A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Constructs Poisson structure on Banach Lie algebroid predual.
First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.
Study first-order locally convex Lie algebroids in Bastiani calculus.
We study a few basic properties of Banach-Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach-Lie groupoid sheds fresh light on earlier research on some infinite-dimensional mani…
This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. T…
Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.
We define the notion of strong projective limit of Banach Lie algebroids. We study the associated structures of Fréchet bundles and the compatibility with the different morphisms. This kind of structure seems to be a convenient framework for various situations.
It is shown that the Euler-Lagrange equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of reduction and the relation with Lagrange multiplier method are also studied.
Under appropriate assumptions, we generalize the concept of linear almost Poisson struc- tures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolution…
Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
The paper defines a new Lie groupoid concept for infinite dimensions.
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
Differential structure on partial isometries over Grassmannian constructed.
Constructing -Lie algebroids via connections
The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.
New structure on unitary group of Hilbert space.
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (…
Lie algebroids are like infinitesimal Lie groupoids.
New invariant real rank identifies constant real Lie algebroids.
This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.
Defines connections on parabolic vector bundles for Lie algebroids.
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
Homotopy invariance proven for twisted Lie algebroid cohomologies.
We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, …
We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obta…
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this…
Integrates transitive Lie algebroids to Lie groupoids, explaining obstructions.
This work explores higher-order algebroids via vector bundle comorphisms.
A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…
In this paper, first we give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its…
New algebraic structures for Lie 2-algebroids and their connections.
The study integrates Banach manifolds into H-manifolds, integrating Lie algebras into H-groups.
A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length …
Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.
This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs …
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in…
This thesis bridges Lie theory and sketch theory using tangent categories.
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.
Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of mo…
Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB - algebroids. In this p…
Almost Lie algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural extension of the bundle and the new anchor allows a Lie algebroid bracket. A cohomo…
Holomorphic Lie algebroid connections on Riemann surfaces are characterized.
The paper presents the geometry of Lie algebroids and its applications to optimal control. The first part deals with the theory of Lie algebroids, connections on Lie algebroids and dynamical systems defined on Lie algebroids (mainly Lagrangian and Hamiltonian systems). In the second part we use the framework of Lie alg…
LA-Courant algebroids link double Lie bialgebroids via Manin triples.
We will prove that the generalized Lie algebroid is a distinguished example by Lie algebroid. The generality of it with respect to the Lie algebroid is similar with the generality of the pull-back vector bundle with respect to the vector bundle. Next, we will prove that the proof of Theorem 3.1 from [15] is a misconcep…
Extends Lie algebroids by Lie algebroids with specific conditions.