Yunge Xu, Yang Han, Wenfeng Jiang
For a truncated quiver algebra over a field of arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finite-dimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.
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Let $A$ be a truncated quiver algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of $A$ is zero in $A$. We give the number of the indecomposable direct summands of the middle term of an almost split sequence for a class of Hochschild extension algebras of $A$ by the standard duality module $D(A)$.
Let $T$ be a Hochschild extension algebra of a finite dimensional algebra $A$ over a field $K$ by the standard duality $A$-bimodule ${\rm Hom}_K(A,\,K)$. In this paper, we determine the ordinary quiver of $T$ if $A$ is a self-injective Nakayama algebra by means of the $\mathbb{N}$-graded second Hochschild homology group $HH_2(A)$ in the sense of Sköldberg.
We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra.
It is shown that a bounded quiver algebra having a 2-truncated oriented cycle is of infinite Hochschild homology dimension and global dimension, which generalizes a result of Solotar and Vigué-Poirrier to nonlocal ungraded algebras having a 2-truncated oriented cycle of arbitrary length. Therefore, a bounded quiver algebra of finite global dimension has no 2-truncated oriented cycles. Note that the well-known "no loops conjecture", which has been proved to be true already, says that a bounded quiver algebra of finite global dimension has no loops, i.e., truncated oriented cycles of length 1. M…
We present the Gerstenhaber algebra structure on the Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. E.L. Green, G. Hartman, E.N. Marcos and O. Solberg provided a canonical way of constructing a minimal projective bimodule resolution of a Koszul quiver algebra. The resolution has a comultiplicative structure which we use to define homotopy lifting maps. We first present a short example that demonstrates the theory. We then study the Gerstenhaber algebra structure on Hochschild cohomology of a family of bound quiver algebras, some m…
A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiver algebra and state necessary and sufficient conditions for the center to be finitely generated as a K-algebra. As an application, necessary and sufficient conditions for finite generation of t…
Let k be a field, q in k. We derive a cup product formula on the Hochschild cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N, where N is the ideal generated by homogeneous nilpotent elements. We explicitly construct non-nilpotent Hochschild cocycles which cannot be generated by lower homological degree elements, thus disproving the Snashall-Solberg finite generation conjecture.
Our aim in this paper is to investigate the first Hochschild cohomology of {\em admissible algebras} which can be seen as a generalization of basic algebras. For this purpose, we study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial…
The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver $Q$, with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that w…
In 1989 Happel conjectured that for a finite-dimensional algebra $A$ over an algebraically closed field $k$, $\gl A< \infty$ if and only if $\hch A < \infty$. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to Happel's conjecture. They found a family of pathological algebra $A_q$ for which $\gl A_q = \infty$ but $\hch A_q=2$. These algebras are pathological in many aspects, however their Hochschild homology behaviors are not pathological any more, indeed one has $\hh A_q = \infty=\gl A_q$. This suggests to pose a seemingly more reasonable conjecture by replacing Hochschild cohomo…
Homotopy lifting maps on Hochschild cohomology and connections to deformation of algebras using reduction systems
该研究利用同伦提升映射刻画Koszul箭图代数Hochschild上同调的Gerstenhaber括号结构，引入递推标量描述相关映射，并通过两种方法确定特定箭图代数的Maurer-Cartan元。
On the first Hochschild cohomology of finite dimensional quiver algebras under gluing idempotents
本文研究粘合幂等元得到的箭图代数与原始代数的第一Hochschild上同调李代数结构，证明其或同构或相差一维理想，并给出稳定等价下的新不变量。
Let $A$ be a truncated polynomial ring over a complete discrete valuation ring $\mathcal{O}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes \mathcal{K}$ is projective as an $A\otimes \mathcal{K}$-module, where $\mathcal{K}$ is the fraction field of $\mathcal{O}$. Then, we may define the stable Auslander-Reiten quiver of the category. We determine the shape of the components of the stable Auslander-Reiten quiver that contain Heller lattices.
We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra A, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.
We consider deformations of quantum exterior algebras extended by finite groups. Among these deformations are a class of algebras which we call truncated quantum Drinfeld Hecke algebras in view of their relation to classical Drinfeld Hecke algebras. We give the necessary and sufficient conditions for which these algebras occur, using Bergman's Diamond Lemma. We compute the relevant Hochschild cohomology to make explicit the connection between Hochschild cohomology and truncated quantum Drinfeld Hecke algebras. To demonstrate the variance of the allowed algebras, we compute both classical type …
Attached to a weight space in an integrable highest weight representation of a simply-laced Kac-Moody algebra $\mathfrak{g}$, there are two natural commutative algebras: the cohomology ring of a quiver variety and the center of a cyclotomic KLR algebra. In this note, we describe a natural geometric map between these algebras in terms of quantum coherent sheaves on quiver varieties. The cohomology ring of an algebraic symplectic variety can be interpreted as the Hochschild cohomology of a quantization of this variety in the sense of Bezrukavnikov and Kaledin. On the other hand, cyclotomic KLR a…
We compute the Hochschild cohomology of the reduced incidence algebras such as the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation on the coalgebra ${\rm Cotor}$-groups of their pre-dual coalgebras. Using the same coalgebraic machinery, we further identify the Hochschild cohomology groups of an incidence algebra associated to a quiver with the ${\rm Ext}$-groups of the incidence algebra associated to a suspension of the quiver.
For a commutative ring $R$ and an ADE Dynkin quiver $Q$, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as $R$-algebras if and only if the bad primes for $Q$ (2 in type D, 2 and 3 for $Q = E_{6}$, $E_{7}$ and 2, 3 and 5 for $Q= E_{8}$) are invertible in $R$. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2, 1/3]$ for $Q = E_{6}$, $E_{7}$ and over $\mathbb{Z}[1/2, 1/3, 1/5]$ for $Q=E_{8}$. Conversely, if some bad prime is not invertible in $R…
We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally we also study the fundamental groups of the bound quivers involved in t…
We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.
We define Calabi-Yau and periodic Frobenius algebras over arbitrary base commutative rings. We define a Hochschild analogue of Tate cohomology, and show that the "stable Hochschild cohomology" of periodic CY Frobenius algebras has a Batalin-Vilkovisky and Frobenius algebra structure. Such algebras include (centrally extended) preprojective algebras of (generalized) Dynkin quivers, and group algebras of classical periodic groups. We use this theory to compute (for the first time) the Hochschild cohomology of many algebras related to quivers, and to simplify the description of known results. Fur…
This article studies the Lie algebra $Diff(KΓ)$ of derivations on the path algebra $KΓ$ of a quiver $Γ$ and the Lie algebra on the first Hochschild cohomology group $H^1(KΓ)$. We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of $Diff(KΓ)$ and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connected matrix and boundary matrix are introduced to study the relations among the…
On the first $τ$-tilting Hochschild cohomology of an algebra
本文引入了有限维代数的1阶$τ$-Hochschild上同调及其“过剩”概念，证明了零过剩有界箭图代数的2阶Hochschild上同调同构定理，并给出了多种代数的过剩计算公式与$τ$-刚性判据。
We construct Artin algebras with vanishing first Hochschild Cohomology but a loop in their ordinary quiver. Some authors guessed earlier that such examples could not exist.
We define $H$-Galois extensions for $k$-linear categories and a Hopf algebra $H$ and prove the existence of a Grothendieck spectral sequence for Hochschild-Mitchell cohomology, related to this situation. This spectral sequence is multiplicative and for a $H$ a group algebra decomposes as a direct sum indexed by the set of conjugacy classes of the group. We also compute some Hochschild-Mitchell cohomology groups of categories with infinite associated quiver.
We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH*$(A)$ when $A$ is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell's resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell's resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.
In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\tilde{\mathbb{A}}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $Λ$ of type $\tilde{\mathbb{A}}$ such that the Gerstenhaber algebra $\operatorname{HH}^*(Λ)$ has non-trivial multiplicative structures. We also show that the derived class of gentle $m$-cluster tilted algebras is not always completely determined by the Hochschild cohomology.
We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cell…
We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.
Hochschild Cohomology of the Symmetric Square of an Annulus with Stops
研究带端点环面平方的部分包裹富卡娅范畴的霍赫希尔德上同调，通过显式生成集和带关系箭图描述dg自同态代数，根据端点数量判断形式性，利用相关方法计算上同调并构造dg变形。
We study quivers with potential (QPs) whose Jacobian algebras are finite dimensional selfinjective. They are an analogue of the `good QPs' studied by Bocklandt whose Jacobian algebras are 3-Calabi-Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under successive mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to b…
We construct an explicit projective bimodule resolution for the Leavitt path algebra of a row-finite quiver. We prove that the Leavitt path algebra of a row-countable quiver has Hochschild cohomolgical dimension at most one, that is, it is quasi-free in the sense of Cuntz-Quillen. The construction of the resolution relies on an explicit derivation of the Leavitt path algebra.
Analogous to a recent result of N. Kowalzig and U. Krähmer for twisted Calabi-Yau algebras, we show that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric algebras. We give a criterion to determine when a Frobenius algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf a…
On the cap product in Hochschild theory
本文对结合单位代数（在交换单位环上投射）的Hochschild理论中的帽积给出了公理化刻画，并通过链映射解释了系数在代数中的帽积，最后对截断多项式代数和多项式代数进行了计算。
Cohen-Macaulay truncated path and Nakayama algebras
本文在Auslander和Reiten[Progr. Math. 95(1991)]定义的意义下，对以下两类有限维代数刻画Cohen-Macaulay代数：(i)截断路代数；(ii)Nakayama代数。对于前者，我们通过普通箭图的刻画表明，若Cohen-Macaulay截断路代数Λ是连通的，则Λ或D(Λ)给出一个对偶化模，其中D表示标准对偶。对于后者，我们证明每个Cohen-Macaulay Nakayama代数Λ都是Iwanaga-Gorenstein的，特别地，Λ总是给出一个对偶化模。因此，在截断路代数和Nakayama代数中不存在非平凡的Cohen-Macaulay代数。
It is shown that path algebras modulo relations of the form $Λ= KQ/I$, where $Q$ is a quiver, $K$ a coefficient field, and $I \subseteq KQ$ the ideal generated by all paths of a given length, can be readily analyzed homologically, while displaying a wealth of phenomena. In particular, the syzygies of their modules, and hence their finitistic dimensions, allow for smooth descriptions in terms of $Q$ and the Loewy length of $Λ$. The same is true for the distributions of projective dimensions attained on the irreducible components of the standard parametrizing varieties for the modules of fixed $…
We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $…
Let $A$ and $B$ be two Morita equivalent finite dimensional associative algebras over a field $\Bbbk$. It is well known that Hochschild cohomology is invariant under Morita equivalence. Since infinitesimal deformations are connected with the second Hochschild cohomology group, we explicitly describe the transfer map connecting $\mathsf{HH}^2(A)$ with $\mathsf{HH}^2(B)$. This allows us to transfer Morita equivalence between $A$ and $B$ to that between infinitesimal deformations of them. As an application, when $\Bbbk$ is algebraically closed, we consider the quotient path algebra associated to …
Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighbouring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.
Let $Λ=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $Λ$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $Λ$ over the enveloping algebra $Λ^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $Ł$ relative to the resolution $\mathbb{P}$ is given as a cup product and als…
Comultiplicative map on projective resolution for a family of algebras one of which is cluster-tilted type D4
本文在一类含cluster-tilted type D4代数的投射双模分解上构造了余乘映射，利用幂等元与链同伦映射描述Hochschild上同调环的乘法结构，并定义星积刻画所有族成员的杯积结构。
Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology
针对无汇点有限箭图，本文通过建立$B_\infty$-代数同伦范畴中的复形同构，验证了根平方零代数的Keller猜想，证明其在单点（余）扩张等变换下不变，结论覆盖所有有限维gentle代数。
The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $Λ$ are explored, with regard to both their geometry and the structure of the modules they encode. Provided that the base field is algebraically closed and has infinite transcendence degree over its prime field, we establish the existence and uniqueness (not up to isomorphism, but in a strong sense to be specified) of a generic module for any irreducible component $\mathcal C$, that is, of a module which displays all categorically defined generic properties of the module…
Multiplication formula for Hernandez and Leclerc's quivers with potentials
该文研究Hernandez和Leclerc的带势箭图表示的$F$-多项式乘法公式，借助其与量子仿射群$U_q(\u2062\backslash\backslashwidehat{\mathfrak{g}})$实单模截断$q$-特征的关联，实现了对两个单模乘积的描述。
In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associ…
For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. The wall-crossing formula leads to a categorification of quantum cluster algebras under some assumption. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a counting formula relating the representation Grassmannians under…
In order to study the Hochschild cohomology of triangular algebras $\mathcal T$, we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with $\mathcal T$, and which converges to $HH^*(\mathcal T)$. We explicit its components, and its differentials which are sums of cup products. In case $n=3$, we study some properties of the differential at level 2. Finally, we apply these results to the paths algebra of a quiver without oriented cycles, and link them with previous results on the incidence algebra of a simplicial complex, and m…
We classify $n$-hereditary monomial algebras in three natural contexts: First, we give a classification of the $n$-hereditary truncated path algebras. We show that they are exactly the $n$-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the $n$-hereditary quadratic monomial algebras. In the case $n=2$, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating $\mathbb A_3\otimes_k \mathbb A_3$, where $\mathbb A_3$ is …
Let $A$ be a monomial associative finite dimensional algebra over a field $\Bbbk$ of characteristic zero. It is well known that the Hochschild cohomology of $A$ can be computed using Bardzell's complex $B(A)$. The aim of this article is to describe an explict $L_\infty$-structure on $B(A)$ that induces a weak equivalence of $L_\infty$-algebras between $B(A)$ and the Hochschild complex $C(A)$ of $A$. This allows us to describe the Maurer-Cartan equation in terms of elements of degree $2$ in $B(A)$. Finally, we make concrete computations when $A$ is a truncated algebra, and we prove that Bardzel…
On $H^1$ of finite dimensional algebras
本文回顾有限维代数（特别是模预生成理想的路径代数）的第一Hochschild上同调，给出箭图无有向圈的单项式代数的维数公式，并提及一般情形的后续研究。
In this article, we prove that for a finite quiver $Q$ the equivalence class of a potential up to formal change of variables of the complete path algebra $\widehat{\mathbb{C} Q}$, is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is an noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra …
On reachability categories, persistence, and commuting algebras of quivers
本文通过箭图的可达性范畴，从范畴与拓扑角度研究其性质，证明交换代数与可达性偏序集的关联代数 Morita 等价，并引入持续性 Hochschild 同调。
Among finite dimensional algebras over a field $K$, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras $A$, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of $A$ in terms of some values of its Avella-Alaminos--Geiss invariants. This in turn implies a cohomological mean…
Support $τ$-tilting posets and Hochschild reconstruction for matrix centralizer algebras
该研究确定矩阵中心化子代数的支持τ-倾斜偏序集与Hochschild不变量，给出其Morita等价准则，实现对应代数类的Morita重构。
For any truncated path algebra $Λ$, we give a structural description of the modules in the categories ${\cal P}^{<\infty}(Λ\text{-mod})$ and ${\cal P}^{<\infty}(Λ\text{-Mod})$, consisting of the finitely generated (resp. arbitrary) $Λ$-modules of finite projective dimension. We deduce that these categories are contravariantly finite in $Λ\text{-mod}$ and $Λ\text{-Mod}$, respectively, and determine the corresponding minimal ${\cal P}^{<\infty}$-approximation of an arbitrary $Λ$-module from a projective presentation. In particular, we explicitly construct - based on the Gabriel quiver $Q$ and th…
Bounds on Frobenius dimension
本文研究结合代数弗罗贝尼乌斯维数，一方面改进有限维代数弗罗贝尼乌斯维数上界，指出达界代数是特定根基代数；另一方面明确计算低维代数及截断路代数的弗罗贝尼乌斯维数，依据基础箭图路径。
We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology.
Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras
本文利用 Auslander-Reiten 三角与 Grothendieck 群上的双线性型刻画自内射代数完美复形在 AR 箭图中的位置，揭示同调分布规律，并在对称代数情形获得 Hermite 型计算方法。
We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\sf g}-vector cone ${\sf G}_{\Diamond_l}$ of these cluster algebras controls the $2$-truncated Kronecker products for all symmetric functions of degree no greater than $l$. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all ${\sf G}_{\Diamond_l}$'s. …
Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value of every additive invariant at A only depends on the number of simple modules. Examples of additive invariants include algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Along the way we establish two results of general interest. The first one concerns the compatibility between bilinear pairings and Galois actio…
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