Weil Algebra Research Articles

Differential Forms and Cohomology on Weil Bundles

Let M be a smooth manifold and A a Weil algebra. We discuss the differential forms in the Weil bundles M A , π , M , and we established a link between differential forms in M A and M as well as their cohomology. We also discuss the cohomology in.

Holonomy Pseudogroups of Manifolds over Weil Algebras

Holonomy Pseudogroups of Manifolds over Weil Algebras

Modules and representations up to homotopy of Lie n-algebroids

This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general nin {mathbb {N}}. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.

Prolongations of G-structures related to Weil bundles and some applications

Let M be a smooth manifold of dimension m ≥ 1 and P be a G-structure on M , where G is a Lie subgroup of linear group GL(m). In [8], it has been defined the prolongations of G-structures related to tangent functor of higher order and some properties have been established. The aim of this paper is to generalize these prolongations to a Weil bundles. More precisely, we study the prolongations of G-structures on a manifold M , to its Weil bundle TAM (A is a Weil algebra) and we establish some properties. In particular, we characterize the canonical tensor fields induced by the A-prolongation of some classical G-structures.

Momentum section on Courant algebroid and constrained Hamiltonian mechanics

We propose a generalization of the momentum map on a symplectic manifold with a Lie algebra action to a Courant algebroid structure. The theory of a momentum section on a Lie algebroid is generalized to the theory compatible with a Courant algebroid. As an example, we identify the momentum section in a constrained Hamiltonian mechanics with Courant algebroid symmetry. Moreover, we construct cohomological formulations by considering the BFV and BV formalism of this Hamiltonian system. The Weil algebra for this structure is constructed.

Towards an M5‐Brane Model II: Metric String Structures

AbstractIn this paper, we develop the mathematical formulation of metric string structures. These play a crucial role in the formulation of certain six‐dimensional superconformal field theories and we believe that they also underlie potential future formulations of the (2,0)‐theory. We show that the connections on non‐abelian gerbes usually introduced in the literature are problematic in that they are locally gauge equivalent to connections on abelian gerbes. Connections on string structures form an exception and we introduce the general concept of an adjusted Weil algebra leading to potentially interacting connections on higher principal bundles. Considering a special case, we derive the metric extension of string structures and the corresponding adjusted Weil algebra. The latter lead to connections that were previously constructed by hand in the context of gauged supergravities. We also explain how the Leibniz algebras induced by an embedding tensor in gauged supergravities fit into our picture.

Functorial Differential Spaces and the Infinitesimal Structure of Space-Time

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding a ring of smooth functions into the category of loci. This permits us to define a category of functorial differential spaces. By suitably choosing various algebras as "stages" in this category, one obtains various classes of differential spaces, both known from the literature and many so far unknown. In particular, if one chooses a Weil algebra, infinitesimals are produced. We study the case with some Weil algebra which allows us to fully develop the corresponding differential geometry with infinitesimals. To test the behavior of infinitesimals, we construct a simplified RWFL cosmological model. As it should be expected, infinitesimals remain latent during the entire macroscopic evolution (regarded backwards in time), and come into play only when the universe attains infinitesimal dimensions. Then they penetrate into the structure of the initial singularity.

The Weil Algebra of a Double Lie Algebroid

Abstract Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

Twisted Weil Algebras for the String Lie 2‐Algebra

AbstractIn this article, we give a concise summary of ‐algebras viewed in terms of Chevalley–Eilenberg algebras, Weil algebras and invariant polynomials and their use in defining connections in higher gauge theory. Using this, we discuss the example of the string Lie 2‐algebra in both the skeletal and the loop model. In both cases, we show how to arrive at the twisted Weil algebras which were used in [1] to construct a non‐abelian self‐dual string soliton, see also [2–4].

Lifting double linear vector fields to Weil like functors on double vector bundles

AbstractThe complete description is given of the product preserving gauge bundle functors F on the category of double vector bundles in terms of the ‐bilinear maps , where are Weil algebras and are finite dimensional (over ) ‐modules. Then the so‐called “iteration” problem is solved (i.e. by means of and is computed). That any two product preserving gauge bundle functors on commute is derived. All gauge natural operators lifting double linear vector fields to product preserving gauge bundle functors F are completely described.

Some functorial prolongations of general connections

We consider the problem of prolongating general connections on arbitrary fibered manifolds with respect to a product preserving bundle functor. Our main tools are the theory of Weil algebras and the Frolicher-Nijenhuis bracket.

Weil diffeology I: Classical differential geometry

Topos theory is a category-theoretic axiomatization of set theory. Model categories are a category-theoretical framework for abstract homotopy theory. They are complete and cocomplete categories endowed with three classes of morphisms (called fibrations, cofibrations and equivalences) satisfying certain axioms. Functors from the category of Weil algebras to the category of sets are called Weil spaces by Wolfgang Bertram and form the Weil topos} after Eduardo J. Dubuc. The Weil topos is endowed intrinsically with the Dubuc functor, a functor from a larger category containing Weil algebras to the Weil topos standing for the incarnation of each algebraic entity of the category in the Weil topos. The Weil functor and the canonical ring object are to be defined in terms of the Dubuc functor. The principal object in this paper is to present a category-theoretical axiomatization of the Weil topos with the Dubuc functor intended to be an adequate framework for axiomatic classical differential geometry and hopefully comparable with model categories. We will give an appropriate formulation and a rather complete proof of a generalization of the familiar and desired fact that the tangent space of a microlinear Weil space is a module over the canonical ring object.

Duality for Graded Manifolds

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of $N$-manifolds of degree 2, announced in the literature.

English

Let M be an m-dimensional manifold and A = Dkr/I = R⊕NA a Weil algebra of height r. We prove that any A-covelocity TxAf ∈ TxA*M, x ∈ M is determined by its values over arbitrary max{widthA,m} regular and under the first jet projection linearly independent elements of TxAM. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result TA*M ~ Tr*M without coordinate computations, which improves and generalizes the partial result obtained in Tomas (2009) from m ≥ k to all cases of m.

Axiomatic Differential Geometry III-3-Its Landscape-Chapter 3: The Old Kingdom of Differential Geometers

The principal objective of this paper is to study the relationship between the old kingdom of differential geometry (the category of smooth manifolds) and its new kingdom (the category of functors on the category of Weil algebras to some smooth category). It is shown that the canonical embedding of the old kingdom into the new kingdom preserves Weil functors.

Higher Order Frames and Material Symmetries

The paper is focused on mathematical approach to materials with microstructure, the special case of them are Cosserat media. Frame bundles are described with respect to their role in continuum mechanics and the structure jet groups. General (r-th order) holonomic, nonholonomic and semiholonomic microstructure configuration are introduces and the algebraic approach using Weil algebras is presented, too.

A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

Using a K-theory point of view, Bott related the Atiyah–Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.

Projective families of Dirac operators on a Banach Lie groupoid

We introduce a Banach Lie group G of unitary operators subject to a natural trace condition. We compute the homotopy groups of G , describe its cohomology and construct an S^1 -central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid G \ltimes \frak k , where \frak k denotes the Hilbert space of self-adjoint Hilbert–Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator \mathbb D in a suitable completion of the quantum Weil algebra \mathcal{U}(\frak{g}) \otimes Cl(\frak{k}) , which is subsequently extended to a projective family of self-adjoint operators \mathbb D_A on G \ltimes \frak k . While the kernel of \mathbb D_A is infinite-dimensional, we show that there is still a notion of finite reducibility at every point, which suggests a generalized definition of twisted K-theory for action Lie groupoids.

Two general examples of Weil algebras having the group of automorphisms connected

The paper is devoted to the algebraic background of Weil contact elements: in particular, we investigate an eventual absence of orientation reversing maps by the description of Weil algebras having their group of automorphisms with only one connected component.

On the Van Est homomorphism for Lie groupoids

The Van Est homomorphism for a Lie groupoid G \rightrightarrows M , as introduced by Weinstein–Xu, is a cochain map from the complex C^\infty (BG) of groupoid cochains to the Chevalley–Eilenberg complex \mathsf C(A) of the Lie algebroid A of G . It was generalized by Weinstein, Mehta, and Abad-Crainic to a morphism from the Bott–Shulman–Stasheff complex \Omega (BG) to a (suitably defined) Weil algebra \mathsf W(A) . In this paper, we will give an approach to the Van Est map in terms of the Perturbation Lemma of homological algebra. This approach is used to establish the basic properties of the Van Est map. In particular, we show that on the normalized subcomplex, the Van Est map restricts to an algebra morphism.