In a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a gradedmultiplication and amulti-Courant bracket on the space of sections of a multi-Dirac structu...

The conservation of the Hamiltonian structures in Whitham's method of averaging. Abstract The work is devoted to the proof of the conservation of local field-theoretical Hamiltonian structures in Whitham's method of averaging. The consideration is based on the procedure of averaging of local Pois-son bracket, proposed by B.A.Dubrovin and S.P.Novikov. Using the Dirac procedure of restriction of ...

We construct a differential and a Lie bracket on the space {Hom(A, A)},k,l≥0 for any associative algebra A. The restriction of this bracket to the space {Hom(A, A)},k≥0 is exactly the Gerstenhaber bracket. We discuss some formality conjecture related with this construction. We also discuss some applications to deformation theory. 0. There exists a well-known way (due to Jim Stasheff) to define ...

Starting from an extension of the Poisson bracket structure and Kubo-MartinSchwinger-property of classical statistical mechanics of continuous systems to spin systems, defined on a lattice, we derive a series of, as we think, new and interesting bounds on correlation functions for general lattice systems. Our method is expected to yield also useful results in Euclidean Field Theory. Furthermore...

We study deformations of the standard embedding of the Lie algebra Vect(S 1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T * S 1 (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of Vect(S 1) into the Lie algebra of functions on ˙ T * S 1 := T * S 1 \S 1 ...

Starting from an extension of the Poisson bracket structure and Kubo-Martin-Schwinger-property of classical statistical mechanics of continuous systems to spin systems, deened on a lattice, we derive a series of, as we think, new and interesting bounds on correlation functions for general lattice systems. Our method is expected to yield also useful results in Euclidean Field Theory. Furthermore...

We first extend the Peierls algebra of gauge invariant functions from the space S of classical solutions to the space H of histories used in path integration and some studies of decoherence. We then show that it may be generalized in a number of ways to act on gauge dependent functions on H. These generalizations (referred to as class I) depend on the choice of an “invariance breaking term,” wh...

A paradigm for describing dynamical systems that have both Hamiltonian and dissipative parts is presented. Features of generalized Hamiltonian systems and metric systems are combined to produce what are called metriplectic systems. The phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric Poisson bracket part and a symmetric dissipative part. Flows a...