Adjoint symmetries for graded vector fields
Abstract
Suppose that 
is a graded manifold and consider a direct subsheaf 
of 
and a graded vector field 
on 
, both satisfying certain conditions. is used to characterize the local expression of . 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 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 and Lagrangians is studied.
is a graded manifold and consider a direct subsheaf of and a graded vector field on , both satisfying certain conditions. is used to characterize the local expression of . 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 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 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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