Integration of derivations for Lie $2$-algebras
In this paper, for a Lie 2-algebra $\g$, we construct the automorphism 2-group $\Aut(\g)$, which turns out to be an integration of the derivation Lie 2-algebra $\Der(\g)$.
math.DG↗
arXiv subjects
Yunhe Sheng
Publications and source records attributed to Yunhe Sheng.
On hom-Lie algebras
In this paper, first we show that $(\g,[\cdot,\cdot],α)$ is a hom-Lie algebra if and only if $(Λ\g^*,α^*,d)$ is an $(α^*,α^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie algebra is a hom-Leibniz algebra, or a hom-Lie 2-algebra.
math-ph↗
Remarks on Leibniz algebras
In this paper, first we construct a Lie 2-algebra associated to every Leibniz algebra via the skew-symmetrization. Furthermore, we introduce the notion of the naive representation for a Leibniz algebra in order to realize the abstract operations as a concrete linear operation. At last, we study some properties of naive cohomologies.
math.RT↗
Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations
In this paper, we introduce the notion of a pre-Lie 2-algebra, which is a categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\infty$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. $\mathcal O$-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras.
math-ph↗
Left-symmetric algebroids
In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from $\mathcal O$-operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left-symmetric algebroids. Representations of left-symmetric algebroids are studied in detail. At last, we study deformations of left-symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.
math.DG↗
Deformations of Lie 2-algebras
In this paper, we consider deformations of Lie 2-algebras via the cohomology theory. We prove that a 1-parameter infinitesimal deformation of a Lie 2-algebra $\g$ corresponds to a 2-cocycle of $\g$ with the coefficients in the adjoint representation. The Nijenhuis operator for Lie 2-algebras is introduced to describe trivial deformations. We also study abelian extensions of Lie 2-algebras from the viewpoint of deformations of semidirect product Lie 2-algebras.
math-ph↗
Lie 2-bialgebras
In this paper, we study Lie 2-bialgebras, with special attention to coboundary ones, with the help of the cohomology theory of $L_\infty$-algebras with coefficients in $L_\infty$-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras and symplectic Lie algebras.
math-ph↗
A new approach to hom-Lie bialgebras
In this paper, we introduce a new definition of a hom-Lie bialgebra, which is equivalent to a Manin triple of hom-Lie algebras. We also introduce a notion of an $\mathcal O$-operator and then construct solutions of the classical hom-Yang-Baxter equation in terms of $\mathcal O$-operators and hom-left-symmetric algebras.
math-ph↗
Hom-Lie 2-algebras
In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, $HL_\infty$-algebras, which is the hom-analogue of $L_\infty$-algebras, and crossed modules of hom-Lie algebras. We prove that the category of hom-Lie 2-algebras and the category of 2-term $HL_\infty$-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.
math-ph↗
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.
math-ph↗
On deformations of Lie algebroids
For any Lie algebroid A, its 1-jet bundle JA is a Lie algebroid naturally and there is a representation π: JA ->DA. Denote by dJ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex (Γ(Hom(^\bulletJA,A)DA), dJ) of the cochain complex (Γ(Hom(^\bulletJA,A)), dJ).
math.DG↗
Integration of Lie 2-algebras and their morphisms
Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.
math-ph↗
Non-abelian Extensions of Lie 2-algebras
In this paper, we introduce the notion of derivations of Lie 2-algebras and construct the associated derivation Lie 3-algebra. We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.
math-ph↗
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.
math.DG↗
Leibniz 2-algebras and twisted Courant algebroids
In this paper, 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 \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term $L_\infty$-algebras and geometric structures behind them are exactly $H$-twisted Lie algebroids introduced in \cite{Grutzmann}.
math-ph↗
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 $TM\oplus\wedge^nT^*M$ 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 $\wedge^nT^*M$. 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. $\dMω=0$. Furthermore, there is a Lie algebroid structure on the admissible bundle $A\subset\wedge^{n}T^*M$. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in \cite{baez:classicalstring}.
math.DG↗
On Double Vector Bundles
In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theory of double vector bundles in term of the corresponding short exact sequences. Examples including the jet bundle and the Atiyah algebroid are discussed.
math.DG↗
Dirac structures of omni-Lie algebroids
Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid $\dev E\oplus \jet E$ is necessarily a Lie algebroid together with a representation on $E$. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in $\huaT=TM\oplus E$; we establish the relation between the normalizer $N_{L}$ of a reducible Dirac structure $L$ and the derivation algebra $\Der(\pomnib (L))$ of the projective Lie algebroid $\pomnib (L)$; we study the cohomology group $\mathrm{H}^\bullet(L,ρ_{L})$ and the relation between $N_{L}$ and $\mathrm{H}^1(L,ρ_{L})$; we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure $L$, which is related with $\mathrm{H}^2(L,ρ_{L})$.
math.DG↗