{"id":2939,"date":"2013-01-22T09:00:29","date_gmt":"2013-01-22T08:00:29","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=2939"},"modified":"2013-01-22T09:00:29","modified_gmt":"2013-01-22T08:00:29","slug":"odd-jacobi-manifolds-and-loday-poisson-brackets","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2013\/01\/22\/odd-jacobi-manifolds-and-loday-poisson-brackets\/","title":{"rendered":"Odd Jacobi manifolds and Loday-Poisson brackets"},"content":{"rendered":"\n\n\n
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I have a new preprint posted on the arXiv; “Odd Jacobi manifolds and Loday-Poisson brackets”. It is a continuation of my studies of odd Jacobi structures on supermanifolds. <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Odd Jacobi manifolds and Loday-Poisson brackets
\nAndrew James Bruce
\n(Submitted on 21 Jan 2013)
\n arXiv:1301.4799 [math-ph]<\/a><\/p>\n

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Interestingly, these relations are identical to the Cartan identities.<\/p>\n

———————————————————–<\/p>\n

There are some subtle differences between even and odd structures and this preprint discusses one such difference. In particular, one can use the derived bracket formalism [3] to construct a Poisson<\/a>-like bracket on the supermanifold mod the skewsymmetry.<\/p>\n

The Loday-Poisson bracket<\/strong><\/p>\n

An odd Jacobi manifold is a supermanifold equipped with an almost Schouten structure and a homological vector field that satisfy some relations. The relations are not important for this discussion. See [1] for details.<\/p>\n

From this data one can construct an odd Jacobi bracket, that is an odd version of a Poisson bracket with a modified Leibniz rule. The adjoint operator is a first order differential operator as opposed to a vector field.<\/p>\n

Furthermore, by using the fact that the homological vector field is a Jacobi vector, that is it a “derivation over the odd Jacobi bracket” one can construct an even bracket using the derived bracket construction.<\/p>\n

The resulting bracket satisfies a version of the Jacobi identity, but is not skewsymmetric. It also satisfies the Leibniz rule (from the left). Lie brackets mod the skewsymmetry were first examined by Loday, and so I call Loday brackets + Leibniz rule “Loday-Poisson brackets”.<\/p>\n

This is in contrast to classical manifolds, where due to the work of Grabowski and Marmo [2], we know that the Jacobi identity<\/a> and the Leibniz rule<\/a> force the skewsymmetry. On supermanifolds we have nilpotent<\/a> functions and this invalidates the assumptions of Grabowski and Marmo.<\/p>\n

Furthermore, on an even Jacobi supermanifold there is no canonical choice of homological vector field to use, if one exists at all.<\/p>\n

In the preprint I present several relations between the Hamiltonian vector fields with respect to the initial odd Jacobi structure and the derived Loday-Poisson structure. I note the similarity with the standard Cartan calculus<\/a>.<\/p>\n

The derived product<\/strong><\/p>\n

I have discussed the derived product on a Q-manifold here<\/a>. As odd Jacobi manifolds come with a homological vector field as part of the structure, they are also Q-manifolds and have a derived product.<\/p>\n

Interestingly, the Loday-Poisson bracket not only satisfies the Leibniz rule (from the left) for the usual product of functions on a supermanifold, but also the derived product.<\/p>\n

That is we have a kind on non-skewsymmetric bracket that satisfies a version of the Jacobi identity and a version of the Leibniz rule over a Grassmann odd noncommutative form of multiplication. To my knowledge, these kinds of noncommutative Poisson algebras have not been studied.<\/p>\n

References <\/strong>
\n[1] Andrew James Bruce. Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids. Extracta Math.<\/em> 27<\/strong>(1) (2012), 91-123<\/p>\n

[2] J. Grabowski and G. Marmo. Non-antisymmetric versions of Nambu-Poisson and Lie algebroid brackets. J. Phys. A: Math. Gen.<\/em> 34<\/strong> (2001), 3803\u20133809.<\/p>\n

[3] Yvette Kosmann\u2013Schwarzbach. Derived brackets. Lett. Math. Phys.<\/em>, 69<\/strong> (2004), 61-87.<\/p>\n","protected":false},"excerpt":{"rendered":"

I have a new preprint posted on the arXiv; “Odd Jacobi manifolds and Loday-Poisson brackets”. It is a continuation of my studies of odd Jacobi structures on supermanifolds. Odd Jacobi manifolds and Loday-Poisson brackets Andrew James Bruce (Submitted on 21 Jan 2013) arXiv:1301.4799 [math-ph] In this paper we construct a non-skewsymmetric version of a Poisson … Continue reading Odd Jacobi manifolds and Loday-Poisson brackets<\/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":[5,13],"tags":[],"class_list":["post-2939","post","type-post","status-publish","format-standard","hentry","category-general","category-research-work"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/2939","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=2939"}],"version-history":[{"count":0,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/2939\/revisions"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=2939"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=2939"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=2939"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}