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Roland Berger

Publications and source records attributed to Roland Berger.

17 recordsLinked to original sources
arXiv2025

Calabi-Yau property in derived Koszul calculus

A Poincaré Van den Bergh duality theorem for strong Kc-Calabi-Yau algebras was obtained by R. Taillefer and the first author under the assumption that the derived functors of functors involved in the statement exist. We prove the existence of these derived functors by showing that the dg category defining the derived Koszul calculus is isomorphic to a dg category of dg modules over a dg algebra. Therefore we get a definition of strong Kc-Calabi-Yau algebras and a corresponding duality theorem without any existence assumption. We prove that a polynomial algebra is strong Kc-Calabi-Yau.

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arXiv2022

A cup-cap duality in Koszul calculus

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 the framework of DG algebras and DG bimodules. Various enriched structures are developed in a weaker situation corresponding to N>2.

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arXiv2020

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.

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arXiv2017

Koszul calculus for N-homogeneous algebras

We extend the Koszul calculus defined on quadratic algebras by Berger, Lambre and Solotar, to N-homogeneous algebras. When N>2, the Koszul cup and cap products are defined by specific expressions, and they are compatible with the Koszul differentials, providing associative products on (co)homology classes. The N-Koszul calculus is calculated for the truncated polynomial algebras.

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arXiv2017

A Koszul sign map

We define a Koszul sign map encoding the Koszul sign convention. A cohomological interpretation is given.

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arXiv2017

N-Cartan calculus

We examine the N-Koszul calculus for the N-symmetric algebras. The case N=2 corresponds to the Elie Cartan calculus. We conjecture that, as in the case N=2, the N-Cartan calculus extends to manifolds when N>2, which would provide a new type of noncommutative differential geometry.

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arXiv2017

Koszul Calculus

We present a new calculus which is well-adapted to quadratic algebras. This calculus consists in Koszul (co)homology, together with Koszul cup and cap products. Some applications are given. Koszul duality for Koszul (co)homology is proved for any quadratic algebra.

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arXiv2013

A criterion for homogeneous potentials to be 3-Calabi-Yau

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.

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arXiv2009

Combinatorics and N-Koszul algebras

The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem.

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arXiv2008

Gerasimov's theorem and N-Koszul algebras

The paper is devoted to graded algebras having a single homogeneous relation. Using Gerasimov's theorem, a criterion to be N-Koszul is given, providing new examples. An alternative proof of Gerasimov's theorem for N=2 is given. Some related results on Calabi-Yau algebras are proved.

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arXiv2006

Poincare-Birkhoff-Witt Deformations of Calabi-Yau Algebras

Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra (with relations) which is Calabi-Yau of dimension 3 is defined from a homogeneous potential W. In this paper, we prove that if we add to W any potential of smaller degree, we get a Poincare-Birkhoff-Witt deformation of A. Such PBW deformations are Calabi-Yau and are characterised among all the PBW deformations of A. Various examples are presented.

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arXiv2005

Inhomogeneous Yang-Mills algebras

We determine all inhomogeneous Yang-Mills algebras and super Yang-Mills algebras which are Koszul. Following a recent proposal, a non-homogeneous algebra is said to be Koszul if the homogeneous part is Koszul and if the PBW property holds. In this paper, the homogeneous parts are the Yang-Mills algebra and the super Yang-Mills algebra.

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arXiv2004

Koszul and Gorenstein properties for homogeneous algebras

Koszul property was generalized to homogeneous algebras of degree N>2 in [5], and related to N-complexes in [7]. We show that if the N-homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincare duality between Hochschild homology and cohomology of A, as for N=2.

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arXiv2003

Koszulity for nonquadratic algebras II

It has been shown recently, in a joint work with Michel Dubois-Violette and Marc Wambst (see math.QA/0203035), that Koszul property of $N$-homogeneous algebras (as defined in the original paper) becomes natural in a $N$-complex setting. A basic question is to define the differential of the bimodule Koszul complex of an $N$-homogeneous algebra, e.g., for computing its Hochschild homology. The differential defined here uses $N$-complexes. That puts right the wrong differential presented in the original paper in a 2-complex setting. Actually, as we shall see, it is impossible to avoid $N$-complexes in defining the differential, whereas the bimodule Koszul complex is a 2-complex.

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arXiv2002

Homogeneous algebras

Various concepts associated with quadratic algebras admit natural generalizations when the quadratic algebras are replaced by graded algebras which are finitely generated in degree 1 with homogeneous relations of degree N. Such algebras are referred to as {\sl homogeneous algebras of degree N}. In particular it is shown that the Koszul complexes of quadratic algebras generalize as N-complexes for homogeneous algebras of degree N.

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