]$ hence a BV-structure on $HH\_*(A,A)[m]$. Moreover we prove that the product and coproduct on $HH\_*(A,A)[m]$ satisfy the Frobenius compatibility condition i.e. $HH\_*(A,A)[m…

同调群上的更高结构

Higher structures on homology groups

本文将 operad 乘法诱导 Gerstenhaber 代数的经典结论对偶到 cooperad，证明带余乘法的 cooperad 赋予同调群 Gerstenhaber 余代数结构，并应用于 Tor 群与 Hochschild 同调。

Algebraic structures in comodule categories over weak bialgebras

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category o…

The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective

For $A$ a Hopf algebra of arbitrary dimension over a field $K$, it is well-known that if $A$ has nonzero integrals, or, in other words, if the coalgebra $A$ is co-Frobenius, then the space of integrals is one-dimensional and the antipode of $A$ is bijective. Bulacu and Caenepeel recently showed that if $H$ is a dual quasi-Hopf algebra with nonzero integrals, then the space of integrals is one-dimensional, and the antipode is injective. In this short note we show that the antipode is bijective.