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Variational multivectors and brackets in the geometry of jet spaces. (English) Zbl 1122.17021

Nikitin, A.G.(ed.) et al., Proceedings of the fifth international conference on symmetry in nonlinear mathematical physics, Kyïv, Ukraine, June 23–29, 2003. Part 3. Kyïv: Institute of Mathematics of NAS of Ukraine (ISBN 966-02-3227-6). Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Mathematics and its Applications. 50(3), 1335-1342 (2004).
An algebraic model in the framework of jet superspace geometry is further developed. First of all, notations and terminology related to graded vector spaces, graded algebras, and differential operators over such algebras are set up. Then a graded Lie algebra structure on a space of multilinear maps on a vector space is introduced generalizing the Nijenhuis-Richardson and Schouten brackets. The following theorem is formulated.
On the space of \(G\)-graded Lie algebra there exists a unique structure with two properties concerning to relationships between subspaces of this graded algebra and multilinear mappings.
Considering multilinear maps as differential operators on polynomial functions the authors provide an algebraic counterpart for a class of local variational differential operators. Next, they say that a vector bundle \(\pi\colon E\to M\) is a superbundle if it is the direct sum \(\pi^0\oplus \pi^1\) of two vector bundles \(\pi^0\colon E_0\to M\) and \(\pi^1\colon E_1\to M\). Then, using general above-mentioned construction, a (super)symmetric bracket on multilinear maps of the differential Cartan forms is introduced as well as many another algebraic structures (a horizontal module, a universal linearization, a variational (super)symmetric multivectors and so on). An application of the constructed algebraic model to the field theory is given for finding of Hamiltonian structures for evolution equations.
For the entire collection see [Zbl 1088.17002].

MSC:

17B70 Graded Lie (super)algebras
58A20 Jets in global analysis
53D17 Poisson manifolds; Poisson groupoids and algebroids
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