Yau Algebras Research Articles

Skew polynomial algebras with coefficients in Koszul Artin–Schelter regular algebras

Let A be a Koszul Artin–Schelter regular algebra with Nakayama automorphism ξ. We show that the Yoneda Ext-algebra of the skew polynomial algebra A[z;ξ] is a trivial extension of a Frobenius algebra. Then we prove that A[z;ξ] is Calabi–Yau; and hence each Koszul Artin–Schelter regular algebra is a subalgebra of a Koszul Calabi–Yau algebra. A superpotential wˆ is also constructed so that the Calabi–Yau algebra A[z;ξ] is isomorphic to the derivation quotient of wˆ. The Calabi–Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin–Schelter regular algebra is also discussed.

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w = fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.

Batalin–Vilkovisky structures on Ext and Tor

Abstract This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and étale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on E x t U ( A , A ) $ \textup {Ext}_U(A,A)$ are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan–Rinehart homotopy formula whose essence is that T o r U ( M , A ) $\textup {Tor}^U(M,A)$ becomes for suitable right U-modules M a Batalin–Vilkovisky module over E x t U ( A , A ) $ \textup {Ext}_U(A,A)$ , or in the words of Nest, Tamarkin, Tsygan and others, that E x t U ( A , A ) $ \textup {Ext}_U(A,A)$ and T o r U ( M , A ) $\textup {Tor}^U(M,A)$ form a differential calculus. As an illustration, we show how the well-known operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi–Yau algebra is a Batalin–Vilkovisky algebra to twisted Calabi–Yau algebras.

Deformations of Koszul Artin–Schelter Gorenstein algebras

We compute the Nakayama automorphism of a Poincaré–Birkhoff–Witt (PBW)-deformation of a Koszul Artin–Schelter (AS) Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi–Yau algebra to be Calabi–Yau. The relations between the Calabi–Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul AS–Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi–Yau algebra is still Calabi–Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this article is elementary and based on linear algebra. The results obtained in this article will be applied in a subsequent paper (He et al., Skew polynomial algebras with coefficients in AS regular algebras, preprint, 2011).

On the homology of almost Calabi–Yau algebras associated to [formula omitted] modular invariants

We compute the Hochschild homology and cohomology, and cyclic homology, of almost Calabi–Yau algebras for SU(3)ADE graphs. These almost Calabi–Yau algebras are a higher rank analogue of the preprojective algebras for Dynkin diagrams, which are SU(2)-related constructions. The Hochschild (co)homology and cyclic homology of A can be regarded as invariants for the braided subfactors associated to the SU(3) modular invariants.

Superpotential algebras and manifolds

We study a special class of Calabi–Yau algebras (in the sense of Ginzburg): those arising as the fundamental group algebras of acyclic manifolds. Motivated partly by the usefulness of ‘superpotential descriptions’ in motivic Donaldson–Thomas theory, we investigate the question of whether these algebras admit superpotential presentations. We establish that the fundamental group algebras of a wide class of acyclic manifolds, including all hyperbolic manifolds, do not admit such descriptions, disproving a conjecture of Ginzburg regarding them. We also describe a class of manifolds that do admit such descriptions, and discuss a little their motivic Donaldson–Thomas theory. Finally, some links with topological field theory are described.

Calabi–Yau algebras and weighted quiver polyhedra

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi–Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3-dimensional Calabi–Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded cancellation algebras. For these algebras the CY-3 condition implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold.

The Calabi–Yau property of smash products

Let H be a semisimple Hopf algebra and R a braided Hopf algebra in the category of Yetter–Drinfeld modules over H. When R is a Calabi–Yau algebra, a necessary and sufficient condition for R#H to be a Calabi–Yau Hopf algebra is given. Conversely, when H is the group algebra of a finite group and the smash product R#H is a Calabi–Yau algebra, we give a necessary and sufficient condition for the algebra R to be a Calabi–Yau algebra.

Poincaré–Birkhoff–Witt Deformation of Koszul Calabi–Yau Algebras

The Calabi–Yau property of the Poincare–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincare–Birkhoff–Witt deformation of a Calabi–Yau algebra of dimension 3 is Calabi–Yau under some conditions. The main result in this paper generalizes their result to higher dimensional Koszul Calabi–Yau algebras. As corollaries, the necessary and sufficient condition obtained by He et al. (J Algebra 324:1921–1939, 2010) for the universal enveloping algebra, respectively, Sridharan enveloping algebra, of a finite-dimensional Lie algebra to be Calabi–Yau, is derived.

Skew group algebras of Calabi–Yau algebras

The Calabi–Yau property of skew group algebras is discussed. It is shown that the skew group algebra A # G of a Koszul Calabi–Yau algebra A with a finite subgroup G of automorphisms of A is Calabi–Yau if and only if G is a finite subgroup of the special linear group SL ( A ) , which is defined by means of the homological determinant. Using the A ∞ -algebra structure on the Yoneda algebra, some results in Bocklandt et al. (2010) [BSW] are generalized, say, every connected graded p-Koszul Calabi–Yau algebra is derived from a superpotential. The superpotential for the skew group algebra A # G is also constructed.

N -representation-finite algebras and twisted fractionally Calabi-Yau algebras

In this paper, we study n-representation-finite algebras from the viewpoint of the fractionally Calabi–Yau property. We shall show that all n-representation-finite algebras are twisted fractionally Calabi–Yau. We also show that for any ℓ>0, twisted (n(ℓ−1)/ℓ)-Calabi–Yau algebras of global dimension at most n are n-representation-finite. As an application, we give a construction of n-representation-finite algebras using the tensor product.

Consistency conditions for brane tilings

Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendrői, Mozgovoy and Reineke regarding the Donaldson–Thomas theory of the moduli space of framed cyclic representations of the associated algebra. Using only a natural cancellation-type consistency condition, we show that the algebras are 3-Calabi–Yau, and calculate Donaldson–Thomas type invariants of the moduli spaces. Two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations, and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to the universal cover. These ideas allow us to generalise the above results to all consistent brane tilings on K ( π , 1 ) surfaces. We also prove a converse: no consistent brane tiling on a sphere gives rise to a 3-Calabi–Yau algebra.

Dualité de Van den Bergh et structure de Batalin–Vilkoviskiĭ sur les algèbres de Calabi–Yau

The abstract notion of Tamarkin–Tsygan calculus with duality gives Batalin–Vilkoviskiǐ structures in a general setting. We apply this technique to the case of Van den Bergh duality for algebras to prove that Calabi–Yau algebras are BV-algebras.

Helices on del Pezzo surfaces and tilting Calabi–Yau algebras

We study tilting for a class of Calabi–Yau algebras associated to helices on Fano varieties. We do this by relating the tilting operation to mutations of exceptional collections. For helices on del Pezzo surfaces the algebras are of dimension three, and using an argument of Herzog, together with results of Kuleshov and Orlov, we obtain a complete description of the tilting process in terms of quiver mutations.

Superpotentials and higher order derivations

We consider algebras defined from quivers with relations that are k th order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for GL n for all n , and four-dimensional Sklyanin algebras. More generally, we show that any N -Koszul, (twisted) Calabi–Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yields an equivalence between N -Koszul twisted Calabi–Yau algebras A and algebras defined by a superpotential ω such that an associated complex is a bimodule resolution of A . Finally, we apply these results to give a description of the moduli space of four-dimensional Sklyanin algebras using the Weil representation of an extension of SL 2 ( Z / 4 ) .

A solvability criterion for the Lie algebra of derivations of a fat point

We consider the Lie algebra of derivations of a zero-dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result was motivated by and generalizes the solvability of the Yau algebra of an isolated hypersurface singularity.

On the noncommutative Donaldson–Thomas invariants arising from brane tilings

Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson–Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi–Yau algebra. We also formulate a rationality conjecture for the generating functions of the Donaldson–Thomas type invariants.

Quivers with potentials and their representations I: Mutations

We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi–Yau algebras, cluster algebras.

On symmetric, smooth and Calabi–Yau algebras

One possible definition for a Calabi–Yau algebra is a symmetric smooth PI algebra. Our main purpose here is to prove some necessary and sufficient criteria for verifying the (local) symmetric property, in smooth PI algebras. Many known smooth PI algebras are shown to have this property. In particular quantum enveloping algebras of complex semi-simple Lie algebras, in the root of unity case and the enveloping algebra of sl ( n ) with ( p , n ) = 1 , in characteristic p, are typical examples. A surprising result is that the inj . dim T , is finite, where T is the trace ring of m, n × n generic matrices over a field of zero characteristic.

Graded Calabi Yau algebras of dimension 3

In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d , the set of good superpotentials of degree d , i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.