Yunhe Sheng

Representations of Hom-Lie Algebras

In this paper, we study representations of hom-Lie algebras. In particular, the adjoint representation and the trivial representation of hom-Lie algebras are studied in detail. Derivations, deformations, central extensions and derivation extensions of hom-Lie algebras are also studied as an application.

Hom-Lie 2-algebras☆

Abstract In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, HL ∞ -algebras, which is the hom-analogue of L ∞ -algebras, and crossed modules of hom-Lie algebras. We prove that the category of hom-Lie 2-algebras and the category of 2-term HL ∞ -algebras are equivalent. We give a detailed study on skeletal hom-Lie 2-algebras. In particular, we construct the hom-analogues of the string Lie 2-algebras associated to any semisimple involutive hom-Lie algebras. We also proved that there is a one-to-one correspondence between strict hom-Lie 2-algebras and crossed modules of hom-Lie algebras. We give the construction of strict hom-Lie 2-algebras from hom-left-symmetric algebras and symplectic hom-Lie algebras.

Higher extensions of Lie algebroids

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integrating an exact Courant algebroid.

INTEGRATION OF SEMIDIRECT PRODUCT LIE 2-ALGEBRAS

The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite dimensional case.

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 the higher-order Dorfman bracket iff ω is a premultisymplectic structure of order n, i.e., dω = 0. Furthermore, there is a Lie algebroid structure on the admissible bundle A ⊂ ∧nT*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).

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].

DIRAC STRUCTURES OF OMNI-LIE ALGEBROIDS

Omni-Lie algebroids are generalizations of Alan Weinsteins omni-Lie algebras. A Dirac structure in an omni-Lie algebroid

On Hom–Lie algebras

dev Eoplus jet E

Omni-Lie 2-algebras and their Dirac structures☆

is necessarily a Lie algebroid together with a representation on