arXiv subjects

Dragoş Ştefan

Publications and source records attributed to Dragoş Ştefan.

8 recordsLinked to original sources

arXiv2024

On deformation theory of associative algebras in monoidal categories

We extend the classical concept of deformation of an associative algebra, as introduced by Gerstenhaber, by using monoidal linear categories and cocommutative coalgebras as foundational tools. To achieve this goal, we associate to each cocommutative coalgebra $C$ and each linear monoidal category $\M$, a $\Bbbk$-linear monoidal category $\M_{C}$. This construction is functorial: any coalgebra morphism $ι:C\to\widetilde{C}$ induces a strict monoidal functor $ι^{*}:\M_{\widetilde{C}}\to\M_{C}$. An $ι$-deformation of an algebra $(A,m)$ is defined as an algebra $(A,\widetilde{m})$ in the fiber of $ι^{*}$ over $(A,m)$. Within this framework, the deformations of $(A,m)$ are organized into a presheaf, which is shown to be representable. In other words, there exists a universal deformation satisfying a specific universal property. It is well established that classical deformation theory is deeply connected to Hochschild cohomology. We identify and analyze the cohomology theory that governs $ι$-deformations in the second part of the paper. Additionally, several particular cases and applications of these results are examined in detail.

math.RA↗

arXiv2017

$PBW$-deformations of graded rings

We prove in a very general framework several versions of the classical Poincaré-Birkhoff-Witt Theorem, which extend results from [BeGi, BrGa, CS, HvOZ, WW]. Applications and examples are discussed in the last part of the paper.

math.QA↗

arXiv2016

Further Properties and Applications of Koszul Pairs

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.

math.KT↗

arXiv2016

On Koszulity of Finite Posets

We prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of $\Tor_n^A(R,R)$. As an application of these characterizations we investigate the Koszulity of the incidence rings for finite graded posets. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets whose incidence rings are Koszul). Specific examples of Koszul posets are included.

math.KT↗

arXiv2016

Koszul pairs and applications

Let $R$ be a semisimple ring. A pair $(A,C)$ is called almost-Koszul if $A$ is a connected graded $R$-ring and $C$ is a compatible connected graded $R$-coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation $(A,C)$ is said to be Koszul. One proves that a connected $R$-ring $A$ is Koszul if and only if there is a connected $R$-coring $C$ such that $(A,C)$ is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fröberg Theorem, are discussed in the last part of the paper.

math.KT↗

arXiv2011

On de Rham Cohomology of Linear Categories

We define the Chern map from the Grothendieck group of a linear category C to the de Rham cohomology of C with coefficients in a DG-category. In order to achieve our goal, we define the notion of connection on a C-module, and we show that the trace of the curvature of a connection is a de Rham cocycle, whose cohomology class does not depend on the choice of the connection.

math.KT↗

arXiv2011

Factorizable enriched categories and applications

We define the twisted tensor product of two enriched categories, which generalizes various sorts of `products' of algebraic structures, including the bicrossed product of groups, the twisted tensor product of (co)algebras and the double cross product of bialgebras. The key ingredient in the definition is the notion of simple twisting systems between two enriched categories. To give examples of simple twisted tensor products we introduce matched pairs of enriched categories. Several other examples related to ordinary categories, posets and groupoids are also discussed.

math.CT↗

arXiv2009

On the classification of twisting maps between $K^n$ and $K^m$

We define the notion of admissible pair for an algebra $A$, consisting on a couple $(Γ,R)$, where $Γ$ is a quiver and $R$ a unital, splitted and factorizable representation of $Γ$, and prove that the set of admissible pairs for $A$ is in one to one correspondence with the points of the variety of twisting maps $\mathcal{T}_A^n:=\mathcal{T}(K^n,A)$. We describe all these representations in the case $A=K^m$.

math.RA↗