Calabi Yau Algebras Research Articles

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups

We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.

Cohomology of Graded Twisting of Hopf Algebras

Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms.

On Yang-Baxter equation and Calabi-Yau property of Koszul algebras

–A well-known result of Beidar, Fong, and Stolin yields a description of symmetric Frobenius algebras through solutions of the quantum Yang-Baxter equation. We will prove a graded version of this theorem and produce a new characterization of Koszul Calabi-Yau algebras. We give another proof of the Calabi-Yau property for three and four-dimensional Sklyanin algebras as an application.

A CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS

AbstractIn earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$ -Calabi–Yau.

Calabi–Yau property of quantum groups of GL(2) representation type

In this paper, we prove that the quantum groups [Formula: see text] introduced by Mrozinski in [C. Mrozinski, Quantum groups of [Formula: see text] representation type, J. Noncommut. Geom. 8(1) (2014) 107–140] and their Hopf–Galois objects are twisted Calabi–Yau algebras, and give their Nakayama automorphisms explicitly.

Serre Functors and Graded Categories

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.

Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras

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 a 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 is finite dimensional. These two theorems can be viewed as a first step towards the singularity theory of noncommutative power series. As an application, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential.

Auslander’s theorem for dihedral actions on preprojective algebras of type A

AbstractGiven an algebra R and G a finite group of automorphisms of R, there is a natural map $\eta _{R,G}:R\#G \to \mathrm {End}_{R^G} R$ , called the Auslander map. A theorem of Auslander shows that $\eta _{R,G}$ is an isomorphism when $R=\mathbb {C}[V]$ and G is a finite group acting linearly and without reflections on the finite-dimensional vector space V. The work of Mori–Ueyama and Bao–He–Zhang has encouraged the study of this theorem in the context of Artin–Schelter regular algebras. We initiate a study of Auslander’s result in the setting of nonconnected graded Calabi–Yau algebras. When R is a preprojective algebra of type A and G is a finite subgroup of $D_n$ acting on R by automorphism, our main result shows that $\eta _{R,G}$ is an isomorphism if and only if G does not contain all of the reflections through a vertex.

A note on the discriminant of reflection Hopf algebras

We provide a formula for computing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra. This method is applied to study the Jacobian and discriminant for reflection Hopf algebras.

From β to η: a new cohomology for deformed Sasaki-Einstein manifolds

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups {H}_{overline{partial}}^{left(p,0right)}(k) graded by their charge under the Reeb vector. We then introduce a new cohomology, η-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing η ≡ df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d mathcal{N} = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the η-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the η-cohomology groups. We verify that this relation is satisfied in the case of S5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of ℙ2, ℙ1× ℙ1 and the del Pezzo surfaces.

Cluster categories of formal DG algebras and singularity categories

AbstractGiven a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau and realise its cluster category as a triangulated hull of an orbit category of a derived category and as the singularity category of a finite-dimensional Iwanaga-Gorenstein algebra. Along the way, we give two results that stand on their own. First, we show that the derived category of coherent sheaves over a Calabi-Yau algebra has a natural cluster tilting subcategory whose dimension is determined by the Calabi-Yau dimension and thea-invariant of the algebra. Second, we prove that two DG orbit categories obtained from a DG endofunctor and its homotopy inverse are quasi-equivalent. As an application, we show that the higher cluster category of a higher representation infinite algebra is triangle equivalent to the singularity category of an Iwanaga-Gorenstein algebra, which is explicitly described. Also, we demonstrate that our results generalise the context of Keller–Murfet–Van den Bergh on the derived orbit category involving a square root of the AR translation.

2-Segal objects and algebras in spans

We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).

GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

AbstractThis is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

Quivers supporting twisted Calabi-Yau algebras

We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form A = k Q / I , where Q is a quiver and I is an ideal of relations coming from taking partial derivatives of a twisted superpotential on Q . We define the type ( M , P , d ) of such an algebra A , where M is the incidence matrix of the quiver, P is the permutation matrix giving the action of the Nakayama automorphism of A on the vertices of the quiver, and d is the degree of the superpotential. We study the question of what possible types can occur under the additional assumption that A has polynomial growth. In particular, we are able to give a nearly complete answer to this question when Q has at most 3 vertices.

Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras

In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.

Lecture notes on quivers with superpotential and their representations

These lecture notes are based on a mini-course presented at the fifth version of the Workshop Geometry in Algebra and Algebra in Geometry held in Medellin–Colombia in October 2019. The aim is to provide the background necessary to understand the theory of quivers with relations given by superpotentials. A heavy emphasis is placed throughout on examples to illustrate the applicability of the theory. The motivations for the lectures come from several sources: superpotentials in physics, Calabi–Yau algebras, and noncommutative resolutions.

Calabi-Yau algebras and the shifted noncommutative symplectic structure

In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [15], on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss [3], have a shifted symplectic structure in the sense of Pantev-Toën-Vaquié-Vezzosi [28].

Preprojective algebra structure on skew-group algebras

We give a class of finite subgroups G<SL(n,k) for which the skew-group algebra k[x1,…,xn]#G does not admit the grading structure of a higher preprojective algebra. Namely, we prove that if a finite group G<SL(n,k) is conjugate to a subgroup of SL(n1,k)×SL(n2,k), for some n1,n2≥1, then the skew-group algebra k[x1,…,xn]#G is not Morita equivalent to a higher preprojective algebra. This is related to the preprojective algebra structure on the tensor product of two Koszul bimodule Calabi–Yau algebras. We prove that such an algebra cannot be endowed with a grading structure as required for a higher preprojective algebra.Moreover, we construct explicitly the bound quiver of the higher preprojective algebra over a finite-dimensional Koszul algebra of finite global dimension. We show in addition that preprojective algebras over higher representation-infinite Koszul algebras are derivation-quotient algebras whose relations are given by a superpotential.

D-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander–Reiten theory of self-injective algebras. We give a systematic construction of (weakly) d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most d. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for k-linear categories to the setting of higher-dimensional Auslander–Reiten theory.Applications include n-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be d-representation-finite in many cases. We also get a complete classification of all d-representation-finite self-injective Nakayama algebras for arbitrary d.

Smash products of Calabi–Yau algebras by Hopf algebras

Let H be a Hopf algebra and A be an H -module algebra. This article investigates when the smash product A\sharp H is (skew) Calabi–Yau, has Van den Bergh duality or is Artin–Schelter regular or Gorenstein. In particular, if A and H are skew Calabi–Yau, then so is A\sharp H and its Nakayama automorphism is expressed using the ones of A and H . This is based on a description of the inverse dualising complex of A\sharp H when A is a homologically smooth dg algebra and H is homologically smooth and with invertible antipode. This description is also used to explain the compatibility of standard constructions of Calabi–Yau dg algebras with taking smash products.