Adjoint symmetries  for graded vector fields

doi:10.1285/i15900932v39n1p33

Adjoint symmetries for graded vector fields

Esmaeil Azizpour, Dordi Mohammad Atayi

Abstract

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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Note di Matematica

Adjoint symmetries for graded vector fields

Abstract