Adjoint symmetries for graded vector fields


Suppose that {\mathcal{M}}=(M,{\mathcal A}_M) is a graded manifold and consider a direct subsheaf \cd of { Der {\mathcal A}_M} and a graded vector field \Gamma on {\mathcal{M}}, both satisfying certain conditions. \cd is used to characterize the local expression of \Gamma. Thus we review some of the basic definitions, properties, and geometric structures related to the theory of adjoint symmetries on a graded manifold and discuss some ideas from Lagrangian supermechanics in an informal fashion. In the special case where {\mathcal{M}} is the tangent supermanifold, we are able to find a generalization of the adjoint symmetry method for time-dependent second-order equations to the graded case. Finally, the relationship between adjoint symmetries of \Gamma and Lagrangians is studied.

DOI Code: 10.1285/i15900932v39n1p33

Keywords: supermanifold; involutive distribution; second-order differential equation field; Lagrangian systems; adjoint symmetry

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