scholarly journals Complete lifting of double-linear semi-basic tangent valued forms to Weil like functors on double vector bundles

2021 ◽  
pp. 351-363
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Bundles ◽  
Double Vector Bundles

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.

On double vector bundles

2014 ◽  
Vol 30 (10) ◽  
pp. 1655-1673 ◽  
Author(s):  
Zhuo Chen ◽  
Zhang Ju Liu ◽  
Yun He Sheng
Keyword(s):  
Vector Bundles ◽  
Double Vector Bundles

scholarly journals On the concept of a filtered bundle

2017 ◽  
Vol 15 (01) ◽  
pp. 1850013 ◽  
Author(s):  
Andrew James Bruce ◽  
Katarzyna Grabowska ◽  
Janusz Grabowski
Keyword(s):  
Field Theory ◽  
Vector Bundles ◽  
Classical Field Theory ◽  
Classical Field ◽  
Geometric Mechanics ◽  
Jet Bundles ◽  
Local Coordinates ◽  
Polynomial Transformation ◽  
Double Vector Bundles ◽  
Partial Polarization

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined.

scholarly journals Quotients of double vector bundles and multigraded bundles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Eckhard Meinrenken
Keyword(s):  
Vector Bundles ◽  
Normal Bundles ◽  
Double Vector Bundles

<p style='text-indent:20px;'>We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.</p>

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, 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.

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