{"id":5238,"date":"2017-05-08T07:20:29","date_gmt":"2017-05-08T07:20:29","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=5238"},"modified":"2017-05-08T07:20:29","modified_gmt":"2017-05-08T07:20:29","slug":"representations-theory-of-lie-algebroids-and-weighted-lie-algebroids","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2017\/05\/08\/representations-theory-of-lie-algebroids-and-weighted-lie-algebroids\/","title":{"rendered":"Representations theory of Lie algebroids and weighted Lie algebroids"},"content":{"rendered":"\n\n\n
\"board\"<\/td>\n Weighted Lie algebroids are Lie algebroids in the category of graded bundles, or vice versa. It is well known that VB- algebroids (vector bundles in the category of Lie algebroids, or vice versa) are related to 2-term representations up to homotopy of Lie algebroids. Thus, it is natural to wonder if a similar relation holds for weighted Lie algebroids as these are a wide generalization fo VB-algebroids. <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In a preprint<\/a> entitled “Graded differential geometry and the representation theory of Lie algebroids” with Janusz Grabowski and Luca Vitagliano, we look at the relation between weighted Lie algebroids [1], Lie algebroid modules [2] and representations up to homotopy of Lie algebroids [3]. We show that associated with any weighted Lie algebroid is a series of canonical Lie algebroid modules over the underlying weight zero Lie algebroid. Moreover, we know, due to Mehta [4], that a Lie algebroid module is (up to isomorphisms classes) equivalent to a representation up to homotopy of the Lie algebroid.<\/p>\n

Weighted Lie groupoids were first defined and studied in [5] and offer a wide generalisation of the notion of a VB-groupoid. We show that a refined version of the Van Est theorem [6] holds for weighted Lie groupoids, and in fact follows from minor adjustments to the ideas and proofs presented by Cabrera & Drummond [7].<\/p>\n

References<\/strong>
\n[1] Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101 (2016), 71\u201399.<\/p>\n

[2] Vaintrob A.Yu., Lie algebroids and homological vector fields, Russ. Math. Surv. 52 (1997), 428\u2013429.<\/p>\n

[3] Abad C.A., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew.Math, 663 (2012), 91\u2013126.<\/p>\n

[4] Mehta R.A., Lie algebroid modules and representations up to homotopy. Indag. Math. (N.S.) 25 (2014), no. 5, 1122\u20131134.<\/p>\n

[5] Bruce A.J., Grabowska K., Grabowski J., Graded Bundles in the Category of Lie Groupoids, SIGMA 11 (2015), 090, 25 pages.<\/p>\n

[6] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv, 78 (2003), 681\u201372.<\/p>\n

[7] Cabrera A., Drummond T., Van Est isomorphism for homogeneous cochains, Pacific J. Math. 287 (2017), 297\u2013336<\/p>\n","protected":false},"excerpt":{"rendered":"

Weighted Lie algebroids are Lie algebroids in the category of graded bundles, or vice versa. It is well known that VB- algebroids (vector bundles in the category of Lie algebroids, or vice versa) are related to 2-term representations up to homotopy of Lie algebroids. Thus, it is natural to wonder if a similar relation holds … Continue reading Representations theory of Lie algebroids and weighted Lie algebroids<\/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-5238","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\/5238","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=5238"}],"version-history":[{"count":10,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5238\/revisions"}],"predecessor-version":[{"id":5248,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5238\/revisions\/5248"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=5238"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=5238"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=5238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}