{"id":5425,"date":"2018-10-11T09:20:26","date_gmt":"2018-10-11T09:20:26","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=5425"},"modified":"2018-10-11T09:20:26","modified_gmt":"2018-10-11T09:20:26","slug":"connections-adapted-to-graded-bundles","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2018\/10\/11\/connections-adapted-to-graded-bundles\/","title":{"rendered":"Connections adapted to graded bundles"},"content":{"rendered":"\n\n\n
\"board\"<\/td>\nIn a preprint Connections Adapted to Non-Negativley Graded Structures<\/a> I examine the notion of connections that respect the graded structure. Such connections are akin to linear connections on vector bundles.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Graded Bundles<\/strong>
\nGraded bundles are a particular \u2018species\u2019 of non-negatively graded the manifold that is very well behaved [1,2]. A graded bundle \\(F\\) is a fibre bundle<\/a> for which one can assign a weight of zero to the base coordinates and a non-zero integer weight to the fibre coordinates. Admissible changes of local coordinates respect this assignment of weight. The resulting structure is a polynomial bundle with the typical fibres being \\(\\mathbb{R}^n\\) (for some n). Note that the changes of coordinates for the fibre coordinates are not linear, but rather polynomial. We, in fact, have a series of affine fribrations<\/p>\n

\\(F := F_k \\longrightarrow F_{k-1} \\longrightarrow \\cdots \\longrightarrow F_1 \\longrightarrow F_0 =: M \\,,\\)<\/p>\n

where we have indicated the highest weight\/degree of the coordinates. Note that the arrow on the far right is a vector bundle. Examples of graded bundles include higher order tangent bundles<\/a> and vector bundles<\/a>. The ethos one can take is that graded bundles are \u2018non-linear\u2019 vector bundles, and so the question of connections that in some sense respect the graded structure is a natural one.<\/p>\n

Connections<\/strong>
\nThe notion of a connection in many different guises, such as a covariant derivative<\/a> or a horizontal distribution<\/a>, can be found throughout differential geometry. In physics, connections are central to the notion of gauge fields such as the electromagnetic field. Connections also play a role in geometric approaches to relativistic mechanics, Fedosov\u2019s approach to deformation quantisation, adiabatic evolution via the Berry phase, and so on.<\/p>\n

The initial approach that I take in the preprint is to generalise the notion of a Koszul connection<\/a>. I phrase this in terms of odd vector fields on a particular supermanifold<\/a> build from the graded bundle and a Lie algebroid (it is, up to a shift in parity, the fibre product of the Lie algebroid and the graded bundle).<\/p>\n

Lie Algebroids<\/strong>
\n Loosely, a Lie algebroid<\/a> can be viewed as a mixture of tangent bundles and Lie algebras [3]. A little more carefully, a Lie algebroid is a vector bundle<\/p>\n

\\(\\pi : A \\longrightarrow M\\)<\/p>\n

that comes equipped with a Lie bracket<\/a> on the space of sections<\/a>, together with an anchor map<\/p>\n

\\(\\rho : Sec(A) \\longrightarrow Vect(M) \\)<\/p>\n

that satisfy some natural compatibility conditions. In particular, the anchor map is a Lie algebra homomorphism.<\/p>\n

Lie algebroids also have a very economical description in terms of Q-manifolds, i.e., supermanifolds equipped with an odd vector field that squares to zero [4]. The example to keep in mind here is the tangent bundle, which is canonically a Lie algebroid: the bracket is the standard Lie bracket between vector fields and the anchor is just the identity. Dual to this picture is the de Rham complex<\/a>. We can understand differential forms<\/a> as functions on the shifted or anti- tangent bundle, which is a supermanifold. The homological vector field we recognise as the de Rham differential. Lie algebroids can be defined via their analogue of the de Rham complex. For the case of a Lie algebra (a Lie algebroid over a point) we have the Chevalley\u2013Eilenberg complex<\/a>. The general case is kind of a mix of these two extremes. <\/p>\n

With this in mind, there is a general mantra: whatever you can do with tangent bundles you can do with Lie algebroids<\/em>. This includes the construction of connections.<\/p>\n

Adapted Connections<\/strong>
\nIn the preprint, I define and study connections that take their values in Lie algebroids over the manifold \\(M\\). I define the notion of a connection that respects the structure of a graded bundle (think linear connections and vector bundles) and show that the set of such objects for any graded bundle and Lie algebroid is non-empty. I refer to these as weighted A-connections. I show how one can construct a quai-action of a Lie algebroid on a graded bundle and that this action respects the graded structure. <\/p>\n

The notions also generalise directly to multi-graded bundles, such as double vector bundles. As far as I know, the notion of a connection adapted to a double vector bundle is completely new. <\/p>\n

Potential Applications<\/strong>
\nGiven that graded bundles, Lie algebroids and connections play important roles in geometric mechanics<\/a>, as do double vector bundles, it is possible that weighted A-connections could find applications here. In particular, there could be some scope here in control theory<\/a> and the reduction of higher derivative systems by symmetries. All this remains to be explored. <\/p>\n

References<\/strong>
\n[1] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21\u201336.
\n[2] Th.Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in: Quantization, Poisson Brackets and Beyond, volume 315 of Contemp. Math., pages 131\u2013168. Amer. Math. Soc., Providence, RI, 2002.
\n[3] J. Pradines, Representation des jets non holonomes par des morphismes vectoriels doubles soudes, C. R. Acad. Sci. Paris Ser. A 278 (1974) 152\u20141526.
\n[4] A.Yu. Va\u0131ntrob, Lie algebroids and homological vector fields, Uspekhi Matem. Nauk. 52 (2) (1997) 428\u2013429<\/p>\n","protected":false},"excerpt":{"rendered":"

In a preprint Connections Adapted to Non-Negativley Graded Structures I examine the notion of connections that respect the graded structure. Such connections are akin to linear connections on vector bundles. Graded Bundles Graded bundles are a particular \u2018species\u2019 of non-negatively graded the manifold that is very well behaved [1,2]. A graded bundle \\(F\\) is a … Continue reading Connections adapted to graded bundles<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20,13],"tags":[],"class_list":["post-5425","post","type-post","status-publish","format-standard","hentry","category-post-doc-luxembourg","category-research-work"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5425","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/comments?post=5425"}],"version-history":[{"count":29,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5425\/revisions"}],"predecessor-version":[{"id":5454,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5425\/revisions\/5454"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=5425"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=5425"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=5425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}