Natural analogs of Lie brackets on affine bundles are studied. In particular, a close relation to Lie algebroids and a duality with certain affine analog of Poisson structure is established as well as affine versions of the complete lift and the Cartan exterior calculus.

The aim of this note is to clarify the relevance of “connections up to homotopy” [4, 5] to the theory of characteristic classes, and to present an application to the characteristic classes of Lie algebroids [3, 5, 7] (and of Poisson manifolds in particular [8, 13]). We have already remarked [4] that such connections up to homotopy can be used to compute the classical Chern characters. Here we p...

1 Introduction: The aim of this note is to point out that Chern characters can be computed using curvatures of " connections up to homotopy " , and to present an application to the vanishing theorem for Lie algebroids. Classically, Chern characters are computed with the help of a connection and its curvature. However, one often has to relax the notion of connection so that one gains more freedo...

We interpret the physical B-field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural “Ricci flow” for generalized geometry. Next we show that the B-field renormalization group flow preserves T-duality in a natural sense. As corollaries we obtain new long time existence results for the B-field renormalization group flow.

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid H = (HL,HR)) is cleft if and only if it is HR-Galois and has a normal basis property relative to the base ring L of HL. Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the ...

In this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB-exact algebroid (corresponding the group $E_{n(n)}\times \mathbb{R}^+$, for $n\le 6$) locally has standard form given by tangent bundle. derive possible twists, flat $\mathfrak{gl}(2,\mathbb{R})$-connection, covariantly closed pair of 3-forms, and 5-form, comment physical ...

We introduce nonassociative geometric objects generalizing naturally Lie groupoids and called (smooth) quasiloopoids loopoids. prove that the tangent bundles of smooth loopoids are canonically again (it is nontrivial in case loopoids). show also this not true if cotangent concerned. After providing a few natural constructions, we how Lie-like functor associates with skew algebroids almost discr...