Higher order Toda brackets

Equivalences of Higher Derived Brackets

This note elaborates on Th. Voronov’s construction [V1, V2] of L∞-structures via higher derived brackets with a Maurer–Cartan element. It is shown that gauge equivalent Maurer–Cartan elements induce L∞-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed.

Higher Toda Mechanics and Spectral Curves

For each one of the Lie algebras gln and g̃ln, we constructed a family of integrable generalizations of the Toda chains characterized by two integers m+ and m−. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m+ = m−, we find a symmetric reduction for each general...

Higher Poisson Brackets and Differential Forms

We show how the relation between Poisson brackets and symplectic forms can be extended to the case of inhomogeneous multivector fields and inhomogeneous differential forms (or pseudodifferential forms). In particular we arrive at a notion which is a generalization of a symplectic structure and gives rise to higher Poisson brackets. We also obtain a construction of Koszul type brackets in this s...

Higher Derived Brackets and Deformation Theory

The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basi...

On Sum-normalised Cohomology of Categories, Twisted Homotopy Pairs, and Universal Toda Brackets