Edward L. Green, Sibylle Schroll
In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.
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In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module. We show that all finitely generated modules over a special multiserial algebra are multiserial. In particular, this implies that, in analogy to special biserial algebras being biserial, special multiserial algebras are multiserial. We then show that the class …
We provide a complete classification of all tame and wild symmetric special multiserial algebras in terms of the underlying Brauer configuration. Our classification contains the symmetric special multiserial algebras of finite representation type.
We survey results on multiserial algebras, special multiserial algebras and Brauer configuration algebras. A structural property of modules over a special multiserial algebra is presented. Almost gentle algebras are introduced and we describe some results related to this class of algebras. We also report on the structure of radical cubed zero symmetric algebras.
For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less or equal to three. This implies that the finitistic dimension conjecture holds for all special multiserial algebras.
Tubes Containing String Modules in Symmetric Special Multiserial Algebras
本文借助 Brauer 构型提出 Green 超游走，刻画对称特殊多序列代数稳定 AR 箭图中含字符串模的管及其秩，并补充描述非 Green 超游走产生的秩二管。
In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one, is the trivial extension of an almost gentle algebra. We sho…
About UMP algebras and a special classification case
本文针对有限维代数导出范畴中的UMP代数类，通过定义该类代数、开发代数组合工具，刻画其局部单项式且特殊多串的性质，并基于约束箭图与Brauer图分类对称特殊双串代数中属于UMP代数的情形。
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。
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。
The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules and called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called "special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteiness.
Local geometry of special pieces of nilpotent orbits
本文研究例外类型李代数幂零锥特殊片段的局部几何，证明了Lusztig猜想的局部版本，补全了通用奇点证明，并发现了四种非孤立奇异奇点及一个缺失的共享轨道对。
We give an explicit description of adjoint quotient maps for Jacobson-Witt algebra Wn and special algebra Sn. An analogue of Kostant's differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of adjoint quotient map for Sn and construct the analogs of Kostant's transverse slice. In addition, we analysis the nilpotent elements and generalize some Premet's results to Sn.
The Hecke group algebra $HW_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $W_0$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $HW_0$ is the natural quotient of the affine Hecke algebra through its level 0 representation. We further show that the level 0 representation is a calibrated principal series representation for a …
We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broué, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequences of this case, we will prove that we can determine completely the irreducible representations of this braid group for dimension at most 5, thus reproving a classification of Tuba and Wenzl in a more general framework.
Given a non-unit, non-zero-divisor, central element $x$ of a ring $Λ$, it is well known that many properties or invariants of $Λ$ determine, and are determined by, those of $Λ/ x Λ$ and $Λ_x$. In the present paper, we investigate how the property of "being tilting" behaves in this situation. It turns out that any tilting module over $Λ$ gives rise to tilting modules over $Λ_x$ and $Λ/ x Λ$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over $Λ$ is tilting if its corresponding localization and quotient are tilti…
We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be af…
Magnitude of module categories and special biserial algebras
本文在代数闭域上证明了关于模范畴 magnitude 的猜想：其值至少为简单模数，等号恰在特殊双列代数时成立，并通过吊床理论将一般情形化归为表示有向情形。
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