arXiv subjects

Stefan Waldmann

Publications and source records attributed to Stefan Waldmann.

At least 19 recordsLinked to original sources

arXiv2026

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↗

arXiv2026

Convergent Twist Deformations

This paper establishes a functorial framework for convergence of Drinfeld's Universal Deformation Formula (UDF) on spaces of analytic vectors. This is accomplished by matching the order of the latter with an equicontinuity condition on the Drinfeld twist underlying the deformation. Throughout, we work with representations of finite-dimensional Lie algebras by continuous linear mappings on locally convex spaces. This allows us to establish not only convergence of the formal power series, but the continuity of the deformed bilinear mappings as well as the entire holomorphic dependence on the deformation parameter $\hbar$. Finally, we demonstrate the effectiveness of our theory by applying it to the explicit Drinfeld twists constructed by Giaquinto and Zhang, where we establish both the equicontinuity condition and determine the corresponding spaces of analytic vectors for concrete representations. Thereby we answer a question posed by Giaquinto and Zhang whether a strict version of their formal twists is possible in the positive.

math.QA↗

arXiv2023

Strict Wick-type deformation quantization on Riemann surfaces: Rigidity and Obstructions

Let $X$ be a hyperbolic Riemann surface. We study a convergent Wick-type star product $\star_X$ on $X$ which is induced by the canonical convergent star product $\star_{\mathbb{D}}$ on the unit disk $\mathbb{D}$ via Uniformization Theory. While by construction, the resulting Fréchet algebras $(\mathcal{A}(X),\star_X)$ are strongly isomorphic for conformally equivalent Riemann surfaces, our work exhibits additional severe topological obstructions. In particular, we show that the Fréchet algebra $(\mathcal{A}(X),\star_X)$ degenerates if and only if the connectivity of $X$ is at least $3$, and $(\mathcal{A}(X),\star_X)$ is noncommutative if and only if $X$ is simply connected. We also explicitly determine the algebra $\mathcal{A}_X$ and the star product $\star_X$ for the intermediate case of doubly connected Riemann surfaces $X$. As a perhaps surprinsing result, we deduce that two such Fréchet algebras are strongly isomorphic if and only if either both Riemann surfaces are conformally equivalent to an (not neccesarily the same) annulus or both are conformally equivalent to a punctured disk.

math.CV↗

arXiv2023

Classical KMS Functionals and Phase Transitions in Poisson Geometry

We study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide an almost complete characterization of the convex cone of KMS measures.

math-ph↗

arXiv2021

Convergent Star Products on Cotangent Bundles of Lie Groups

For a connected real Lie group $G$ we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of $G$. This star product trivially converges on polynomial functions on $T^*G$ thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on $T^*G$, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter $\hbar$. This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on $G$ with an appropriate nuclear Fréchet algebra of functions on $\mathfrak{g}^*$. The passage to the Weyl-ordered star product, i.e. the Gutt star product on $T^*G$, is shown to be preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on $\hbar$.

math.QA↗

arXiv2020

A Serre-Swan Theorem for Coisotropic Algebras

Coisotropic algebras are used to formalize coisotropic reduction in Poisson geometry as well as in deformation quantization and find applications in various other fields as well. In this paper we prove a Serre-Swan Theorem relating the regular projective modules over the coisotropic algebra built out of a manifold $M$, a submanifold $C$ and an integrable smooth distribution $D \subseteq TC$ with vector bundles over this geometric situation and show an equivalence of categories for the case of a simple distribution.

math.DG↗

arXiv2020

Deformation and Hochschild Cohomology of Coisotropic Algebras

Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

math.QA↗

arXiv2020

Morita equivalence of formal Poisson structures

We extend the notion of Morita equivalence of Poisson manifolds to the setting of {\em formal} Poisson structures, i.e., formal power series of bivector fields $π=π_0 + λπ_1 +\cdots$ satisfying the Poisson integrability condition $[π,π]=0$. Our main result gives a complete description of Morita equivalent formal Poisson structures deforming the zero structure ($π_0=0$) in terms of $B$-field transformations, relying on a general study of formal deformations of Poisson morphisms and dual pairs. Combined with previous work on Morita equivalence of star products, our results link the notions of Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.

math.SG↗

arXiv2020

BRST Reduction of Quantum Algebras with $^*$-Involutions

In this paper we investigate the compatibility of the BRST reduction procedure with the Hermiticity of star products. First, we introduce the generalized notion of abstract BRST algebras with corresponding involutions. In this setting we define adjoint BRST differentials and as a consequence one gets new BRST quotients. Passing to the quantum BRST setting we show that for compact Lie groups the new quantum BRST quotient and the quantum BRST cohomology are isomorphic in zero degree implying that reduction is compatible with Hermiticity.

math.QA↗

arXiv2019

Coisotropic Triples, Reduction and Classical Limit

Coisotropic reduction from Poisson geometry and deformation quantization is cast into a general and unifying algebraic framework: we introduce the notion of coisotropic triples of algebras for which a reduction can be defined. This allows to construct also a notion of bimodules for such triples leading to bicategories of bimodules for which we have a reduction functor as well. Morita equivalence of coisotropic triples of algebras is defined as isomorphism in the ambient bicategory and characterized explicitly. Finally, we investigate the classical limit of coisotropic triples of algebras and their bimodules and show that classical limit commutes with reduction in the bicategory sense.

math.QA↗

arXiv2019

Convergence of Star Product: From Examples to a General Framework

We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincaré disc.

math.QA↗

arXiv2018

Comparison and Continuity of Wick-type Star Products on certain coadjoint orbits

In this paper we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev-Lachowska and Karabegov and we prove that they agree in general. In the case of the 2-sphere we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

math.QA↗

arXiv2018

A Convergent Star Product on the Poincaré Disc

On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra.

math.CV↗

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↗

arXiv2017

Convergent Star Products for Projective Limits of Hilbert Spaces

Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.

math.QA↗

arXiv2016

Convergence of the Gutt Star Product

In this work we consider the Gutt star product viewed as an associative deformation of the symmetric algebra S^\bullet(g) over a Lie algebra g and discuss its continuity properties: we establish a locally convex topology on S^\bullet(g) such that the Gutt star product becomes continuous. Here we have to assume a mild technical condition on g: it has to be an Asymptotic Estimate Lie algebra. This condition is e.g. fulfilled automatically for all finite-dimensional Lie algebras. The resulting completion of the symmetric algebra can be described explicitly and yields not only a locally convex algebra but also the Hopf algebra structure maps inherited from the universal enveloping algebra are continuous. We show that all Hopf algebra structure maps depend analytically on the deformation parameter. The construction enjoys good functorial properties.

math.QA↗

arXiv2016

Nuclear Group Algebras for Finitely Generated Groups

We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their duals are studied in detail and compared to other approaches. As application we discuss the convergence of the complete growth function introduced by Grigorchuk and Nagnibeda.

math.GR↗

arXiv2016

Obstructions for Twist Star Products

In this short note we point out that not every star product is induced by a Drinfel'd twist by showing that not every Poisson structure is induced by a classical $r$-matrix. Examples include the higher genus symplectic Pretzel surfaces and the symplectic sphere $\mathbb{S}^2$.

math.QA↗