Yau Algebras Research Articles

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.

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.

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.

On 2-Representation Infinite Algebras Arising From Dimer Models

AbstractThe Jacobian algebra arising from a consistent dimer model is a bimodule 3-Calabi–Yau algebra, and its center is a 3-dimensional Gorenstein toric singularity. A perfect matching (PM) of a dimer model gives the degree, making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation infinite algebra that is a generalization of a representation infinite hereditary algebra. Internal PMs, which correspond to toric exceptional divisors on a crepant resolution of a 3-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Combining this characterization with the theorems due to Amiot–Iyama–Reiten, we show that the stable category of graded maximal Cohen–Macaulay modules admits a tilting object for any 3-dimensional Gorenstein toric isolated singularity. We then show that all internal PMs corresponding to the same toric exceptional divisor are transformed into each other using the mutations of PMs, and this induces derived equivalences of 2-representation infinite algebras.

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.

New $k$-th Yau algebras of isolated hypersurface singularities and weak Torelli-type theorem

New $k$-th Yau algebras of isolated hypersurface singularities and weak Torelli-type theorem

Classification of the nilradical of $k$-th Yau algebras arising from singularities

Classification of the nilradical of $k$-th Yau algebras arising from singularities

Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

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).

Cyclic $A_\infty$-algebras and double Poisson algebras

In this article we prove that there exists an explicit bijection between nice d -pre-Calabi–Yau algebras and d -double Poisson differential graded algebras, where d \in \mathbb{Z} , extending a result proved by N. Iyudu and M. Kontsevich. We also show that this correspondence is functorial in a quite satisfactory way, giving rise to a (partial) functor from the category of d -double Poisson dg algebras to the partial category of d -pre-Calabi–Yau algebras. Finally, we further generalize it to include double P_{\infty} -algebras, introduced by T. Schedler.

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.

Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities

Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $$L_{k}(V)$$ associated to (V, 0). Its dimension is denoted as $$\lambda _{k}(V)$$ . The $$L_{k}(V)$$ is a generalization of the Yau algebra L(V) and $$L_{0}(V)=L(V)$$ . These numbers $$\lambda _{k}(V)$$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $$L_1(V)$$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $$\lambda _{1}(V)$$ . We also formulate a sharp upper estimate conjecture for the $$L_k(V)$$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.

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.

On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities

Let (V,0) be an isolated hypersurface singularity defined by the holomorphic function $f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)$ . The k-th Yau algebra Lk(V ) is defined to be the Lie algebra of derivations of the k-th moduli algebra $A^{k}(V) := \mathcal {O}_{n}/(f, m^{k}J(f))$ , where k ≥ 0, m is the maximal ideal of $\mathcal {O}_{n}$ . I.e., Lk(V ) := Der(Ak(V ),Ak(V )). These new series of derivation Lie algebras are quite subtle invariants since they capture enough information about the complexity of singularities. In this paper we formulate a conjecture for the complete characterization of ADE singularities by using generalized Cartan matrix Ck(V ) associated to k-th Yau algebras Lk(V ), k ≥ 1. In this paper, we provide evidence for the conjecture and give a new complete characterization for ADE singularities. Furthermore, we compute their other various invariants that arising from the 1-st Yau algebra L1(V ).

On two inequality conjectures for the k-th Yau numbers of isolated hypersurface singularities

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function $$f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)$$ . In our previous work, we introduced a series of novel Lie algebras associated to (V, 0), i.e., k-th Yau algebra $$L^k(V), k\ge 0$$ . It was defined to be the Lie algebra of derivations of the k-th moduli algebras $$A^k(V)= {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0$$ , where m is the maximal ideal of $${\mathcal {O}}_n$$ . I.e., $$L^k(V):=\text {Der}(A^k(V), A^k(V))$$ . The dimension of $$L^k(V)$$ was denoted by $$\lambda ^k(V)$$ . The number $$\lambda ^k(V)$$ , which was called k-th Yau number, is a subtle numerical analytic invariant of (V, 0). Furthermore, we formulated two conjectures for these k-th Yau number invariants: a sharp upper estimate conjecture of $$\lambda ^k(V)$$ for weighted homogeneous isolated hypersurface singularities (see Conjecture 1.2) and an inequality conjecture $$\lambda ^{(k+1)}(V) > \lambda ^k(V), k\ge 0$$ (see Conjecture 1.1). In this article, we verify these two conjectures when k is small for large class of singularities.

Variation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularities

Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.

Koszul calculus of preprojective algebras

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A 1 and A 2 , vanishes in any (co)homological degree p > 2 . Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees p and 2 − p , and we prove a generalised version of the 2-Calabi–Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi–Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi–Yau property and we say that they are Koszul complex Calabi–Yau (Kc–Calabi–Yau) of dimension 2. For Kc–Calabi–Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincaré Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.

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.

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.