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

2019 ◽  
Vol 292 (9) ◽  
pp. 2092-2100
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Linear Vector ◽  
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.

scholarly journals Convex topological algebras via linear vector fields and Cuntz algebras

2021 ◽  
Vol 225 (3) ◽  
pp. 106535
Author(s):  
Wolfgang Bock ◽  
Vyacheslav Futorny ◽  
Mikhail Neklyudov
Keyword(s):  
Vector Fields ◽  
Topological Algebras ◽  
Cuntz Algebras ◽  
Linear Vector

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

Limit cycles of linear vector fields on manifolds

Nonlinearity ◽  
2016 ◽  
Vol 29 (10) ◽  
pp. 3120-3131
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Linear Vector

HORIZONTAL LAPLACIAN ON TANGENT BUNDLE OF FINSLER MANIFOLD WITH g-NATURAL METRIC

2012 ◽  
Vol 09 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI ◽  
CHUNPING ZHONG
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Vector Bundles ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Order Differential Operator ◽  
Finsler Manifolds ◽  
Scalar And Vector Fields ◽  
Matsumoto Metric ◽  
Weitzenböck Formula

Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenböck formula of the horizontal Laplacian associated to Ga,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to Ga,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to Ga,b are investigated.

Linear Vector Fields on ~ G   k   (R n )

1980 ◽  
Vol 80 (4) ◽  
pp. 673 ◽  
Author(s):  
Maria Luiza Leite ◽  
Isabel Dotti de Miatello
Keyword(s):  
Vector Fields ◽  
Linear Vector
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