Zhangju Liu

E-Courant Algebroids

In this paper, we introduce the notion of E-Courant algebroids, where E is a vector bundle. It is a kind of generalized Courant algebroid and contains Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its special cases. We explore novel phenomena exhibited by E-Courant algebroids and provide many examples. We study the automorphism groups of omni-Lie algebroids and classify the isomorphism classes of exact E-Courant algebroids. In addition, we introduce the concepts of E-Lie bialgebroids and Manin triples.

Leibniz 2-Algebras and Twisted Courant Algebroids

In this article, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in [26] as well as twisted Courant algebroids by closed 4-forms introduced in [13]. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term L ∞-algebras and geometric structures behind them are exactly H-twisted Lie algebroids introduced in [10].

Omni-Lie 2-algebras and their Dirac structures☆

We introduce the notion of omni-Lie 2-algebra, which is a categorification of Weinstein’s omni-Lie algebras. We prove that there is a one-to-one correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces of a 2-vector space V and Dirac structures on the omni-Lie 2-algebra gl(V)⊕V. In particular, strict Lie 2-algebra structures on V itself one-to-one correspond to Dirac structures of the form of graphs. Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe (non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie 2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which include string Lie 2-algebra structures.

The Pontryagin class for pre-Courant algebroids

In this paper, we show that the Jacobiator J J of a pre-Courant algebroid is closed naturally. The corresponding equivalence class [J ♭ ] [ J ♭ ] is defined as the Pontryagin class, which is the obstruction of a pre-Courant algebroid to be deformed into a Courant algebroid. We construct a Leibniz 2-algebra and a Lie 2-algebra associated to a pre-Courant algebroid and prove that these algebraic structures are isomorphic under deformations. Finally, we introduce the twisted action of a Lie algebra on a manifold to give more examples of pre-Courant algebroids, which include the Cartan geometry.

Generalized lie bialgebras

We generalize Lie bialgebras and Lie bialgebioids to new objects which we call generalized Lie bialgebras. Similar to Lie bialgebras and Lie bial-gebroids, generalized Lie bialgebras are self-dual and generate canonically Hamiltonian structures on their representative spaces. We show that for a generalized Lie bialgebra (E, Ē), a pair (L 1,L 2) of E +Ē is again a generalized Lie bialgebra iff (L 1,L 2) is a Dirac structure pair . Construction of generalized Lie bialgebras by using Poisson tensors and Hamiltonian operators are also discussed in detail and an example relating to an infinite-dimensional integrable system is given.

Quadratic Deformations of Lie–Poisson Structures

In this letter, first we give a decomposition for any Lie–Poisson structure

Quantum integrable systems constrained on the sphere

pi_{mathfrak g}

Omni-Lie algebroids☆

associated to the modular vector. In particular,