Hamiltonian Formalism in Supermechanics

The Hamilton–Cartan formalism in supermechanics is developed, the graded structure on the manifold of solutions of a variational problem defined by a regular homogeneous Berezinian Lagrangian density is determined and its graded symplectic structure is analyzed. The graded symplectic structure on the manifold of solutions of a classical regular Lagrangian is compared with the Koszul–Schouten brackets.     

Convex-compact sets and Banach discs

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E′ of a locally convex space E is the σ(E′,E)-closure of the union of countably many σ(E′,E)-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.     

Poisson-Nijenhuis structures and the Vinogradov bracket

We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frölicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.Partially supported by Fundació Caixa Castelló.Partially supported by the Spanish DGICYT grant #P B91-0324.     

Poisson Brackets of Even Symplectic Forms on the Algebra of Differential Forms

Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established.     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

Existence and uniqueness of solutions to superdifferential equations

We state and prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supervector field, X = X₀ + X₁, has a unique integral flow, Г: ^1¦1 x (M, A_M) → (M, A_M), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an ^1¦1-action is obtained: the homogeneous components, X₀, and, X₁, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on ^1¦1, however, has to be specified: there are three non-isomorphic Lie supergroup structures on ^1¦1, all of which have addition as the group operation in the underlying Lie group . On the other extreme, even if X₀, and X₁ do not close to form a Lie superalgebra, the integral flow of X is uniquely determined and is independent of the Lie supergroup structure imposed on ^1¦1. This fact makes it possible to establish an unambiguous relationship between the algebraic Lie derivative of supergeometric objects (e.g., superforms), and its geometrical definition in terms of integral flows. It is shown by means of examples that if a supergroup structure in ^1¦1 is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an ^1¦1-action of the chosen structure. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given.