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Simone Gutt

Publications and source records attributed to Simone Gutt.

At least 19 recordsLinked to original sources
arXiv2022

Almost complex structures, transverse complex structures, and transverse Dolbeault cohomology

We define a transverse Dolbeault cohomology associated to any almost complex structure $j$ on a smooth manifold $M$. This we do by extending the notion of transverse complex structure and by introducing a natural j-stable involutive limit distribution with such a transverse complex structure. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of (M,j) introduced by Cirici and Wilson, showing that the (p,0) cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure.

math.DG↗
arXiv2022

Some pseudo-K\"ahler Einstein $4$-symmetric spaces with a "twin" special almost complex structure

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-K\"ahler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.

math.DG↗
arXiv2020

On Twistor Almost Complex Structures

In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ defined on the twistor space $J(M,g)$ of an even-dimensional manifold $M$ with additional structures $g$ and $\nabla$ a $g$-connection. We also look at the question of the compatibility of $J^{\pm}_\nabla$ with a natural closed $2$-form $\omega^{J(M,g,\nabla)}$ defined on $J(M,g)$. For $(M,g)$ we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection $\nabla$. In all cases $J(M,g)$ is a bundle of complex structures on the tangent spaces of $M$ compatible with $g$ and we denote by $\pi \colon J(M,g) \longrightarrow M$ the bundle projection. In the case $M$ is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.

math.DG↗
arXiv2019

Group Actions in Deformation Quantisation

This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -- with intersection -- the introduction to formal deformation quantization and group actions, corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are implemented by a moment map and we give indications to build a corresponding quantum moment map. Reduction is a construction in classical mechanics with symmetries which allows to reduce the dimension of the manifold; we describe one of the various quantum analogues which have been considered in the framework of formal deformation quantization. We end up by some considerations about convergence of star products.

math.SG↗
arXiv2018

L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras

We study the L-infinity-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras generated by a vector space of dimension at least 2. We show that for these examples formality in Kontsevich's sense does NOT hold, although some of them allow unconditioned deformability. We compute the L-infinity-structure on the cohomology given by homotopy transfer in certain cases.

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

A possible symplectic framework for Radon-type transforms

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in K\"ahlerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type tranforms, associating to a smooth compactly supported function on a homogeneous manifold $M$, a function on a homogeneous space $N$ of totally geodesic submanifolds of $M$, and vice versa. We describe some spaces $M$ and $N$ in such Radon-type duality with $M$ a model of symplectic symmetric space with Ricci-type canonical connection and $N$ an orbit of totally geodesic symplectic submanifolds.

math.SG↗
arXiv2013

On Mpc-structures and Symplectic Dirac Operators

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group Mpc and classify G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space.

math.SG↗
arXiv2011

Symplectic Dirac Operators and Mpc-structures

Given a symplectic manifold $(M,\omega)$ admitting a metaplectic structure, and choosing a positive $\omega$-compatible almost complex structure $J$ and a linear connection $\nabla$ preserving $\omega$ and $J$, Katharina and Lutz Habermann have constructed two Dirac operators $D$ and ${\wt{D}}$ acting on sections of a bundle of symplectic spinors. They have shown that the commutator $[ D, {\wt{D}}]$ is an elliptic operator preserving an infinite number of finite dimensional subbundles. We extend the construction of symplectic Dirac operators to any symplectic manifold, through the use of $\Mpc$ structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in $H^2(M,\Z)$. For any $\Mpc$ structure, choosing $J$ and a linear connection $\nabla$ as before, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator $\mathcal{P}$ is elliptic. Using the Fock description of the spinor space allows the definition of a notion of degree and the construction of a dense family of finite dimensional subbundles; the operator $\mathcal{P}$ stabilizes the sections of each of those.

math.SG↗
arXiv2010

Transitive Subgroups of Transvections Acting on Some Symplectic Symmetric Spaces of Ricci Type

Symmetric symplectic spaces of Ricci type are a class of symmetric symplectic spaces which can be entirely described by reduction of certain quadratic Hamiltonian systems in a symplectic vector space. We determine, in a large number of cases, if such a space admits a subgroup of its transvection group acting simply transitively. We observe that the simply transitive subgroups obtained are one dimensional extensions of the Heisenberg group.

math.SG↗
arXiv2009

Involutions and Representations for Reduced Quantum Algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space M_red. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow [BHW] (with a simplified approach) and build a star product *_red on M_red from a strongly invariant star product * on M. The new questions which are addressed in this paper concern the existence of natural *-involutions on the reduced quantum algebra and the representation theory for such a reduced *-algebra. We assume that * is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C = J^{-1}(0), with some equivariance property, defines a *-involution for *_red on the reduced space. Looking into the question whether the corresponding *-involution is the complex conjugation (which is a *-involution in the Marsden-Weinstein context) yields a new notion of quantized unimodular class. We introduce a left *-submodule and a right *_red-submodule C^\infty_cf(C)[[lambda]] of C^\infty(C)[[lambda]]; we define on it a C^\infty(M_red)[[lambda]]-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C^\infty(M_red)[[lambda]] and the finite rank operators on C^\inftycf(C)[[lambda]]. The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate *-representations of (C^\infty(M_red)[[lambda]], *_red) on pre-Hilbert right D-modules to those of (C^\infty(M)[[lambda]], *), for any auxiliary coefficient *-algebra D over \mathbb{C}[[lambda]].

math.QA↗
arXiv2008

Extrinsic symplectic symmetric spaces

We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural star-quantization on a class of examples.

math.SG↗
arXiv2008

Universal star products

One defines the notion of universal deformation quantization: given any manifold $M$, any Poisson structure $¶$ on $M$ and any torsionfree linear connection $\nabla$ on $M$, a universal deformation quantization associates to this data a star product on $(M,¶)$ given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor $¶$, the curvature tensor $R$ and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poisson cohomology.

math.SG↗
arXiv2007

Symplectic Connections of Ricci Type and Star Products

In this article we relate the construction of Ricci type symplectic connections by reduction to the construction of star product by reduction yielding rather explicit descriptions for the star product on the reduced space.

math.QA↗
arXiv2006

Symplectic connections

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to noncommutative symplectic symmetric spaces.

math.SG↗
arXiv2005

Reduction,Induction and Ricci flat symplectic connections

In this paper we present a construction of Ricci-flat connections through an induction procedure. Given a symplectic manifold $(M,ω)$ of dimension $2n$, we define induction as a way to construct a symplectic manifold $(P,μ)$ of dimension $2n+2$. Given any symplectic connection $\nabla$ on $(M,ω)$, we define an induced connection $\nabla^P$ which is a Ricci-flat symplectic connection on $(P,μ)$.

math.SG↗
arXiv2003

Construction of Ricci-type connections by reduction and induction

Given the Euclidean space $\R^{2n+2}$ endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold of dimension greater than 2 endowed with a symplectic connection of Ricci-type is locally given by a local version of such a reduction. We also consider the reverse of this reduction procedure, an induction procedure: we construct globally on a symplectic manifold endowed with a connection of Ricci-type $(M,ω,\nabla)$ a circle or a line bundle which embeds in a flat symplectic manifold $(P,μ,\nabla^1)$ as the zero set of a function whose third covariant derivative vanishes, in such a way that $(M,ω,\nabla)$ is obtained by reduction from $(P,μ,\nabla^1)$. We further develop the particular case of symmetric symplectic manifolds with Ricci-type connections.

math.DG↗