Hypercomplex structures on Courant algebroids

Hypercomplex structures on Courant algebroids unify holomorphic symplectic structures and usual hypercomplex structures. 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. To cite this article: M. Stiénon, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Reduction of Courant algebroids and generalized complex structures

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on CP².     

Matched pairs of Courant algebroids

We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We also consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.     

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

Introducing Nijenhuis forms on L_∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L_∞-algebra. Then we show that Nijenhuis forms on L_∞-algebras also give a short and e?cient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.     

Lie and Courant algebroids on foliated manifolds

This is an exposition of the subject, which was developed in the author’s papers [19, 20]. Various results from the theory of foliations (cohomology, characteristic classes, deformations, etc.) are extended to subalgebroids of Lie algebroids that generalize the tangent integrable distributions. We also suggest a definition of foliated Courant algebroids and give some corresponding results and constructions.     

On higher analogues of Courant algebroids

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n + 1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-form ω is closed under...     

Levi decomposition of analytic Poisson structures and Lie algebroids

We prove the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.     

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.     

Generalized gradients on Lie algebroids

Generalized $O\!\left( n\right) $ -gradients for connections on Lie algebroids are derived.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

Last multipliers on Lie algebroids

In this paper we extend the theory of last multipliers as solutions of the Liouville’s transport equation to Lie algebroids with their top exterior power as trivial line bundle (previously developed for vector fields and multivectors). We define the notion of exact section and the Liouville equation on Lie algebroids. The aim of the present work is to develop the theory of this extension from the tangent bundle algebroid to a general Lie algebroid (e.g. the set of sections with a prescribed last multiplier is still a Gerstenhaber subalgebra). We present some characterizations of this extension in terms of Witten and Marsden differentials.     

Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators

A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own.     

Foliated Lie and Courant Algebroids

If A is a Lie algebroid over a foliated manifold (M, F){(M, {\mathcal {F}})}, a foliation of A is a Lie subalgebroid B with anchor image TF{T{\mathcal {F}}} and such that A/B is locally equivalent with Lie algebroids over the slice manifolds of F{\mathcal F}. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's supervector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid A as a bracket–closed, isotropic subbundle B with anchor image TF{T{\mathcal {F}}} and such that B ^{^} /B{B^{ \bot } /B} is locally equivalent with Courant algebroids over the slice manifolds of F{\mathcal F}. Examples that motivate the definition are given.     

Stability Constants for Courant Approximations

We study stability constants for Courant approximations in the metric of the spaces L₂()and W ₂ ¹ () in classes of triangular grids with uniformly bounded angles and with angles approaching zero. In the first case, we give sufficiently precise numerical estimates for the constants, and in the second case, we give asymptotics of the behavior of the boundaries for the corresponding quadratic forms.     

Left counital Hopf algebras on free Nijenhuis algebras

Factorization in algebra is an important problem. In this paper, we first obtain a unique factorization in free Nijenhuis algebras. By using of this unique factorization, we then define a coproduct and a left counital bialgebraic structure on a free Nijenhuis algebra. Finally, we prove that this left counital bialgebra is connected and hence obtain a left counital Hopf algebra on a free Nijenhuis algebra.