A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P ), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an ‘integral’ of the Koszul-Schouten bracket [ , ]P of differential forms in the sense that the exterior d...

We define Gerstenhaber’s graded Lie bracket directly on complexes other than the bar complex, under some conditions, resulting in a practical technique for explicit computations. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic gr...

We propose a new, infinite class of brackets generalizing the Fr\"olicher--Nijenhuis bracket. This can be reduced to family generalized Nijenhuis torsions recently introduced. In particular, Haantjes bracket, first example our construction, is relevant in characterization moduli operators. shall also prove that vanishing higher-level torsion given operator sufficient condition for integrability...

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute b...

In this short review article we sketch some developments which should ultimately lead to the analogy of the Chern-Weil homomorphism for principle bundles in the realm of non commutative differential geometry. Principal bundles there should have Hopf algebras as structure ‘cogroups’. Since the usual machinery of Lie algebras, connection forms, etc., just is not available in this setting, we base...

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Frölicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of ”traceless”...

Abstract We introduce algebraic structures on the polyvector fields of an torus that serve to compute multiplicities in tropical and log Gromov–Witten theory while also connecting mirror symmetry dual deformation complex structures. Most notably these include a quantum field $L_{\infty }$-structure. The latter is instance Getzler’s gravity algebra, $l_2$-bracket restriction Schouten–Nijenhuis b...

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frölicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity of microcubes.