Searcharxiv⌕ Search

arXiv subjects

Jonas Schnitzer

Publications and source records attributed to Jonas Schnitzer.

18 recordsLinked to original sources
arXiv2026

On the Quantization-Dequantization Correspondence for (co)Poisson Hopf Algebras

In this paper, we construct a functorial quantization of (co)Poisson Hopf algebras within a broad categorical framework. We further introduce categories naturally associated with (co)Poisson Hopf algebras, namely Drinfeld-Yetter modules. These categories provide a canonical setting in which we define explicit dequantization functors that are inverse to the quantization functors. Using this framework, we also establish functorial (de)quantization results for the corresponding module categories. Finally, we recover the classical results of Etingof and Kazhdan as special cases of our construction and discuss applications to deformation quantization \`a la Tamarkin.

math.QA↗
arXiv2025

Homotopies for Lagrangian field theory

Consider the variational bicomplex for $\mathcal{E}$ the space of sections of a graded, affine bundle. Local functionals $\mathcal{F}$ are defined as an equivalence class of density-valued functionals, which represent Lagrangian densities. A choice of a $k$-symplectic local form $\omega$ on $\mathcal{E}$ induces a Lie$[k]$ algebra structure on (Hamiltonian) local functionals $(\mathcal{F}_{\mathrm{ham}},\{\cdot,\cdot\}_{\mathrm{ham}})$. For any $\omega$ and any choice of a cohomological vector field $Q$ compatible with $\omega$, we build three explicit $L_\infty$ algebras on a resolution of $\mathcal{F}_{\mathrm{ham}}$, which are all $L_\infty$ quasi-isomorphic to a dgL$[k]$a $(\mathcal{F}_{\mathrm{ham}},d_{\mathrm{ham}},\{\cdot,\cdot\}_{\mathrm{ham}})$. In particular, one of our equivalent $L_\infty$ algebras is a dgL$[k]$ algebra. In the case $k=-1$, this provides an explicit lift of the standard Batalin--Vilkovisky framework to local forms enriched by the $L_\infty$ structure, in terms of local homotopies, which interprets the modified classical master equation as a Maurer--Cartan equation for the distinguished dgL$[k]$a we construct. We conjecture that the data of a lift to local forms of a BV theory contains a homotopy moment map on the cohomology of the Koszul complex of the underlying Lagrangian field theory.

math-ph↗
arXiv2025

Classification and Reduction of Homogeneous Star Products

We present a classification of homogeneous star products on duals of Lie algebroids in terms of the second Lie algebroid cohomology. Moreover, we extend this classification to projectable star products, i.e., to quantizations compatible with (coisotropic) reduction. This implies that quantization commutes with reduction in the considered setting.

math.QA↗
arXiv2025

Quantization of infinitesimal braidings and pre-Cartier quasi-bialgebras

In this paper we extend Cartier's deformation theorem of braided monoidal categories admitting an infinitesimal braiding to the non-symmetric case. The algebraic counterpart of these categories is the notion of a pre-Cartier quasi-bialgebra, which extends the well-known notion of quasitriangular quasi-bialgebra given by Drinfeld. Our result implies that one can quantize the infinitesimal $\mathcal{R}$-matrix of any Cartier quasi-bialgebra. We further discuss the emerging concepts of infinitesimal quantum Yang-Baxter equation and Cartier ring, the latter containing braid groups with additional generators that correspond to infinitesimal braidings. Explicit deformations of the representation categories of the gauge deformed quasitriangular quasi-bialgebras $E(n)$ are provided.

math.QA↗
arXiv2024

Global Homotopies for Differential Hochschild Cohomologies

We construct explicit global homotopies for differential Hochschild cochains in differential geometry, thereby upgrading the classical Hochschild-Kostant-Rosenberg map to a deformation retract. Our approach combines two key techniques: a symbol calculus from differential geometry and a coalgebraic version of the van Est theorem. To demonstrate its effectiveness, we develop deformation retracts in several related settings, including principal bundles and invariant contexts. As a byproduct, we recover the classical Hochschild-Kostant-Rosenberg theorem and compute previously inaccessible Hochschild cohomologies.

math.DG↗
arXiv2024

Deformations of Lagrangian $NQ$-submanifolds

In this paper we prove graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem in order to study the deformation theory of Lagrangian $NQ$-submanifolds of degree $n$ symplectic $NQ$-manifolds. Using Weinstein's Lagrangian tubular neighbourhood Theorem, we attach to every Lagrangian $NQ$-submanifold an $L_\infty$-algebra, which controls its deformation theory. The main examples are coisotropic submanifolds of Poisson manifolds and (higher) Dirac structures with support in (higher) Courant algebroids.

math.SG↗
arXiv2022

An Introduction to $L_\infty$-Algebras and their Homotopy Theory

In this review we give a detailed introduction to the theory of (curved) $L_\infty$-algebras and $L_\infty$-morphisms. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of $L_\infty$-algebras and $L_\infty$-modules. In particular, one can interpret $L_\infty$-morphisms and morphisms of $L_\infty$-modules as Maurer-Cartan elements in certain $L_\infty$-algebras, and we show that twisting the morphisms with equivalent Maurer-Cartan elements yields homotopic morphisms.

math.QA↗
arXiv2022

The Strong Homotopy Structure of BRST Reduction

In this paper we propose a reduction scheme for polydifferential operators phrased in terms of $L_\infty$-morphisms. The desired reduction $L_\infty$-morphism has been obtained by applying an explicit version of the homotopy transfer theorem. Finally, we prove that the reduced star product induced by this reduction $L_\infty$-morphism and the reduced star product obtained via the formal Koszul complex are equivalent.

math.QA↗
arXiv2021

Weak Dual Pairs in Dirac-Jacobi Geometry

Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. 1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac-Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac-Jacobi transversals. 2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).

math.DG↗
arXiv2021

The Homotopy Class of twisted $L_\infty$-morphisms

The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between $L_\infty$-morphisms that are twisted with gauge equivalent Maurer-Cartan elements.

math.QA↗
arXiv2020

The Strong Homotopy Structure of Poisson Reduction

In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_\infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up a suitable deformation retract of DGLA's. We first obtained an explicit formula for the $L_\infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_\infty$-morphism. Finally, we perfom a comparison with other reduction procedures.

math.QA↗
arXiv2020

A proof of Tsygan's formality conjecture for Hamiltonian actions

In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivariant multivector fields extends to an $L_\infty$-quasi-isomorphism. We construct this $L_\infty$-quasi-isomorphism using the $G$-invariant formality constructed by Dolgushev. This result has immediate consequences in deformation quantization, since it allows to obtain a quantum moment map from a classical momentum map with respect to a $G$-invariant Poisson structure.

math.QA↗
arXiv2019

Characteristic (Fedosov-)class of a twist constructed by Drinfel'd

In a seminal paper Drinfel'd explained how to associate to every classical r-matrix for a Lie algebra $\lie g$ a twisting element based on $\mathcal{U}(\lie g)[[\hbar]]$, or equivalently a left invariant star product of the corresponding symplectic structure $ω$ on the 1-connected Lie group G of g. In a recent paper, the authors solve the same problem by means of Fedosov quantization. In this short note we provide a connection between the two constructions by computing the characteristic (Fedosov) class of the twist constructed by Drinfel'd and proving that it is the trivial class given by $ \frac{[ω]}{\hbar}$.

math.QA↗
arXiv2019

The Local Structure of Generalized Contact Bundles

Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

math.DG↗
arXiv2019

Normal Forms for Dirac-Jacobi bundles and Splitting Theorems for Jacobi Structures

The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alternative proof of the splitting theorem of homogeneous Poisson structures.

math.DG↗
arXiv2018

Regular Jacobi Structures and Generalized Contact Bundles

A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5-dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure.

math.DG↗
arXiv2017

A Universal Construction of Universal Deformation Formulas, Drinfel'd Twists and their Positivity

In this paper we provide an explicit construction of star products on U(g)-module algebras by using the Fedosov approach. This construction allows us to give a constructive proof to Drinfel'd theorem and to obtain a concrete formula for Drinfel'd twist. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley-Eilenberg cohomology of the Lie algebra g. Finally, we show that for Lie algebras with Kähler structure we obtain a strongly positive universal deformation of *-algebras by using a Wick-type deformation. This results in a positive Drinfel'd twist.

math.QA↗