Tag - A-infinity structures

David Fernández: Non-commutative Poisson geometry and pre-Calabi-Yau algebras

2nd December, 2024

In order to define suitable non-commutative Poisson structures, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras. Furthermore, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras; certain cyclic A∞-algebras which can be seen as non-commutative versions of shifted Poisson manifolds. In this talk, I will present an extension of the Iyudu-Kontsevich correspondence to the differential graded setting. I will also explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras.

Paolo Stellari: Comparing the homotopy categories of dg categories and of A∞-categories

9th May, 2024

In this talk, we show that the homotopy category of (small) dg categories and the homotopy category of A∞-categories are equivalent (even from a higher categorical viewpoint). We will discuss several issues related to the various notions of unity and provide several applications. The main ones are about the uniqueness of enhancements for triangulated categories and a full proof of a claim by Kontsevich and Keller concerning a description of the category of internal Homs for dg categories.

Marion Boucrot: The relation between A∞-morphisms and pre-Calabi-Yau morphisms

2nd May, 2023

Pre-Calabi-Yau algebras were introduced in the last decade by  M. Kontsevich, A. Takeda and Y. Vlassopoulos using the necklace bracket. This notion is equivalent to a cyclic A∞-algebra for the natural bilinear form in the finite-dimensional case. Moreover, W-K. Yeung showed that double Poisson DG structures provide an example of pre-Calabi-Yau structures. In 2020, D. Fernandez and E. Herscovich proved that given a morphism of double Poisson DG algebras from A to B, one can produce a cyclic A∞-algebra and A∞-morphisms between the latter and the cyclic A∞-algebras associated to A and B. I will explain how to generalize this result to pre-Calabi-Yau algebras by doing an explicit construction of a (cyclic) A∞-algebra and A∞-morphisms given a pre-Calabi-Yau morphism.

Laurent Côté: Action filtrations associated to smooth categorical compactifications

9th July, 2021

There is notion of a smooth categorical compactification of dg/A∞ categories: for example, a smooth compactification of algebraic varieties induces a smooth categorical compactification of the associated bounded dg categories of coherent sheaves. In symplectic topology, wrapped Fukaya categories of Weinstein manifolds admit smooth compactifications by partially wrapped Fukaya categories. The goal of this talk is to explain how to associate an "action filtration" to a smooth categorical compactifications, which is invariant (up to appropriate equivalence) under zig-zags of smooth compactifications. I will then discuss applications to symplectic topology and categorical dynamics.

Patrick Le Meur: Equivariant models for graded algebras and application to Calabi-Yau duality

16th July, 2020

Skew Calabi-Yau algebras are generalizations of Calabi-Yau algebras due to Reyes, Rogalski, and Zhang. Within the graded (associative and unital) algebras over a field k, they form the non-commutative analogues of the regular algebras. As a special feature, such an algebra A is equipped with its so-called Nakayama automorphism φ. The talk will present ongoing investigations on the presentations of these algebras by generators and relations taking into account their homological specificities. Such presentations are well-known for Calabi-Yau algebras (after Ginzburg, Bocklandt and van den Bergh) and also for Koszul skew Calabi-Yau algebras (after Bocklandt, Wemyss and Schedler). The general situation involves the interaction of the A∞-Yoneda algebra E(A) := ExtA(k,k) with the Nakayama automorphism φ, and also the A∞-Yoneda algebra E(A[x,φ]) of the Ore extension A[x,φ] of A by φ. More precisely, one is particularly intereseted in minimal models of these A∞-algebras. After having presented all these concepts, I will discuss the relationship between these minimal models as well as consequences in terms of presentations of A.