We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable algebroids are special cases of local algebroids, which still general enough to include many physically motivated such as Lie, Courant, metric higher-Courant They create a natural framework for generalizations differential geometric smooth manifold. The symmetrization...

In this paper we study the cohomology H• st (E) of a Courant algebroid E. We prove that if E is transitive, H• st (E) coincides with the naive cohomology H• naive (E) of E as conjectured by Stiénon and Xu [SX08]. For general Courant algebroids E we define a spectral sequence converging to H• st (E). If E is with split base, we prove that there exists a natural transgression homomorphism T3 (wit...

Inspired by [3] we introduce the concept of extended Hopf algebra and consider their cyclic cohomology in the spirit of Connes-Moscovici [3, 4, 5]. Extended Hopf algebras are closely related, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscov...

We propose a new framework for constructing geometric and physical models on spacetimes provided with Lie algebroid symmetry, i.e. manifolds provided with additional anchor and generalized Lie algebra commutator structures. The approach is related to the geometry of moving nonholonomic frames with associated nonlinear connections. A strict application of such geometric methods to spinor fields ...

Motivated by properties of higher tangent lifts geometric structures, we introduce concepts weighted structures for various objects on a manifold F equipped with homogeneity structure. The latter is smooth action the monoid $$(\mathbb {R},\cdot )$$ multiplicative reals. Vector bundles are particular cases and them call $$\mathrm{V\!B}$$ -structures. In case Lie algebroids groupoids, include -al...

We introduce the notion of -algebroid, generalising both Lie and Courant algebroids, as well algebroids used in exceptional generalised geometry for . Focusing on case, we prove a classification “exact” translate related Leibniz parallelisable spaces into tractable algebraic problem. After discussing general Poisson–Lie duality, show that U-duality is compatible with equations motion supergravi...

We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.