arXiv subjects

Tristan Bozec

Publications and source records attributed to Tristan Bozec.

12 recordsLinked to original sources

arXiv2025

Functorial constructions related to double Poisson vertex algebras

For any double Poisson algebra, we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions, with the aim of comparing the different corresponding categories. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.

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arXiv2025

Koszul complexes and derived intersections

The aim of this article is to provide a complementary understanding to some results of the second author using the machinery of Koszul complexes, and to explain how this approach can provide a new description of projective derived intersections.

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arXiv2024

Topological field theories associated with Calabi-Yau categories

We construct symmetric monoidal higher categories of iterated Calabi-Yau cospans, that are noncommutative analogs of iterated lagrangian correspondences. We actually give a general (and functorial) procedure that applies to iterated nondegenerate cospans on certain comma categories. This allows us to factor the AKSZ fully extended TFT associated with the moduli of objects of a Calabi-Yau category (taking values in iterated lagrangian correspondences) through a fully extended TFT taking values in iterated Calabi-Yau cospans.

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arXiv2024

Calabi-Yau structures on (quasi-)bisymplectic algebras

We show that relative Calabi--Yau structures on noncommutative moment maps give rise to (quasi-)bisymplectic structures, as introduced by Crawley-Boevey-Etingof-Ginzburg (in the additive case) and Van den Bergh (in the multiplicative case). We prove along the way that the fusion process (a) corresponds to the composition of Calabi-Yau cospans with "pair-of-pants" ones, and (b) preserves the duality between non-degenerate double quasi-Poisson structures and quasi-bisymplectic structures. As an application we obtain that Van den Bergh's Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras coincide with the ones induced by the 2-Calabi-Yau structures on (dg-versions of) these algebras.

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arXiv2024

Relative critical loci and quiver moduli

In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.

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arXiv2023

Calabi-Yau structures for multiplicative preprojective algebras

In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.

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arXiv2021

MV Polytopes and Masures

We realize affine Mirkovi{\'c}-Vilonen polytopes using Littelmann's path model in the framework of masures. We are also able to read the decorations on the paths in the case of sl2.

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arXiv2021

Irreducible components of the global nilpotent cone

This paper gives a combinatorial description of the set of irreducible components of the semistable locus of the global nilpotent cone, in genus $\ge2$. The first main result of this paper states that the set of irreducible components of the global nilpotent cone is given by the very natural decomposition in twisted Jordan strata, which are smooth. Then we move on to the semistable locus and obtain purely combinatorial conditions on the twisted Jordan type to be semistable. The proof uses an analogous result obtained in the context of moduli stacks of chains, and shows that semistability can be tested on the most `simple' subsheaves - the ones built with iterated kernels and images. The proof is constructive and do not rely on the coprimality of the rank and degree, in particular we get that the attracting cells are irreducible in any case. The last section describes this set of semistable components in terms of integral polytopes.

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arXiv2017

Addendum to "Quivers with loops and perverse sheaves"

We prove a character formula for the Hopf algebra defined in arXiv:1401.5302 that generalizes quantum groups, as well as for the simple modules associated to dominant integral weights defined in arXiv:1403.0846.

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arXiv2016

Quivers with loops and Lagrangian subvarieties

In this article we define a generalization of Lusztig Lagrangian varieties in the case of arbitrary quivers, possibly carrying loops. As opposed to the Lagrangian varieties constructed by Lusztig, which consisted in nilpotent representations, we have to consider here slightly more general representations. That this is necessary is already clear from the Jordan quiver case. Our proof of the Lagrangian character is based on induction, but with non trivial first steps, consisting in the study of quivers with one vertex but possible loops. From our proof emerges a new combinatorial structure on the set of irreducible components, which is more general than the usual crystals, in that there are now more operators associated to a vertex with loops. We finally consider a convolution algebra of constructible functions on our varieties, and construct a family of constructible functions naturally attached to the irreducible components.

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arXiv2014

Quivers with loops and generalized crystals

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

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arXiv2014

Quivers with loops and perverse sheaves

In this article we consider the general definition of Lustig sheaves for arbitrary quivers, possibly carrying loops. We answer a conjecture raised by Lusztig asking if the more "simple" Lusztig perverse sheaves are enough to span the whole Grothendieck group usually built with these sheaves. Our proof is based on induction, with the help of restriction and induction functors, but with non trivial first steps, consisting in the study of quivers with one vertex but possible loops. We also need to consider regularity conditions on the support of our perverse sheaves to perform efficient restrictions at imaginary vertices. From our proof emerges a new combinatorial structure on our generalized canonical basis, which is more general than the usual crystals, in that there are now more operators associated to a vertex with loops. In a second part, we construct and study a Hopf algebra which generalizes the usual quantum groups. The geometric study previously made leads to a natural definition, which includes countably infinite sets of generators at imaginary roots, with higher order Serre relations and commutativity conditions imposed by the Jordan quiver case. We finally prove that the positive part of this algebra is isomorphic to our Grothendieck group, thanks to the study of a nondegenerate Hopf pairing.

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