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...

We study a Lagrangian extension of the 5d Martínez Alonso–Shabat equation \(\mathcal {E}\)$$\begin{aligned} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{aligned}$$that coincides with cotangent {T}^{*}\mathcal {E}\) to latter. describe Lie algebra structure its symmetries (which happens be quite nontrivial and is described in terms deformations) construct two families recursion operators for symme...

The central part of calculus on manifolds is usually the calculus of differential forms and the best known operators are exterior derivative, Lie derivatives, pullback and insertion operators. Differential forms are a graded commutative algebra and one may ask for the space of graded derivations of it. It was described by Frölicher and Nijenhuis in [1], who found that any such derivation is the...

In this note, we prove the equivalence of two characterizations of hypercomplex structures on Courant algebroids, one in terms of Nijenhuis concomitants and the other in terms of (almost) torsionfree connections for which each of the three complex structures is parallel. A Courant algebroid [4] consists of a vector bundle π : E → M , a nondegenerate symmetric pairing 〈, 〉 on the fibers of π, a ...

In this paper we re-express the Schouten-Nijenhuis, the Frölicher-Nijenhuis and the Nijenhuis-Richardson brackets on a symplectic space using the extended Poisson brackets structure present in the path-integral formulation of classical mechanics.

In geometry of nonlinear partial differential equations, recursion operators that act on symmetries an equation $$\mathcal{E}$$ are understood as Bäcklund auto-transformations the $$\mathcal{TE}$$ tangent to . We apply this approach a natural two-component extension 3D Pavlov–Mikhalev $$u_{yy}=u_{tx}+u_{y}u_{xx}-u_{x}u_{xy}.$$ describe Lie algebra for extension, construct two (one them was know...

Let ? 2 be a fixed natural number. The complete description is given of the product preserving gauge bundle functors F on category F?VB flag vector bundles K = (K;K1,...,K?) length in terms systems I (I1,..., I??1) A-module homomorphisms Ii : Vi+1 Vi for Weil algebras A and finite dimensional (over R) A-modules V1,...,V?. so called iteration problem investigated. affinors FK are classified. gau...

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

In the present paper we prove a statement closely related to the cyclic formality conjecture [Sh]. In particular, we prove that for a volume form Ω and a Poisson bivector field π on Rd such that divΩ π = 0, the Kontsevich star-product [K] with the harmonic angle function is cyclic, i.e. ∫ Rd (f ∗ g) · h · Ω = ∫ Rd (g ∗ h) · f · Ω for any three functions f, g, h on Rd (for which the integrals ma...