arXiv subjects
Publications and source records attributed to Zhangju Liu.
Drinfeld classified Poisson homogeneous spaces of a Poisson Lie group in terms of Dirac structures of the Lie bialgebra. In this paper, we study homogeneous spaces of a 2-group and develop Drinfeld theorem in the Poisson 2-group context.
Berwick-Evens and Lerman recently showed that the category of vector fields on a geometric stack has the structure of a Lie $2$-algebra. Motivated by this work, we present a construction of graded weak Lie $2$-algebras associated with quasi-Poisson groupoids based on the space of multiplicative forms on the groupoid and differential forms on the base manifold. We also establish a morphism between the Lie $2$-algebra of multiplicative multivector fields and the weak Lie $2$-algebra of multiplicative forms, allowing us to compare and relate different aspects of Lie $2$-algebra theory within the context of quasi-Poisson geometry. As an infinitesimal analogy, we explicitly determine the associated weak Lie $2$-algebra structure of IM $1$-forms along with differential $1$-forms on the base manifold for any quasi-Lie bialgebroid.
A strict Lie $2$-algebra $Γ(\wedge^\bullet A) \stackrel{T}{\rightarrow} \mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ is associated with any Lie groupoid $\mathcal{G}$. Here, $Γ(\wedge^\bullet A)$ is the Schouten algebra of the tangent Lie algebroid $A$ of $\mathcal{G}$ and $\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ is the space of multiplicative multivectors on $\mathcal{G}$. The quotient ${R}_{\mathrm{mult}}^\bullet:=\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})/\mathrm{Img} T$, a Morita invariant of $\mathcal{G}$, is called the reduced space of multiplicative multivectors. We prove a canonical decomposition formula of elements in $\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ and establish a key relation between ${R}_{\mathrm{mult}}^k$ and the cohomology $\mathrm{H} ^1(\mathfrak{J} \mathcal{G},\wedge^k A)$ where $\mathfrak{J} \mathcal{G}$ is the jet groupoid of $\mathcal{G}$ and $1\leqslant k\leqslant \mathrm{rank} A$. We also study ${R}_{\mathrm{diff}}^\bullet $, the reduced space of Lie algebroid differentials on $A$. By taking infinitesimals, $\barδ: $ ${R}_{\mathrm{mult}}^\bullet $ $\to $ ${R}_{\mathrm{diff}}^\bullet $, the two reduced spaces are related. We find that the kernel of $\barδ$ is isomorphic to the kernel of the Van Est map $\mathrm{H}^1(\mathcal{G},\wedge^k \kerρ)\to \mathrm{H}^1(A,\wedge^k \kerρ)$, where $ρ$ is the anchor of $A$.
Given a Lie groupoid $\mathcal{G}$ over $M$, $A$ the tangent Lie algebroid of $\mathcal{G}$, and $ρ: A\rightarrow TM$ the anchor map, we provide a formula that decomposes an arbitrary multiplicative $k$-form $Θ$ on $\mathcal{G}$ into two parts. The first part is $e$, a $1$-cocycle of $\mathfrak{J}\mathcal{G}$ valued in $\wedge^k T^*M$, and the second part is $θ\in Γ(A^*\otimes (\wedge^{k-1} T^*M))$ which is $ρ$-compatible, meaning that $ι_{ρ(u)}θ(u)=0$ for all $u\in A$. We call this pair of data $(e,θ)$ the $(0,k)$-characteristic pair of $Θ$. Next, we prove that if $\mathcal{G}$ is a Poisson Lie groupoid, then the space $Ω^{\bullet}_{\mathrm{mult}}(\mathcal{G})$ of multiplicative forms on $\mathcal{G}$ has a differential graded Lie algebra (DGLA) structure. Furthermore, when combined with $Ω^\bullet(M)$, which is the space of forms on the base manifold $M$, $Ω^{\bullet}_{\mathrm{mult}}(\mathcal{G})$ forms a canonical DGLA crossed module. This supplements a previously known fact that multiplicative multivector fields on $\mathcal{G}$ form a DGLA crossed module with the Schouten algebra $Γ(\wedge^\bullet A)$ stemming from the tangent Lie algebroid $A$.
In this paper, first we introduce the notion of an omni-representation of a Leibniz algebra $\g$ on a vector space $V$ as a Leibniz algebra homomorphism from $\g$ to the omni-Lie algebra $\gl(V)\oplus V$. Then we introduce the omni-cohomology theory associated to omni-representations and establish the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.
We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using Čech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.
We define double principal bundles (DPBs), for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples. It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and connections in DPBs are investigated.
We first recall two equivalent definitions of Lie $2$-algebras, categorification of Lie algebras and $2$-term $L_\infty$-algebras. Then we present four different kinds of Lie $2$-algebras from $2$-plectic manifolds, Courant algebroids, homotopy Poisson manifolds and affine multivector fields on a Lie groupoid respectively. Moreover, we recall the cohomology theory of Lie $2$-algebras and analyze its lower degree cases. The integration of strict Lie $2$-algebras to strict Lie $2$-groups is also discussed.
We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra $L$ and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of $L$ are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra $L_{\operatorname{Lie}}$.
Motivated by Kapranov's discovery of an sh Lie algebra structure on the tangent complex of a Kähler manifold and Chen-Stiénon-Xu's construction of sh Leibniz algebras associated with a Lie pair, we find a general method to construct sh Leibniz algebras. Let $\mathcal{A}$ be a commutative dg algebra. Given a derivation of $\mathcal{A}$ valued in a dg module $Ω$, we show that there exist sh Leibniz algebra structures on the dual module of $Ω$. Moreover, we prove that this process establishes a functor from the category of dg module valued derivations to the category of sh Leibniz algebras over $\mathcal{A}$.
Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields has a natural graded strict Lie 2-algebra structure and affine (1,1)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.
For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^*$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$, which coincide with the groupoid orbits of the symplectic groupoid $T^*\mathcal{G}$ over $A^*$. It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang-Mills field.
Given a Leibniz algebra L with left center Z, we work on C(L,Z,S(Z)), the Z-standard complex of L with coefficients in S(Z). We construct the derived bracket for a fat Leibniz algebra in terms of a certain 3-cocycle and a Poisson algebra structure on the space of so-called "representable cochains".
In this paper, we show that the Jacobiator $J$ of a pre-Courant algebroid is closed naturally. The corresponding equivalence class $[J^\flat]$ 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.
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 $\Der(\g)$.
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.
The notion of crossed modules for Lie 2-algebras is introduced. We show that, associated to such a crossed module, there is a strict Lie 3-algebra structure on its mapping cone complex and a strict Lie 2-algebra structure on its derivations. Finally, we classify strong crossed modules by means of the third cohomology group 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.