Karin Erdmann, Thorsten Holm, Osamu Iyama, Jan Schröer
Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.
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We investigate the inequality ${\rm Findim}\ \! Λ^{op} \leq {\rm dell}\ \! Λ$ between the finitistic dimension and the delooping level of an Artin algebra $Λ$, and whether equality holds in general. We prove that equality ${\rm Findim}\ \! Λ^{op} = {\rm dell}\ \! Λ$ always holds for Artin algebras with radical square zero.
Let A be a finite dimensional symmetric cllular algebras. We construct a nilpotent ideal in A. The ideal connects the radicals of cell modules with the radical of the algebra. It also reveals some information on the dimensions of simple modules of A.
Radical preservation and the finitistic dimension
本文引入根保持概念与拟一致Loewy长度的有界箭图代数，证明相关同态可反映维数有限性，且这类代数及半准素环有有限大有限维数，还构造了现有结果无法解释的代数族。
We prove that the Crisp and Gow's quiver operation on a finite quiver Q produces a new quiver Q' with fewer vertices, such that the finite dimensional algebras kQ/J^2 and kQ'/J^2 are singularly equivalent. This operation is a general quiver operation which includes as specific examples some operations which arise naturally in symbolic dynamics (e.g., (elementary) strong shift equivalent, (in-out) splitting, source elimination, etc.).
We conjecture that the injective dimension of the Jacobson radical equals the global dimension for Artin algebras. We provide a proof of this conjecture in case the Artin algebra has finite global dimension and in some other cases.
The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let $B\subseteq A$ be an extension of Artin algebras and $I$ an ideal of $B$ such that the full subcategory of $B/I$-modules is $B$-syzyg…
Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical $J$ vanishes. We determine the irreducible components of the module variety $\text{Mod}_{\bf d}(Λ)$ for any dimension vector $\bf d$. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when $Λ$ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops. For any algebra $Λ$ with $J^2 = 0$, our criteria for identifying the com…
We study pairs of associative algebras and linear functionals. New results together with corrected proofs for previously published material are presented. In particular, we prove the identity ind Mat(n) (#) A=n * ind A$ for finite-dimensional unital associative algebra A with index 1. [ Symbol (#) denotes tensor product ]
We study the behaviour of the Igusa-Todorov functions for radical square zero algebras. We show that the left and the right $ϕ$-dimensions coincide, in this case. Some general results are given, but we concentrate more in the radical square zero algebras. Our study is based on two notions of hearth and member of a quiver $Q$. We give some bounds for the $ϕ$ and the $ψ$-dimensions and we describe the algebras for which the bound of $ψ$ is obtained. We also exhibit modules for which the $ϕ$-dimension is realised.
Trace radicals and cocenters of free products
研究含幺结合代数\(A\)和\(B\)自由积\(A*B\)的迹根，通过结合余中心分解与有限维表示构造，证明\(A*B\)的迹根是\(A\)和\(B\)迹根的直和，得出\(A*B\)是迹RFD的充要条件是\(A\)和\(B\)都是迹RFD。
Let $Λ$ be an artin algebra. We obtain that $Λ$ is syzygy-finite when the radical layer length of $Λ$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the category $\mod Λ$ of finitely generated right $Λ$-modules in terms of the projective of certain class of simple right $Λ$-modules and also get the left big finitistic dimension conjecture holds.
Big Finitistic Dimensions of Radical-Cube-Zero Algebras
本文构造十维立方根零代数，给出反例否证big finitistic维数猜想，并分类相关模及建立有限性准则，证明投射维数二类的野性。
We show that an artin algebra having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube.
Associative/Lie radical (co)invariance under tracial (co)actions
该研究针对霍普夫代数上的有限维结合或李（共）模代数，在特征合适且（共）作用保迹的条件下，证明了多种根基的（共）不变性结果，推广了已有准则并提供统一框架。
Given a finite dimensional algebra $A$ over an algebraically closed field we study the relationship between the powers of the radical of a morphism in the module category of the algebra $A$ and the induced morphism in the module category of the endomorphism algebra of a tilting $A$-module. We compare the nilpotency indices of the radical of the mentioned module categories. We find an upper bound for the nilpotency index of the radical of the module category of iterated tilted algebras of Dynkin type.
One of our main results is a classification all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade's Lemma. Along the way we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.
Let $A$ be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of $Q_A$ are suficient to be consider in order to compute the nilpotency index of the radical of the module category of $A$. In many cases, we give a formula to compute such index taking into account the ordinary quiver of the given algebra.
We prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is Hermitian, pairs of bilinear forms in which the second is symmetric (skew-symmetric), and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.
We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers.
In this paper, using pseudo path algebras, we generalize Gabriel's Theorem on elementary algebras to left Artinian algebras over a field $k$ when it is splitting over its radical, in particular, when the dimension of the quotient algebra decided by the $n$'th Hochschild cohomology is less than 2 (for example, $k$ is finite or char$k=0$). Using generalized path algebras, the generalized Gabriel's Theorem is given for finite dimensional algebras with 2-nilpotent radicals which is splitting over its radical. As a tool, the so-called pseudo path algebras are introduced as a new generalization of p…
Given a finite dimensional representation $M$ of a finite dimensional algebra, two hierarchies of degenerations of $M$ are analyzed in the context of their natural orders: the poset of those degenerations of $M$ which share the top $M/JM$ with $M$ - here $J$ denotes the radical of the algebra - and the sub-poset of those which share the full radical layering $\bigl(J^lM/J^{l+1}M\bigr)_{l \ge 0}$ with $M$. In particular, the article addresses existence of proper top-stable or layer-stable degenerations - more generally, it addresses the sizes of the corresponding posets including bounds on the …
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…
A generalization of the nilpotency index of the radical of the module category of an algebra
本研究推广CGS的结果，确定了代数闭域上有限维表示有限的单项式代数与toupie代数中，计算其模范畴根的幂零指数时$Q_A$需考虑的顶点，放宽了Auslander-Reiten箭图需为具长度分支的限制。
We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-r…
The Auslander-Reiten conjecture for algebras with radical cube zero
本文证明根立方为零的有限维代数满足Auslander-Reiten猜想，即每个自正交生成元为投射模，通过构造非投射模在1到3s+1度间的非零自扩张实现。
This paper consists of three parts. In the first part we prove that Zhu's and $C_2$-algebras in type $A$ have the same dimensions. In the second part we compute the graded decomposition of the $C_2$-algebras in type $A$, thus proving the Gaberdiel-Gannon's conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part.
Let $\La$ be an elementary locally bounded linear category over a field with radical squared zero. We shall show that the bounded derived category $D^b(\ModbLa)$ of finitely supported left $\La$-modules admits a Galois covering which is the bounded derived category of almost finitely co-presented representations of a gradable quiver. Restricting to the bounded derived category $D^b({\rm mod}^b\hspace{-2pt}\La)$ of finite dimensional left $\La$-modules, we shall be able to describe its indecomposable objects, obtain a complete description of the shapes of its Auslander-Reiten components, and cl…
Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of the combinatorial properties of the set partition combinatorics of the full uni-triangular groups, including combinatorial indexing sets, dimensions, and computable character formulas. Associated with these supercharacter theories is also a family of polytopes whose integer lattice points give the theories geometric underpinnings.
In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying and analysing structural properties of Brauer configurations and Brauer configuration algebras, we show that a Brauer configuration algebra is multiserial; that is, its Jacobson radical is a sum of uniserial modules whose pairwise intersection is eith…
Simple transitive 2-representations of bimodules over radical square zero Nakayama algebras via localization
本文研究了根方零Nakayama代数上双模2-范畴的简单传递2-表示分类问题，通过定义2-表示的局部化并利用构造的泛性质，证明了具有有限顶点的简单传递2-表示均等价于胞腔2-表示的局部化，从而实现了完整分类。
We introduce the concept of a prime band in a string algebra $Λ$ and use it to associate to $Λ$ its finite bridge quiver. Then we introduce a new technique of `recursive systems' for showing that a graph map between finite dimensional string modules lies in its stable radical. Further we study two classes of non-domestic string algebras in terms of some connectedness properties of its bridge quiver. `Meta-$\bigcup$-cyclic' string algebras constitute the first class that is essentially characterized by the statement that each finite string is a substring of a band. Extending this class we have …
Irreducible components of varieties of representations of a class of finite dimensional algebras
针对一类三次截断且带额外关系的非交换多项式代数，本文通过证明固定根分层或底座分层子簇的不可约性并比较两类覆盖，确定了其表示簇的不可约分量。
Thick subcategories over weakly symmetric algebras with radical cube zero
本文分类了根立方零弱对称代数的厚子范畴，发现谱半径为2时可由Serre子范畴分类，否则仅平凡厚子范畴，并首次构造了非完全交的交换诺特Gorenstein优势局部环。
To study the set of torsion classes of a finite dimensional basic algebra, we use a decomposition, called sign-decomposition, parametrized by elements of $\{\pm1\}^n$ where $n$ is the number of simple modules. If $A$ is an algebra with radical square zero, then for each $ε\in \{\pm1\}^n$ there is a hereditary algebra $A_ε^!$ with radical square zero and a bijection between the set of torsion classes of $A$ associated to $ε$ and the set of faithful torsion classes of $A_ε^!$. Furthermore, this bijection preserves the property of being functorially finite. As an application in $τ$-tilting theory…
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.
The primary purpose is to introduce and explore projective varieties, $\text{GRASS}_{\bf d}(Λ)$, parametrizing the full collection of those modules over a finite dimensional algebra $Λ$ which have dimension vector $\bf d$. These varieties extend the smaller varieties previously studied by the author; namely, the projective varieties encoding those modules with dimension vector $\bf d$ which, in addition, have a preassigned top or radical layering. Each of the $\text{GRASS}_{\bf d}(Λ)$ is again partitioned by the action of a linear algebraic group, and covered by certain representation-theoreti…
For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic c…
We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.
On the stable radical of the module category for special biserial algebras
本文证明了关于特殊双列代数模范畴稳定根的猜想，给出了计算稳定根的算法，并构造了具有任意给定稳定根的 tame 代数。
We prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional (apart from the roots of unity). We also find necessary and sufficient conditions for the radical to be zero, which is a q-generalization of Opdam's formula for the singular k-parameters with the multiple zero-eigenvalues of the corresponding Dunkl operators.
Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $Γ= (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A…
We give an upper bound for the dimension of the bounded derived categories of $(m,n)$-Igusa-Todorov algebras which is a generalization of $n$-Igusa-Todorov algebras, where $m,n$ are two nonnegative integers. As an applications, we get a new upper bound for the dimension of bounded derived categories in terms of the projective dimensions of certain of simple modules as well as radical layer length of artin algebra $Λ$.
Simple algebras and exact module categories
本文验证了Etingof-Ostrik猜想：有限张量范畴中代数对象精确当且仅当为简单代数的有限直积，并引入Jacobson根类比，应用于不可压缩有限对称张量范畴。
We classify all the simple modules for the algebra of relations on a finite set, give their dimension, and find the dimension of the Jacobson radical of the algebra.
Let $Λ$ be a commutative local uniserial ring of length at least seven with radical factor ring $k$. We consider the category $S(Λ)$ of all possible embeddings of submodules of finitely generated $Λ$-modules and show that $S(Λ)$ is controlled $k$-wild with a single control object $I\in S(Λ)$. In particular, it follows that each finite dimensional $k$-algebra can be realized as a quotient $\End(X)/\End(X)_I$ of the endomorphism ring of some object $X\in S(Λ)$ modulo the ideal $\End(X)_I$ of all maps which factor through a finite direct sum of copies of $I$.
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