Yang Han
We give two sufficient and necessary conditions for a Hochschild extension of a finite dimensional algebra by its dual bimodule and a Hochschild 2-cocycle to be a symmetric algebra.
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We show how to compute the low Hochschild cohomology groups of a partial relation extension algebra.
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.
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 $D(A)$.
We introduce the Hochschild extensions of dg algebras, which are $A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric $A_\infty$-algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau comple…
We formulate the Gerstenhaber algebra structure of Hochschild cohomology of finite group extensions of some quantum complete intersections. When the group is trivial, this work characterizes the graded Lie brackets on Hochschild cohomology of these quantum complete intersections, previously only known for a few cases. As an example, we compute the algebra structure for two generator quantum complete intersections extended by select groups.
We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite.
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 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.
In this article we prove derived invariance of Hochschild-Mitchell homology and cohomology and we extend to $k$-linear categories a result by Barot and Lenzing concerning derived equivalences and one-point extensions. We also prove the existence of a Happel long exact sequence and we give a generalization of this result which provides an alternative approach.
Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.
Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions
本文针对具有双射反极的弱Hopf代数H，研究受限忠实平坦弱H-Galois扩张A/B的AS-Gorenstein性质，建立联系A与B的Hochschild上同调的谱序列，证明当A为诺特仿射PI代数、B为AS-Gorenstein时，A也具有该性质。
For a perfect field $k$ of characteristic $p>0$ and for a finite dimensional symmetric $k$-algebra $A$ Külshammer studied a sequence of ideals of the centre of $A$ using the $p$-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If $A$ is symmetric then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work. In the present paper we follow this program and propose an analogue of the dual to the Külshammer ideal structure on t…
Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras
该研究将群论中的三个经典结果推广到乘性李代数的非交换二阶上同调场景，证明了对应定理、Wells定理并建立一维Lyndon-Hochschild-Serre正合序列，特殊情形下可复现已有结论。
It is shown that the cochain complex of relative Hochschild A-valued cochains of a depth two extension A | B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End {}_BA_B over the centralizer R = A^B with grouplike element 1_S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S,S)-bicomodule R^e. This specializes to finite dimensional algebras, H-separable extensions and Hopf-Galois extensions.
Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf
本文构造显式链映射，在温和假设下证明Hopf-Galois扩张的Hochschild上同调与H-不变子代数同构，推广群作用经典结果，并计算量子(-1)-平面与Kac-Paljutkin Hopf代数smash积的上同调及杯积。
Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by E= Ext2(DC,C), then we prove that HH1(B) is isomorphic, as a vector space, to the direct sum of HH1(C) with HH1(B,E). This yields homological interpretations for results of the first and the fourth author with M.J. Redondo.
In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories…
It is known that Hochschild cohomology groups are represented by crossed extensions of associative algebras. In this paper, we introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$, for $n \geq 2$. The equivalence classes of such extensions are represented by the $(n+1)$-th Chevalley-Eilenberg cohomology group $H^{n+1}_{CE} (\mathfrak{g}, M).$
We consider associative algebras L over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of L. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.
We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.
We prove that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple, if and only if it is commutative and semisimple, if and only if the restricted Lie algebra $P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes, in the special but essential situation with finiteness assumption added.
We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology $\HH^{\ast}(A)$ of an associative conformal algebra $A$, and show that the Lie bracket together with the cup product is a Gerstenhaber algebra on the Hochschild cohomology of an associative conformal algebra. Moreover, we consider the Hochschild cohomology of split extension conformal algebra $A\hat{\oplus}M$ of $A$ with a conformal bimodule $M$, and show that there exist an…
Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras
本文将群的对称上同调推广到余交换Hopf代数，建立对称上同调与对称Hochschild上同调的同构，并研究其与经典上同调一致的条件。
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。
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。
A subring pair B < A has right depth 2n if the n+1'st relative Hochschild bar resolution group is isomorphic to a direct summand of a multiple of the n'th relative Hochschild bar resolution group as A-B-bimodules; depth 2n+1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n-1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B. In this paper the H-depth of semisimple subalgebra pairs is derived from the transpose inclusion m…
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…
Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild cohomology group $HH^1(TA)$. From these a formula for the dimension of $HH^1(TA)$ can be derived.
A minimum depth d^I(S --> R) is assigned to a ring homomorphism S --> R and a R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion mapping. Ideal depth gives lower bounds for d(S,R) in case of group C-algebra pair or semisimple complex algebra extensions. If R | S is a QF extension of finite depth, minimum left and right even depth are shown to coincide. If R < S is moreover a Frobenius extension with R a right S-generator, its subring depth is shown to coincide with its tower depth. In the proc…
For an extension of associative algebras $B\subset A$ over a field and an $A$-bimodule $X$, we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of $A$ and $B$, and the relative Hochschild homology, all of them with coefficients in $X$. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness.
Derivations and the first Hochschild cohomology group of the quantum grassmannian
本文利用分次技术、正规元与去齐次化局部化等式，将量子 Grassmann 代数导子联系到量子矩阵代数导子，并显式确定其第一 Hochschild 上同调群。
A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$, with the known cohomology of $n$-Lie algebras. The abelian extensions and infinitesimal deformations of $n$-Lie algebras, on the other hand, are shown to be characterized by the usual cohomology of $n$-Lie algebras. Furthermore, the Hochschild-Serre spectral sequence of the Lie algebra cohomology is upgraded to the level of $n$-Lie algebras, and is applied …
We introduce the notion of compatible actions in the context of split extensions of finite dimensional Lie algebras over a field. Using compatible actions, we construct a new resolution to compute the cohomology of semi-direct products of Lie algebras. We also give an alternative way to construct the Hochschild-Serre spectral sequence associated to a split extension of finite dimensional Lie algebras and obtain a sharper bound for the length of this spectral sequence.
Homological properties of extensions of algebras
本文提出强投射有界扩张概念，证明左整体维数有限性、Hochschild 同调支集及大左有限维数有限性在该扩张下保持，并构造了有限相对整体维数的新扩张类，推广了 Cibils 等人的结果。
A Survey on Han's Conjecture
本文综述了Han猜想的研究进展，介绍了Hochschild同调与代数整体维数的关系，并总结了满足该猜想的代数实例及保持该猜想的扩张。
Extensions of Multilinear Module Expansions
该文研究多线性算子扩张模构成的代数簇中的扩张问题，验证了阿贝尔理想确定的扩张参数化与既有泛代数扩张研究的一致性，建立了Well型定理与低维Hochschild-Serre正合序列。
Given any twisting cochain t:C -->A, where C is a connected, coaugmented chain coalgebra and A is an augmented chain algebra over an arbitrary PID R, we construct a twisted extension of chain complexes A --> H(t) --> C. We show that both the well-known Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra are special cases of H(t), which we therefore call the Hochschild complex of t. We explore the extent of the naturality of the Hochschild complex construction and apply the results of this exploration to determining conditions under which H(t) …
We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. We show that these characteristic classes can be seen as obstructions for the vanishing of differentials in the spectral sequence and prove a decomposition theorem. We also interpret our results in the settings of group and Lie algebra extensions and prove some interesting corollaries concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes.
Replacing bar-like resolutions in a simplicial setting
针对 Hochschild（上）同调推广中各维模结构变化的问题，本文提出一种在单纯情形下替换类 bar 分解的方法。
The paper concerns perfect diassociative algebras and their implications to the theory of central extensions. It is first established that perfect diassociative algebras have strong ties with universal central extensions. Then, using a known characterization of the multiplier in terms of a free presentation, we obtain a special cover for perfect diassociative algebras, as well as some of its properties. The subsequent results connect and build on the previous topics. For the final theorem, we invoke an extended Hochschild-Serre type spectral sequence to show that, for a perfect diassociative a…
We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra $\Lg$, an ideal $\Lh$ of $\Lg$ and the quotient $\Lg / \Lh$ with coefficients in a $\Lg$-module. The existence of such a sequence follows from the Hochschild-Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in…
Extensions realizing affine datum : central extensions
本文在具差分项的簇中研究实现仿射基准的中心扩张，推广群论经典定理，建立一维Hochschild-Serre序列、推导Schur-Hopf公式，证明覆盖存在性并刻画完美代数，其结果可特化至模同余格代数等簇。
Cohomological rigidity of conformal Galilei algebras and their central extensions
本文研究实共形伽利略代数复化及其中心扩张的第二伴随上同调，利用霍赫希尔德-塞尔分解定理计算相关空间，得出自旋为半整数时d≠2的共形伽利略代数具上同调刚性的结论，完成特定参数范围的形式刚性刻画。
In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to commutativity of the cup-product on Hochschild cohomology, a morphism from the graded algebra Sym(k^n) to Ext^*(A,A), the Ext-algebra in the category of A-bimodules. We prove that any extension of the n-parameter first order deformation of A to an INFINITE ORDER formal deformation provides a canonical `lift' of the graded algebra morphism above to a dg-algebra…
We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, seminormal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case when M=N^n gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen an…
In this paper, we introduce the notions of crossed module of associative conformal algebras, 2-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the 3-th Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra, and prove that the Deligne groupoid of this differential graded Lie algebra corresponds one to on…
We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorifed versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.
Let $L/K$ be a finite Galois extension of fields with group $Γ$. Associated to each Hopf-Galois structure on $L/K$ is a group $G$ of the same order as the Galois group $Γ$. The type of the Hopf-Galois structure is by definition the isomorphism type of $G$. We investigate the extent to which general properties of either of the groups $Γ$ and $G$ constrain those of the other. Specifically, we show that if $G$ is nilpotent then $Γ$ is soluble, and that if $Γ$ is abelian then $G$ is soluble. The proof of the latter result depends on the classification of finite simple groups. In contrast to these …
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.
We investigate criteria for algebra extensions that are of Galois type with respect to the coaction of a Hopf algebra or, more generally, a one-sided quotient of a Hopf algebra, or with respect to an entwining. We study the module- and comodule-theoretical properties of such extensions.
Explicit descriptions of the subfields $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$ of $NL$ and new explicit criteria for $NL = (NL)^{pi}(NL)^{sep}$
本文利用多项式系数和数值不变量，显式描述了纯不可分扩张下子域的结构，并给出了域分解的新显式判据。
Ştefan spectral sequence for faithfully flat Hopf Galois extensions is multiplicative
本文证明忠实平坦Hopf Galois扩张的Ştefan谱序列具有乘法性，引入相关形式体系构造对应谱序列，结合比较定理完成等同，在多类代数结构上得到新乘法性结果。
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