Λ-modules and holomorphic Lie algebroid connections

2012 ◽  
Vol 10 (4) ◽  
pp. 1422-1441 ◽  
Author(s):  
Pietro Tortella
Keyword(s):  
Lie Algebroid

scholarly journals On local integration of Lie brackets

2020 ◽  
Vol 2020 (760) ◽  
pp. 267-293 ◽  
Author(s):  
Alejandro Cabrera ◽  
Ioan Mărcuţ ◽  
María Amelia Salazar
Keyword(s):  
Vector Field ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Theory ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Local Integration ◽  
Lie Brackets ◽  
Complete Account

AbstractWe give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

scholarly journals On the gauge features of gravity on a Lie algebroid structure

2014 ◽  
Vol 55 (3) ◽  
pp. 032502 ◽  
Author(s):  
S. Fabi ◽  
B. Harms ◽  
S. Hou
Keyword(s):  
Lie Algebroid

scholarly journals LIE ALGEBROIDS AS SPACES

2018 ◽  
Vol 19 (2) ◽  
pp. 487-535 ◽  
Author(s):  
Ryan Grady ◽  
Owen Gwilliam
Keyword(s):  
Tangent Bundle ◽  
Symplectic Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Faithful Functor ◽  
Derived Geometry

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

scholarly journals The Weil Algebra of a Double Lie Algebroid

2020 ◽  
Author(s):  
Eckhard Meinrenken ◽  
Jeffrey Pike
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Lie Algebroid ◽  
Weil Algebra ◽  
Algebra Of Functions ◽  
Double Vector Bundles ◽  
Definition Of ◽  
Deformation Complex ◽  
Double Vector Bundle ◽  
Gerstenhaber Bracket

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.

scholarly journals (Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations

2014 ◽  
Vol 25 (12) ◽  
pp. 1450116 ◽  
Author(s):  
Constantin M. Arcuş ◽  
Esmaeil Peyghan
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Ricci Tensor ◽  
Tensor Field ◽  
General Framework ◽  
Bianchi Type ◽  
Einstein Equations ◽  
Lie Algebroid ◽  
Linear Connections ◽  
Kaluza Klein

Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

scholarly journals A Lie algebroid framework for non-holonomic systems

2005 ◽  
Vol 38 (5) ◽  
pp. 1097-1111 ◽  
Author(s):  
Tom Mestdag ◽  
Bavo Langerock
Keyword(s):  
Lie Algebroid ◽  
Holonomic Systems

scholarly journals Lie Algebroid Yang–Mills with matter fields

2009 ◽  
Vol 59 (12) ◽  
pp. 1613-1623 ◽  
Author(s):  
C. Mayer ◽  
T. Strobl
Keyword(s):  
Lie Algebroid ◽  
Yang Mills ◽  
Matter Fields

scholarly journals GEOMETRICAL MECHANICS ON ALGEBROIDS

2006 ◽  
Vol 03 (03) ◽  
pp. 559-575 ◽  
Author(s):  
KATARZYNA GRABOWSKA ◽  
PAWEŁ URBAŃSKI ◽  
JANUSZ GRABOWSKI
Keyword(s):  
Vector Bundle ◽  
Lie Algebroids ◽  
Legendre Transform ◽  
Analytical Mechanics ◽  
Lie Algebroid ◽  
Noether Theorem ◽  
Geometric Framework ◽  
Hamiltonian Functions

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.

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