Mikhail Antipov, Alexandra Zvonareva
In this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use the rank of the maximal torus of the identity component $Out^0(A)$ of the group of outer automorphisms of a symmetric stably biserial algebra $A$.
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Derived equivalences of Brauer graph algebras
本文综述 Brauer 图代数的导出等价分类及其在导出等价下封闭性的证明脉络。
Generalized Kauer moves and derived equivalences of Brauer graph algebras
本文将Kauer移动推广到任意有限条边，基于切割粘贴操作构造广义移动，并证明其产生Brauer图代数的导出等价，可用silting突变解释。
The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied. Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider $A_{\infty}$-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived …
Classification of Brauer graph algebras under stable equivalence of Morita type
本文证明布拉耶尔图代数的导出等价与Morita型稳定等价等价，并将此结论应用于稳定Picard群轨道的研究，同时给出该类代数在Morita型稳定等价下对半单直和项封闭的新证明。
Invariants of derived equivalences for admissible fractional Brauer graph algebras
本文研究可容许分数Brauer图代数，通过组合结构给出导出等价的几个易于检验的 invariants，并证明这类代数可视为温和代数的重复代数与r重平凡扩张。
Tilting mutations as generalized Kauer moves for (skew) Brauer graph algebras with multiplicity
本文将广义Kauer移动与倾斜突变诱导导出等价的结果推广到任意重数Brauer图代数及斜Brauer图代数，并证明重数为1的斜Brauer图代数是斜gentle代数的平凡扩张。
Virtual Morita equivalences and Brauer character bijections
本文将 Kessar 和 Linckelmann 关于 Morita 等价与 Brauer 特征双射的定理推广至虚拟 Morita 等价，并由此证明 Navarro 对 Alperin 权猜想的精化在多种特定缺陷群及线性群块中成立。
These lecture notes on Brauer graph algebras are the result of a series of four lectures given at the CIMPA research school in Mar del Plata, Argentina, in March 2016. After motivating the study of Brauer graph algebras by relating them to special biserial algebras, the definition of Brauer graph algebras is given in great detail with many examples to illustrate the concepts. This is followed by a short section on the interpretation of Brauer graphs as decorated ribbon graphs. A section on gentle algebras and their graphs, trivial extensions of gentle algebras, admissible cuts of Brauer graph …
We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial al…
Brauer graph algebras are closed under stable equivalence of Morita type
该研究证明Brauer图代数在Morita型稳定等价下关于半单直和项封闭，借此重证其在导出等价下封闭，还解决了相关重构问题。
Brauer pairs for splendid Rickard equivalences
本文定义了链复形的Brauer对，揭示了splendid Rickard等价与$p$-置换等价之间的结构关系，并展示了其在块代数中的应用。
In this paper we construct non-negative gradings on a basic Brauer tree algebra $A_Γ$ corresponding to an arbitrary Brauer tree $Γ$ of type $(m,e)$. We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra $A_S$, whose tree is a star with the exceptional vertex in the middle, to $A_Γ$. The grading on $A_S$ comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green's walk around $Γ$ (cf. [\ref{Zak}]). By computing endomorphism rings of these tilting complexes …
Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
针对具有公共中心 p-子群的有限群，本文提出构造 Morita 型相对 Z-稳定等价的方法，推广 Linckelmann 的相关结论，并引入相对 Brauer 不可分解性概念及其等价判定条件。
In this paper we establish a connection between ribbon graphs and Brauer graphs. As a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related Brauer graph algebras reflecting this geometrical move.
We introduce group graded basic Morita equivalences between algebras deter- mined by blocks of normal subgroups, and by using the extended Brauer quotient, we show that they induce graded basic Morita equivalences at local levels.
In this paper we show that the derived Picard group $TrPic(A)$ of the Brauer star algebra of type $(n,t)$ is generated by shift, $Pic(A)$ and equivalences $\{H_i\}_{i=1}^n$ in the case $t>1$, where $H_i$ were shown to satisfy the relations of the braid group on the affine diagram $\widetilde{A}_{n-1}$ by Schaps and Zakay-Illouz. In the multiplicity free case we show that $TrPic(A)$ is generated by a slightly bigger set.
Fixed points of extended tensor products
本文研究了扩展张量积的不动点与布劳尔构造的交换性，引入新的方块图以连接 $p$-置换等价性与局部等价性，并提出新的技术工具以证明其交换性。
We give a handy way to have a situation that the $kG$-Scott module with vertex $P$ remains indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $p>0$. The motivation is that the Brauer indecomposability of a $p$-permutation bimodule is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method, that then can possibly lift to a splendid derived equivalence. Further our result explains a hidden reason why the Brauer indecomposability of the Scot…
There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhd…
Let $k$ be an algebraically closed field of prime characteristic $p$, and let $P$ be a $p$-subgroup of a finite group $G$. We give sufficient conditions for the $kG$-Scott module $\mathrm{Sc}(G,P)$ with vertex $P$ to remain indcomposable under the Brauer construction with respect to any subgroup of $P$. This generalizes similar results for the case where $P$ is abelian. The background motivation for this note is the fact that the Brauer indecomposability of a $p$-permutation bimodule is a key step towards showing that the module under consideration induces a stable equivalence of Morita type, …
We give a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing met…
We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined \linebreak by R.~Kessar, N.~Kunugi and N.~Mitsuhashi. The motivation of \linebreak this paper is a fact that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of t…
Kernel of Scott modules and Brauer indecomposability
本文研究了Scott模块的布拉eur不可分解性与其模块核的关系，提出了一般化的判据，并证明在某些情况下，Scott模块在p-局部子群上的不可分解性可以提升到原群上。
Fractional Brauer configuration algebras I: definitions and examples
本文通过推广Brauer配置为分数Brauer配置，引入分数Brauer配置代数，并证明S型和MS型分别对应局部有界Frobenius代数及特殊多序列Frobenius代数，且在代数闭域上刻画了S型表示有限类与标准自内射代数类的重合关系。
We show that any Brauer tree algebra has precisely $\binom{2n}{n}$ $2$-tilting complexes, where $n$ is the number of edges of the associated Brauer tree. More explicitly, for an external edge $e$ and an integer $j\neq0$, we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is $\binom{2n-|j|-1}{n-1}$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of $T$. To prove this, we use a geometric model of Brauer graph algebras on the closed oriented marked surfaces and a classification of $2$-tilting complexes due to Adachi-Aihara-Chan.
Inspired by the works of Rickard on splendid equivalences and of Chuang and Rouquier on perverse equivalences, we are here interested in the combination of both, a splendid perverse equivalence. This is naturally the right framework to understand the relations between global and local perverse equivalences between blocks of finite groups, as a splendid equivalence induces local derived equivalences via the Brauer functor. We prove that under certain conditions, we have an equivalence between a perverse equivalence between the homotopy category of p-permutation modules and local derived pervers…
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
Using equivalences of categories we provide isomorphisms between the Brauer groups of different Hopf algebras. As an example, we show that when k is a field of characteristic different from 2 the Brauer groups BC(k,H_4,r_t) for every dual quasitriangular structure r_t on Sweedler's Hopf algebra H_4 are all isomorphic to the direct sum of (k,+) and the Brauer-Wall group of k. We provide an isomorphism between the Brauer group of a Hopf algebra H and the Brauer group of the dual Hopf algebra H^* generalizing a result of Tilborghs. Finally we relate the Brauer groups of H and of its opposite and …
The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and strongly derived unbounded algebras naturally. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an …
We introduce the periplectic $q$-Brauer category over an integral domain of characteristic not $2$. This is a strict monoidal supercategory and can be considered as a $q$-analogue of the periplectic Brauer category. We prove that the periplectic $q$-Brauer category admits a split triangular decomposition in the sense of Brundan-Stroppel. When the ground ring is an algebraically closed field, the category of locally finite dimensional right modules for the periplectic $q$-Brauer category is an upper finite fully stratified category in the sense of Brundan and Stroppel. We prove that periplectic…
Quasi-biserial algebras, special quasi-biserial algebras and symmetric fractional Brauer graph algebras
本文研究了准双射代数的性质，并建立了其与带有多重的标记 Ribbon 图之间的对应关系，证明了Kauer 移动诱导了对称特殊准双射代数之间的导出等价。
Two-term tilting complexes of biserial fractional Brauer graph algebras
本文研究双序列分数Brauer图代数的两项倾斜复形和Kauer移动，证明其斜群代数是斜Brauer图代数，并给出倾斜离散性的充要条件及导出等价封闭性。
Let $k$ be an arbitrary field, $Λ$ be a $k$-algebra and $V$ be a $Λ$-module. When it exists, the universal deformation ring $R(Λ,V)$ of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts of $V$ up to $R\otimes_k Λ$, where $R$ is any complete, local commutative Noetherian $k$-algebra with residue field $k$. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type $p$-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special bi…
Let $H$ be a Hopf algebra over a commutative ring $k$ with unity and $σ:H\otimes H\longrightarrow k$ be a cocycle on $H$. In this paper, we show that the Yetter-Drinfeld module category of the cocycle deformation Hopf algebra $H^σ$ is equivalent to the Yetter-Drinfeld module category of $H$. As a result of the equivalence, the "quantum Brauer" group BQ$(k,H)$ is isomorphic to BQ$(k,H^σ)$. Moreover, the group $\Gal(\HR)$ constructed in \cite{Z} is studied under a cocycle deformation.
We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $τ$-tilting modules, is a derived invariant. Moreover, we apply our result to the …
Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $ν$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-sy…
Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A $ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $\mathfrak g$, where $\mathfrak g$ is a direct sum of copies of $\mathfrak {sl}_\infty$ (resp., $ \hat{\mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$)…
In this article, we study the modular representations of the special linear group of degree two over a finite field in defining characteristic. In particular, we study the automorphisms of derived category of representations. We have been able to obtain a new type of autoequivalence. This autoequivalence has some uncommon features. It is more conveniently conceived and proved using the representation theory of its Brauer correspondence but at the same time it can be very neatly described, using a type of derived equivalence called perverse equivalence, in global settings.
We extend the notion of a {$p$-permutation equivalence} between two $p$-blocks $A$ and $B$ of finite groups $G$ and $H$, from the definition in [Boltje-Xu 2008] to a virtual $p$-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of $A$ and $B$ are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that $p$-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of $p$-permutation equivalences between $A$ and $B$ is fini…
We develop the local-global theory of blocks for profinite groups. Given a field $k$ of characteristic $p$ and a profinite group $G$, one may express the completed group algebra $k[[G]]$ as a product $\prod_{i\in I}B_i$ of closed indecomposable algebras, called the blocks of $G$. To each block $B$ of $G$ we associate a pro-$p$ subgroup of $G$, called the defect group of $B$, unique up to conjugacy in $G$. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a Brauer correspondence between the blocks of $G$ with defe…
The computation of the Brauer group BM of modified supergroup algebras is perfomed, yielding, in particular, the computation of the Brauer group of all finite-dimensional triangular Hopf algebras when the base field is algebraically closed and of characteristic zero. The results are compared with the computation of lazy cohomology and with Yinhuo Zhang's exact sequence. As an example, we compute explicitely the Brauer group and lazy cohomology for modified supergroup algebras with (extensions of) Weyl groups of irreducible root systems as a group datum and their standard representation as a re…
Universal deformation rings of a special class of modules over generalized Brauer tree algebras
本文计算了广义Brauer树代数上周期弦模的泛变形环，并对一类特定广义Brauer树代数中含单模的Ω-稳定分支中的模完成了泛变形环分类，综合运用了Brauer图代数的表示论工具及导出等价理论。
In this paper, we characterize a Rickard complex, which induces a Rickard equivalence between the block algebras of a block $b$ and its Brauer correspondent and whose vertices have the same order as defect groups of the block $b$. The homology of such a Rickard complex vanishes at all degree but degree $q$, and the homology at degree $q$ induces a basic Morita equivalence between the block algebras in the sense of Puig.
Differential graded Brauer groups
本文定义由微分分次中心单代数按稳定同构等价形成的 dg Brauer 群，并证明它与域 K 的经典 Brauer 群同构。
In this paper, we first show that the irreducible characters of a quotient table algebra modulo a normal closed subset can be viewed as the irreducible characters of the table algebra itself. Furthermore, we define the character products for table algebras and give a condition in which the products of two characters are characters. Thereafter, as a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras.
A Proof of the Jung-Kim Conjecture on the Centers of Walled Brauer Algebras
本文在二级张量积范畴化框架下统一阐述五类Khovanov弧代数，证明特征零代数闭域上带壁Brauer代数中心由其Jucys-Murphy元素的超对称多项式生成，还给出中心的齐次基并重新研究混合Schur-Weyl对偶性。
We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. In particular, this provides a new fusion procedure for the symmetric group involving an arbitrary parameter. The R-matrices are solutions of the Yang--Baxter equation associated with the classical Lie algebras g_N of types B, C and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g_N) to U(g_N) and show that the fusion procedure provi…
Hom-orthogonal modules and brick-Brauer-Thrall conjectures
本文利用 Hom-正交性及代数几何工具刻画同维无限 bricks 族的存在条件，证明新代数族上的 bBT 猜想，并推广已有预投射分支结果。
Let k be an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GL_n-modules to the category of modules for the walled Brauer algebra. As a corollary we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook lengt…
We describe how traceless projection of tensors of a given rank can be constructed in a closed form. On the way to this goal we invoke the representation theory of the Brauer algebra and the related Schur-Weyl dualities. The resulting traceless projector is constructed from purely combinatorial data involving Young diagrams. By construction, the projector manifestly commutes with the symmetric group and is well-adapted to restrictions to $GL$-irreducible tensor representations. We develop auxiliary computational techniques which serve to take advantage of the obtained results for applications.…
A Proof of the Second brick-Brauer-Thrall Conjecture for $E$-infinite algebras
本文证明对代数闭域上的有限维代数，$E$-无限性蕴含第二 brick-Brauer-Thrall 猜想，并由此推出稳定离散性与 $E$-有限性等价，同时证明 Demonet 猜想蕴含稳定版本猜想，解决了多个开放问题。
Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\ge1, r\ge0 and m\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogon…
We give a proof, based on the rigidity of tilting complexes, that the class of self-injective finite-dimensional algebras over an algebraically closed field is closed under derived equivalence.
We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which…
Categorical actions and derived equivalences for finite odd-dimensional orthogonal groups
本文证明了 Broué 的阿贝尔亏群猜想对有限奇数维正交群成立，通过约化定理、构造 Kac-Moody 代数的范畴作用及研究孤立 RoCK 块，最终建立了块及其 Brauer 对应间的导出等价。
Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system. Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra and the blocks of the so-called $p$-local Mackey algebra $μ_{R}^{1}(G)$. Let $b$ be a block of $RG$ with abelian defect group $D$. Let $b'$ be its Brauer correspondant in $N_{G}(D)$. It is conjectured by Broué that the blocks $RGb$ and $RN_{G}(D)b'$ are derived equivalent. Here we look at equivalences between the corresponding blocks of $p$-local Mackey algebras. We prove that an analogue of the Broué's conjecture is true for the $p…
On the source algebra equivalence class of blocks with cyclic defect groups, III
本文研究了具有循环缺陷群的p-块的源代数等价类，通过分类不变量W(B)来确定其等价类。
A note on simple-minded systems and weakly simple-minded systems over self-injective algebras
本文研究自入射代数上简单心系统与弱简单心系统的关系，给出了正交系统成为 domestic Brauer 图代数上简单心系统的充要条件，并构造了2-domestic Brauer 图代数上的一类简单心系统。
On the source algebra equivalence class of blocks with cyclic defect groups, II
本文研究了循环p-群的端排列模块在拟单群p-块下的分类，通过将问题转化为一般线性群和单位ary群的分类来完成。
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