{"id":725,"date":"2011-11-19T17:31:26","date_gmt":"2011-11-19T16:31:26","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=725"},"modified":"2011-11-19T17:31:26","modified_gmt":"2011-11-19T16:31:26","slug":"jacobi-algebroids-and-quasi-q-manifolds","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2011\/11\/19\/jacobi-algebroids-and-quasi-q-manifolds\/","title":{"rendered":"Jacobi algebroids and quasi Q-manifolds"},"content":{"rendered":"

In “Jacobi algebroids and quasi Q-manifolds”\u00a0 (arXiv:1111.4044v1<\/a> [math-ph]) I reformulate the notion of a Jacobi algebroid (aka generalised Lie algebroid or Lie algebroid in the presence of a 1-cocycle) in terms of an odd Jacobi structure of weight minus one\u00a0 on the total space of the “anti-dual bundle” \\(\\Pi E^{*}\\). This mimics the weight minus one Schouten structure associated with a Lie algebroid. The weight is assigned as zero to the base coordinates ans one to the (anti-)fibre coordinates.<\/p>\n

Recall that a Lie algebroid can be understood as a weight one homological vector field\u00a0 on the “anti-bundle” \\(\\Pi E\\). What is the corresponding situation for Jacobi algebroids?<\/p>\n

Well, this leads to a new notion, what I call a quasi Q-manifold…<\/p>\n

A quasi Q-manifold is a supermanifold equipped with an odd vector field \\(D\\) and an odd function \\(q\\) that satisfy the following<\/p>\n

\\(D^{2}= \\frac{1}{2}[D,D] = q \\: D\\)<\/p>\n

and<\/p>\n

\\(D[q]=0\\).<\/p>\n

The extreme examples here are<\/p>\n

    \n
  1. Q-manifolds, that is set \\(q=0\\). Then \\(D^{2}=0\\).<\/li>\n
  2. Supermanifolds with a distinguished (non-zero) odd function, that is set \\(D=0\\).\u00a0 (This includes the cotangent bundle of\u00a0 Schouten and higher Schouten\u00a0 manifolds)<\/li>\n
  3. The entire category of supermanifolds if we set \\(D=0\\) and \\(q =0\\).<\/li>\n<\/ol>\n

     <\/p>\n

    The theorem here is that a Jacobi algebroid,\u00a0 understood as a weight minus one Jacobi structure on \\(\\Pi E^{*}\\) is equivalent to\u00a0 \\(\\Pi E\\) being a weight one\u00a0 quasi Q-manifold.\u00a0 I direct the interested reader to the preprint for details.<\/p>\n

    A nice example is \\(M:= \\Pi T^{*}N \\otimes \\mathbb{R}^{0|1}\\), where \\(N\\) is a pure even classical manifold.\u00a0 The supermanifold \\(M\\) is in fact an odd contact manifold<\/em> or equivalently an odd Jacobi manifold of weight minus one, see arXiv:1101.1844v3<\/a> [math-ph]. Then\u00a0 it turns out that \\(M^{*} := \\Pi TN\\otimes \\mathbb{R}^{0|1}\\)\u00a0 is a weight one quasi Q-manifold. It is worth recalling that \\(\\Pi T^{*}N\\) has a canonical Schouten structure (in fact odd symplectic) and that \\(\\Pi TN\\) is a Q-manifold where the homological vector field is identified with the de Rham differential on \\(N\\).\u00a0 Including the “extra odd direction” deforms these structures.<\/p>\n

    As far as I can tell quasi Q-manifolds are a new class of supermanifold that generalises Q-manifolds and Schouten manifolds.\u00a0 It is not know if other examples of such structures outside the theory of Lie and Jacobi algebroids are interesting. Only time will tell…<\/p>\n","protected":false},"excerpt":{"rendered":"

    In “Jacobi algebroids and quasi Q-manifolds”\u00a0 (arXiv:1111.4044v1 [math-ph]) I reformulate the notion of a Jacobi algebroid (aka generalised Lie algebroid or Lie algebroid in the presence of a 1-cocycle) in terms of an odd Jacobi structure of weight minus one\u00a0 on the total space of the “anti-dual bundle” \\(\\Pi E^{*}\\). This mimics the weight minus … Continue reading Jacobi algebroids and quasi Q-manifolds<\/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":[13],"tags":[],"class_list":["post-725","post","type-post","status-publish","format-standard","hentry","category-research-work"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/725","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=725"}],"version-history":[{"count":0,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/725\/revisions"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=725"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}