Yuly Billig, Michael Lau
Thin coverings are a method of constructing graded-simple modules from simple (ungraded) modules. After a general discussion, we classify the thin coverings of (quasifinite) simple modules over associative algebras graded by finite abelian groups. The classification uses the representation theory of cyclotomic quantum tori. We close with an application to representations of multiloop Lie algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbuesch.
Support $τ$-tilting modules correspond to some classes of categorical objects bijectively, such as two-term tilting complexes for any finite dimensional symmetric algebra. This fact motivates us to classify support $τ$-tilting modules over blocks of finite groups. Therefore we classify support $τ$-tilting modules over particular blocks of finite groups by using the modular representation theoretical approaches including Clifford theory, Green's indecomposability theorem and so on.
Integral models of Harish-Chandra modules of the finite covering groups of PU(1,1)
本文针对 PU(1,1) 的有限覆盖群，计算其实诱导与上同调诱导对应的 Harish-Chandra 模整模型。
We prove that all quasi-cotilting modules are pure-injective and cofinendo. It follows that the class CogenM is always a covering class whenever M is a quasi-cotilting module. Some characterizations of quasi-cotilting modules are given. As a main result, we prove that there is a bijective correspondence between the equivalent classes of quasi-cotilting modules and torsion-free covering classes.
Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence between the Galois coverings with group G of A and the Galois coverings with group G of B. The hypothesis on T is expressed using the Hasse diagram of basic tilting A-modules and is always verified if A is of finite representation type. Then, we use the above correspondence to prove that A is simply connected if and only if B is simply connected, under the sam…
We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.
Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.
Global Weyl modules for thin Lie algebras are finite-dimensional
本文将Weyl模定义推广到权格分次李代数，证明满足薄性条件的该类李代数的整体Weyl模为有限维，以平面多项式哈密顿向量场李代数为示例，还引入模范畴分层并确定分层范畴。
The paper introduces the notion of a representation $k$-graph $(Δ,α)$ for a given $k$-graph $Λ$. It is shown that any representation $k$-graph for $Λ$ yields a module for the Kumjian-Pask algebra $KP(Λ)$, and the representation $k$-graphs yielding simple modules are characterised. Moreover, the category $RG(Λ)$ of representation $k$-graphs for $Λ$ is investigated using the covering theory of higher-rank graphs.
For $\imath$quantum covering groups $(\mathbf{U}, \mathbf{U}^\imath)$ of super Kac-Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as $\mathbf{U}^\imath$-modules, as well as a canonical basis for the modified form of the $\imath$quantum group $\mathbf{U}^\imath$, using the $\imath^π$-divided powers, rank one canonical basis for $\mathbf{U}^\imath$.
It is shown that, the quasi-Koszulities of algebras and modules are Morita invariance. A finite-dimensional $K$-algebra $A$ with an action of $G$ is quasi-Koszul if and only if so is the skew group algebra $A \ast G$, where $G$ is a finite group satisfying $\char K \nmid |G|$. A finite-dimensional $G$-graded $K$-algebra $A$ is quasi-Koszul if and only if so is the smash product $A # G^*$, where $G$ is a finite group satisfying $\char K \nmid |G|$. These results are applied to prove that, if a finite-dimensional connected quiver algebra is Koszul then so are its Galois covering algebras with fi…
The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear spans. We show how stability spaces of thin modules are related to order polytopes. In the case of non-thin modules, we show how the stability spaces of string and band modules are related to the stability spaces of the thin modules corresponding to the abstract string and band. We use this to analyse the way in which the stability space of a band module is …
Imaginary modules arising from tensor products of snake modules
针对簇代数检测虚模的局限，基于表示论框架，通过蛇模张量积的socle描述与覆盖关系推广，构造并证明了新的单虚模，扩展了量子环面代数中的虚模类。
Enochs Conjecture for cotorsion pairs and more
该研究针对一般情形下仍未解决的Enochs猜想，证明了由固定n<ω的ℵₙ-表现模构成的类对应的Filt(S)类满足猜想，还证明所有由模集合生成的Filt(S)类满足猜想与ZFC相容，推进了该猜想的证明进程。
We provide a unified approach, via deformations of incidence algebras, to several important types of representations with finiteness conditions, as well as the combinatorial algebras which produce them. We show that over finite dimensional algebras, representations with finitely many orbits, or finitely many invariant subspaces, or distributive coincide, and further coincide with thin modules in the acyclic case. Incidence algebras produce examples of such modules, and we show that algebras which are locally hereditary, and whose projective are distributive, or equivalently, which have finitel…
In this part one of a series of papers, we introduce a new version of quantum covering and super groups with no isotropic odd simple root, which is suitable for the studies of integrable modules, integral forms and bar-involution. A quantum covering group involves a quantum parameter q and a sign parameter pi squaring to 1, and it specializes to a quantum supergroup when pi=-1. Following Lusztig, we formulate and establish various structural results of the quantum covering groups, including bilinear form, quasi-R-matrix, Casimir, character formulas for integrable modules, and higher Serre rela…
In 1996, Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via $G_r T$-modules to representations of the algebraic group $G = \mathrm{GL}_n$. We study analogues of these algebras and their Auslander-Reiten theory for reductive algebraic groups $G$ and Borel subgroups $B$ by considering the categories of polynomial representations of $G_r T$ and $B_r T$ as full subcategories of $\mathrm{mod} \thinspace G_r T$ and $\mathrm{mod}\thinspace B_r T$, respectively. We show that every component $Θ$ of the stable Ausland…
Assume that $K$ is an algebraically closed field, $R$ a locally support-finite locally bounded $K$-category, $G$ a torsion-free admissible group of $K$-linear automorphisms of $R$ and $A=R/G$. We show that the Krull-Gabriel dimension $KG(R)$ of $R$ is finite if and only if the Krull-Gabriel dimension $KG(A)$ of $A$ is finite. In these cases $KG(R)=KG(A)$. We apply this result to determine the Krull-Gabriel dimension of standard selfinjective algebras of polynomial growth. Finally, we show that there are no super-decomposable pure-injective modules over standard selfinjective algebras of domest…
In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show t…
In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique trivial submodule to the case of the regular modules of modular adjacency algebras. We completely determine the $p$-transitive quasi-thin association schemes and the $p$-transitive association schemes with thin thin residue by their structure theory properties.
In this paper, we study the restriction of an irreducible unitary representation $π$ of the universal covering $\widetilde{Sp}_{2n}(\mb R)$ to a Heisenberg maximal parabolic group $\tilde P$. We prove that if $π|_{\tilde P}$ is irreducible, then $π$ must be a highest weight module or a lowest weight module. This is in sharp constrast with the $GL_n(\mathbb R)$ case. In addition, we show that for a unitary highest or lowest weight module, $π|_{\tilde P}$ decomposes discretely. We also treat the groups $U(p,q)$ and $O^*(2n)$.
Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that any irreducible $T$-module $W$ is both thin and dual thin in the sense of Terwilliger. We produce two bases for $W$ and describe the action of $T$ on these bases. We prove that the isomorphism class of $W$ as a $T$-module is determined by two parameters, the dual endpoint and diameter of $W$. We find a recurrence which gives the multiplicities with which the i…
Athinization of irreducible $\widehat{\mathfrak{gl}}_n$-modules with dominant highest weights
研究通过薄化方法将不可约 $\widehat{\mathfrak{gl}}_n$-模块转化为 affine Yangian 上的薄模块，提供新的组合实现。
Half-integral weight modular forms are naturally viewed as automorphic forms on the so-called metaplectic covering of $\operatorname{GL}_2(\mathbf{A}_{\mathbf{Q}})$ -- a central extension by the roots of unity $μ_2$ in $\mathbf{Q}$. For an odd prime number $p$, we give a complete classification of the smooth irreducible genuine mod-$p$ representations of the corresponding covering of $\operatorname{GL}_2(\mathbf{Q}_p)$ by showing that the functor of taking pro-$p$-Iwahori-invariants and its left adjoint define a bijection onto the set of simple right modules of the pro-$p$ Iwahori Hecke algebr…
We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a q-Schur algebra (parameter not a half integer), providing thus character formulas for simple modules. We give some generalization to B_n(d). We prove an ``abstract'' translation principle. These results follow from the unicity of certain highest categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups.
We study the finite dimensional complex representations of the mapping class group $\mathcal{M}_{g,1}$ that are derived from some finite Galois coverings of the compact oriented surface with one boundary component $Σ_{g,1}$. The key ingredients are Fox derivation, Magnus modules and the Skolem-Noether theorem, which enable us to compute the $\mathcal{M}_{g,1}$-action on the module $L_g^η$ very explicitly, where $η$ is a primitive $p$the root of unity for an odd prime $p$.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。