Quotients of double vector bundles and multigraded 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>

The Weil Algebra of a Double Lie Algebroid

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 the concept of a filtered bundle

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.