In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed nondegenerate (n + 1)-form. The case relevant to classical string theory is when n = 2 and is called ‘2-plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2...

We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebro...

We remark that the equations underlying the notion of generalized complex structure have simple geometric meaning when passing to Lie algebroids/groupoids.

We extend R. Fernandes’ construction of the secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism. In [4] R. Fernandes has constructed a sequence of secondary characteristic classes of a Li...

We present a characterization, in terms of torsion-free generalized connections, for the integrability various structures (generalized almost complex structures, hypercomplex Hermitian and hyper-Hermitian structures) defined on Courant algebroids. develop new, self-contained, approach theory Dirac generating operators regular algebroids with scalar product neutral signature. As an application w...

We give a simple characterization of Mackenzie’s double Lie algebroids in terms of homological vector fields. Application to the ‘Drinfeld double’ of Lie bialgebroids is given and an extension to the multiple case is suggested.

Lie algebroids and Lie pseudoalgebras arise from a wide variety of constructions in differential geometry; they have been introduced repeatedly into the geometry, physics and algebra literatures since the 1950s, under some 14 different terminologies. The first main part (Sections 2-5) of this survey describes the four principal classes of examples, emphazising that each arises by means of a gen...