Dirac structures for higher analogues of Courant algebroids

Abstract

In this paper, we introduce the notion of a (p, k)-The (p, 0)-Dirac structures are exactly the higher analogues of Dirac structures of order p introduced by Zambon in [L∞-algebras and higher analogues of Dirac structures and Courant algebroids, J. Symplectic Geom. 10(4) (2012) 563-599]. The (p, p − 1)-Dirac structures are exactly the Nambu-Dirac structures introduced by Hagiwara in [Nambu-Dirac manifolds, J. Phys. A 35(5) (2002) 1263-1281]. In the regular case, such a (p, k)-Dirac structure is characterized by a characteristic pair.

Linearization of the higher analogue of Courant algebroids

“…Recently, the higher analogues of the standard Courant algebroid T M ⊕ ∧ n T * M are widely studied due to applications in Nambu-Poisson structures, multisymplectic structures, L ∞algebra theory and topological field theory [1,4,15,19,22]. In particular, Dirac structures of the higher analogues of the standard Courant algebroid T M ⊕ ∧ n T * M are deeply studied in [2,6,20,42]. In [31], the authors introduced the notion of an omni n-Lie algebra gl(V ) ⊕ ∧ n V and proved that it is the base-linearization of the higher analogue of the standard Courant algebroid T M ⊕ ∧ n T * M .…”

Weak $(p,k)$-Dirac manifolds

“…It is well-known that (pre)symplectic structure can provide a regular (or singular) Poisson structure on M . Y. Bi and Y. Sheng [3] generalize the (1, 0)-Dirac structure to (p, k)-Dirac structures for higher analogues of Courant algebroids by the (p, k)-Lagrangian condition. The weak (p, k)-Dirac structures are more important because they provide a specific view of Numbu-Poisson manifolds, higher-Poisson manifolds and pre-multisymplectic geometry.…”

“…The graph of Nambu-Poisson structure is a Nambu-Dirac structure. In [3] the authors prove that the Nambu-Dirac structure is a (p, p−1)-Dirac structure by the (p, k)-Lagrangian condition. The weak (p, 0)-Dirac manifold has a related application in physics.…”

Higher omni-Lie algebroids

“…Now, fix a positive integer n. There is an "n-form version" of the generalized tangent bundle, called the higher generalized tangent bundle: T M ⊕ ∧ n T * M . The higher generalized tangent bundle and its isotropic and involutive subbundles have been first considered in [1,27] (see also [2,3]). They encode higher Dirac structures, in particular closed n-forms and certain Nambu structures [27].…”