We define a graded twisted-coassociative coproduct on the tensor algebra TW of any Zn-graded vector space W . If W is the desuspension space ↓V of a graded vector space V , the coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1to-1 with sequences πs, s ≥ 1, of s-linear maps on V (resp. Zn-graded Loday structures on V , sequences tha...

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M , and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ is transversal to N . This correspondence relates the Schouten-Nijenhuis bracket of multivector fields o...

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M , and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ is transversal to N . This correspondence relates the Schouten-Nijenhuis bracket of multivector fields o...

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ is transversal to N . This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on ...

It is shown how derived brackets naturally arise in sigma-models via Poissonor antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression for the derived bracket is obtained. The generalized Nijenhuis tensor of generalized complex geometry is shown to coincide up to a de-Rham closed term with the...

Frölicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless, the conceptual status of their discovery has been obscured by the genuinely algebraic techniques they exploited. The principal objective in this paper is to show that the double dualization functor in a Cartesian clo...

This paper is devoted to a method of constructing completely integrable systems based on the micro-local theory of bihamiltonian structures [GZ89, GZ91, Bol91, GZ93, GZ00, Pan00, Zak01]. The main tool are the so-called microKronecker bihamiltonian structures [Zak01], which will be called Kronecker in this paper for short (in [GZ00] the term Kronecker was used for the micro-Kronecker structures ...

We extend the Kontsevich formality L∞-morphism U : T • poly(R ) → D poly(R ) to an L∞-morphism of an L∞-modules over T • poly(R ), Û : C•(A, A) → Ω(R), A = C(R). The construction of the map Û is given in Kontsevich-type integrals. The conjecture that such an L∞-morphism exists is due to Boris Tsygan [Ts]. As an application, we obtain an explicit formula for isomorphism A∗/[A∗, A∗] ∼ → A/{A, A} ...

If a graded Lie algebra is the direct sum of two graded sub Lie algebras, its bracket can be written in a form that mimics a ”double sided semidirect product”. It is called the knit product of the two subalgebras then. The integrated version of this is called a knit product of groups — it coincides with the ZappaSzép product. The behavior of homomorphisms with respect to knit products is invest...

We construct Nijenhuis operators from particular bialgebras called dendriform-Nijenhuis bialgebras. It turns out that such Nijenhuis operators commute with TD-operators, a kind of Baxter-Rota operators, and are therefore closely related to dendriform trialgebras. This allows the construction of associative algebras, called dendriform-Nijenhuis algebras, made out of nine operations and presentin...