Andrea Solotar

Universidad de Buenos Aires | UBA · Department of Mathematics (FCEN)

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if...

We prove the graded braided commutativity of the Hochschild cohomology of $A$ with trivial coefficients, where $A$ is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra of an abelian group, under some finiteness conditions on a projective resolution of $A$ as $A$-bimodule. This is a generalization of a result b...

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers conta...

Let B⊂A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we gi...

Corrigendum: Jacobi–Zariski long nearly exact sequences for associative algebras Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, Andrea Solotar Volume 53, Issue 6, Bulletin of the London Mathematical Society, pages: 1651-1652. First Published online: September 24, 2021

Let $B\subset A$ be a bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that $B$ satisfies Han's conjecture if and only if $A$ does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is bounded. Examples of non split bounde...

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 whic...

We introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application, we prove that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. We propose a conceptual approach that may lead to a proof of the graded commutativity, based on derived categories in...

We study homological properties of a family of algebras called toupie algebras. Our main objective is to obtain the Gerstenhaber structure of their Hochschild cohomology, with the purpose of describing the Lie algebra structure of the first Hochschild cohomology space, together with the Lie module structure of the whole Hochschild cohomology.

In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in [32]. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gen...

We provide a formula for the change of the dimension of the first Hochschild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first Ho...

A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.

We provide a formula for the change of the dimension of the first Hoch\-schild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first...

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers conta...

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers conta...

For each nonzero $h\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independ...

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.

We develop a general framework for studying relative weight representations for certain pairs consisting of an associative algebra and a commutative subalgebra. Using these tools we describe projective and simple weight modules for quantum Weyl algebras for generic values of deformation parameters. We consider two quantum versions: one by Maltsinio...

We solve the isomorphism problem for non noetherian down-up algebras $A(\alpha,0,\gamma)$ by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for $\gamma = 0$ or quantum versions of the Weyl algebra $A_1$ for non zero $\gamma$. In particular we ob...

In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in earlier work by the second author. We give an explicit description of the Lie algebra structure of the first...

After the online publication of the article [1] we learned that Professor Jean-Louis Koszul passed away on January 12th, 2018. We dedicate the article to him.

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 Vir...

We provide a framework connecting several well-known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective reso...

The main objective of this paper is to present a theory for computing the Hochschild cohomology of algebras built on a specific data, namely multi-extension algebras. The computation relies on cohomological functors evaluated on the data, and on the combinatorics of an ad hoc quiver. One-point extensions are occurrences of this theory, and Happel's...

We investigate the behavior of finitely generated projective modules over a down-up algebra. Specifically, we show that every noetherian down-up algebra $A(\alpha,\beta,\gamma)$ has a non-free, stably free right ideal. Further, we compute the stable rank of these algebras using Stafford's Stable Range Theorem and Kmax dimension.

Nichols algebras are an important tool for the classification of Hopf algebras. Within those with finite GK dimension, we study homological invariants of the super Jordan plane, that is, the Nichols algebra $A=B(V(-1,2))$. These invariants are Hochschild homology, the Hochschild cohomology algebra, the Lie structure of the first cohomology space -...

Let $G,H$ be groups, $\phi: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $\phi$ induces an $H$-grading on $A$, and on any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given an injective $G$-graded $A$-module, we give bounds for its injective dimension when seen as $H$-graded $A$-module. Follow...

Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $\Lambda$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at th...

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective reso...

We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi-Yau homogeneous down-up algebras. This family was defined by Benkart and Roby in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and arguments similar to tho...

We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information t...

Given a left noetherian k-algebra A graded by a group G, an injective object I in the category of G-graded A-modules and a morphism from G to another group G', we provide bounds for the injective dimension of I as a G'-graded A-module. For this, we use three change of grading functors. Most of the constructions concerning these functors work in the...

We define the fundamental group of a Hopf algebra over a field. For this we consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf linear categories over a finite group. We compute some examples.

The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell's well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing thus a useful tool for computing invariants associated to the algebras. We illustrate how to use it giving sev...

Categories over a field $k$ can be graded by different groups in a connected way; we consider morphisms between these gradings in order to define the fundamental grading group. We prove that this group is isomorphic to the fundamental group \`a la Grothendieck as considered in previous papers. In case the $k$-category is Schurian generated we prove...

We give a necessary and sufficient condition for an N-Koszul algebra defined by a homogeneous potential, to be 3-Calabi-Yau. As an application, we recover two families of 3-Calabi-Yau algebras recently appeared in the literature, by studying skew polynomial algebras over non-commutative quadrics.

We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of 'quantum' type in all but a few exceptional cases.

In this article we establish an explicit link between the classical theory of deformations \`a la Gerstenhaber -- and a fortiori with the Hochschild cohomology-- and (weak) PBW-deformations of homogeneous algebras. Our point of view is of cohomological nature. As a consequence, we recover a theorem by R. Berger and V. Ginzburg, which gives a precis...

The aim of this article is to describe families of representations of the Yang-Mills algebras YM(n) (n is an element of N->= 2) defined by A. Connes and M. Dubois-Violette. We first describe some irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of YM, big enough to seperate points of...

Let $k$ be a commutative ring. We study the behaviour of coverings of $k$-categories through fibre products and find a criterion for a covering to be Galois or universal.

Let k be a field. We attach a CW-complex to any Schurian k-category and we prove that the fundamental group of this CW-complex is isomorphic to the intrinsic fundamental group of the k-category. This extends previous results by J. C. Bustamante [Commun. Algebra 30, No. 11, 5307–5329 (2002; Zbl 1017.16009)]. We also prove that the Hurewicz morphism...

Let $a$ and $b$ be two integers such that $2\le a<b$. In this article we define the notion of $(a,b)$-Koszul algebra as a generalization of $N$-Koszul algebras. We also exhibit examples and we provide a minimal graded projective resolution of the algebra $A$ considered as an $A$-bimodule, which allows us to compute the Hochschild homology groups fo...

Consider the intrinsic fundamental group \`a la Grothendieck of a linear category using connected gradings. In this article we prove that any full convex subcategory is incompressible, in the sense that the group map between the corresponding fundamental groups is injective. We start by proving the functoriality of the intrinsic fundamental group w...

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k$-algebra $A$, we consider the category of all connected gradings of $A$ by a group $G$ and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundam...

The aim of this article is to compute the Hochschild and cyclic homology groups of Yang-Mills algebras, that have been defined by A. Connes and M. Dubois-Violette. We proceed here the study of these algebras that we have initiated in a previous article. The computation involves the use of a spectral sequence associated to the natural filtration on...

We prove without any assumption on the ground field that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension in not finite. Comment: 9 pages

The aim of this article is to describe families or representations of the Yang-Mills algebras YM(n) (where n>1) defined by Connes and Dubois-Violette. We first describe irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of YM(n), big enough to separate points of the algebra. In order to...

We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges' result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in nonco...

We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. We prove that this group is isomorphic to the inverse limit of the Galois groups associated to Galois coverings. Moreover, the grading deduced from a Galois covering enables us to describe...

We consider $${\mathsf{N}}$$ -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left trunc...

We define H-Galois extensions for k-linear categories 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 group algebra decomposes as a direct sum indexed by conjugacy classes of the group. We also compute some Hochschild–Mitchel...

In this notes we start with the basic definitions of derived categories , derived functors, tilting complexes and stable equivalences of Morita type. Our aim is to show via several examples that this is the best framework to do homological algebra, We also exhibit their usefulness for getting new proofs of well known results. Finally we consider th...

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 alternativ...

Algebras over a field k generalize to categories over k in order to considers Galois coverings. Two theories presenting analogies, namely smash extensions and Galois coverings with respect to a finite group are known to be different. However we prove in this paper that they are Morita equivalent. For this purpose we need to describe explicit proces...

We prove that E x t A ∙ ( k , k ) \mathrm {Ext} ^{\bullet }_A(k,k) is a Gerstenhaber algebra, where A A is a Hopf algebra. In case A = D ( H ) A=D(H) is the Drinfeld double of a finite-dimensional Hopf algebra H H , our results imply the existence of a Gerstenhaber bracket on H G S ∙ ( H , H ) H^{\bullet }_{GS}(H,H) . This fact was conjectured by R...

We study isomorphisms between generalized Weyl algebras, giving a complete answer to the quantum case of this problem for R=k[h].

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 h...

We prove that Ext^*_A(k,k) is a Gerstenhaber algebra, where A is a Hopf algebra. In case A=D(H) is the Drinfeld double of a finite dimensional Hopf algebra H, our results implies the existence of a Gerstenhaber bracket on H^*_{GS}(H,H). This fact was conjectured by R. Taillefer in math.KT0207154. The method consists in identifying Ext^*_A(k,k) as a...

We prove that if C is a cocommutative k-coalgebra such that for all group-like elements , then smoothness of C is equivalent to the condition Hoch∗(C)=0 for all ∗⩾N.

We compute Hochschild homology and cohomology of a class of generalized Weyl algebras (for short GWA, defined by Bavula in St.Petersbourg Math. Journal 1999 4(1) pp. 71-90). Examples of such algebras are the n-th Weyl algebras, U(sl_2), primitive quotients of U(sl_2), and subalgebras of invariants of these algebras under finite cyclic groups of aut...

Soit G un sous-groupe fini de Sp(2n,C) opérant par automorphismes dans l'algèbre de Weyl An(C). Nous calculons les groupes d'homologie et de cohomologie de Hochschild de l'algèbre d'invariants An(C)G. Let G be a finite subgroup of Sp(2n,C) acting by automorphisms in the Weyl algebra An(C). We compute the Hochschild homology and cohomology groups of...

Soit G un sous-groupe fini de Sp (2 n , C ) opérant par automorphismes dans l'algèbre de Weyl A n ( C ). Nous calculons les groupes d'homologie et de cohomologie de Hochschild de l'algèbre d'invariants A n ( C ) G. Let G be a finite subgroup of Sp (2 n , C ) acting by automorphisms in the Weyl algebra A n ( C ). We compute the Hochschild homology a...

Given a k-coalgebra C, two cohomology theories have been associated to it by Doi. One of them, denoted Hoch*, plays the role of Hochschild homology of algebras, and the other one corresponds to Hochschild cohomology. Their resemblance to Hochschild (co)homology includes the fact that they are invariant with respect to Morita-Takeuchi equivalence. W...

For a discrete group G, we define the notion of G-II-algebra whose model is given by the graded group (πn(K))n≥1 corresponding to a topological space K on which G acts. This notion is a natural extension of the notion of II-algebra ([8]) and we use it to construct a spectral sequence relied to the II-algebra IImapGpt(X, Y), associated to the space...

For a discrete group G, we define the notion of G-II-algebra whose model is given by the graded group (pi(n)(K))(n greater than or equal to 1) corresponding to a topological space K on which G acts. This notion is a natural extension of the notion of II-algebra ([8]) and we use it to construct a spectral sequence relied to the IT-algebra Pi map(pt)...

We study how Hilbert bimodules correspond in the algebraic case to hermitian Morita equivalences and consequently we obtain a description of the hermitian Picard group of a commutative involutive algebra A as the semidirect product of the classical hermitian Picard group of A and the automorphisms of A commuting with the involution. We also obtain...

Pour un groupe discret G, nous définissons la notion de G-Π-algèbre dont le modèle est donné par le groupe gradué (π n (K)) n≥1 correspondantàcorrespondantà un espace topologique pointé K sur lequel G opère. Cette notion est une extension naturelle de la notion de Π-algèbre ([8]) et nous l'utilisons pour construire une suite spectrale reliéè a la Π...

In this paper we present a direct proof of what is suggested by Holm’s results (T. Holm, The Hochschild cohomology ring of a modular group algebra: the commutative case, Comm. Algebra 24, 1957–1969 (1996)): there is an isomorphism of algebras HH*(kG,kG) → kG ⊗ H*(G,k) where G is a finite abelian group, k a ring, HH*(kG,kG) is the Hochschild cohomol...

Let A be an associative k-algebra with involution, where k is a commutative ring of characteristic not equal to two. Then the dihedral groups act on the Hochschild complex and, following Loday, a new homological theory, called dihedral homology, can be defined generalizing the notion of cyclic homology defined by Connes. Here we give a model to com...

In this paper we prove the invariance of positive Hochschild ho-mology and dihedral homology with respect to hermitian Morita e-quivalence between involutive algebras. We also define the notion of hermitian k-congruence and prove some results on Morita invariance of HH + * and HD * in this context.

Let A be a commutative k -algebra with 1. We present a characterization of α -derivations, for α: A →> A a morphism of algebras, using α -Taylor series. When S = C[x,x ⁻¹ ,ξ] and α(x) = qx, α(ξ) = qξ , we compare the q -de Rham cohomology of the C -algebra S with the Hochschild homology of D q , the algebra of q -difference operators on C [ x,x ⁻¹...

We compute the cyclic homology of A = k[X]/< f > for an arbitrary commutative ring k and a monic polynomial f.

Le but de cet article est d'étendre au cadre bivariant le théoreme de Jones, Goodwillie et Burghelea-Fiedorowicz (cf. [J], [G], [B-F]) , qui prouve l'isomor-phisrne entre la cohomologie cyclique du complexe singulier d'un Sl-espace X et la cohornologie Sl-équivariante de X. Nous faisons égalernent la cornparaison entre la longue suite exacte de Con...

We compute the cyclic homology of A = k[X]/〈Xⁿ〉 for an arbitrary commutative ring k, and we apply this result to compute the cyclic homology of k[X]/〈f〉, when k is a field and f is an arbitrary polynomial.

The main theorems of this paper show the invariance of this cohomology theory with respect not only to Morita - Takeuchi equivalence, but to $k$-congruences, which is a weaker condition than Morita - Takeuchi equivalence, and state that they are also invariant under Azumaya extensions of $k$. We also show that they are invariant respect to cosepara...

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