arXiv subjects

Honglei Lang

Publications and source records attributed to Honglei Lang.

At least 19 recordsLinked to original sources

arXiv2026

Relative Rota-Baxter operators and crossed homomorphisms on Lie 2-groups

A relative Rota-Baxter operator on Lie 2-groups is introduced as a pair of relative Rota-Baxter operators on the underlying Lie groups which is also a Lie groupoid morphism. Such an operator induces a factorization theorem for Lie 2-groups and gives rise to a categorical solution of the Yang-Baxter equation. We further define relative Rota-Baxter operators on Lie group crossed modules. The well-known one-to-one correspondence between Lie 2-groups and crossed modules is extended to an equivalence between the respective relative Rota-Baxter operators on these two structures. Finally, as the formal inverse of relative Rota-Baxter operators, crossed homomorphisms on Lie 2-groups are also studied.

math-ph↗

arXiv2025

Reflections on Rota-Baxter Lie algebras, the classical reflection equation and Poisson homogeneous spaces

In this paper, first we introduce the notion of reflections on quadratic Rota-Baxter Lie algebras of weight $\lambda$, and show that they give rise to solutions of the classical reflection equation for the corresponding triangular Lie bialgebra ($\lambda=0$) and factorizable Lie bialgebra ($\lambda\neq0$). Then we study reflections on relative Rota-Baxter Lie algebras, and also show that they give rise to solutions of the classical reflection equation for certain Lie bialgebras determined by the relative Rota-Baxter operators. In particular, involutive automorphisms on pre-Lie algebras and post-Lie algebras naturally lead to reflections on the induced relative Rota-Baxter Lie algebras. Finally, we derive Poisson Lie groups and Poisson homogeneous spaces from quadratic Rota-Baxter Lie algebras and relative Rota-Baxter Lie algebras.

math-ph↗

arXiv2025

Poisson homogeneous spaces of Poisson 2-groups

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.

math-ph↗

arXiv2023

The weak Lie 2-algebra of multiplicative forms on a quasi-Poisson groupoid

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.

math.DG↗

arXiv2023

On the reduced space of multiplicative multivectors

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 $A$.

math.AG↗

arXiv2023

Multiplicative forms on Poisson groupoids

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 $A$.

math.DG↗

arXiv2023

The Atiyah class of generalized holomorphic vector bundles

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.

math.AG↗

arXiv2022

Dirac generating operators of split Courant algebroids

Given a vector bundle $A$ over a smooth manifold $M$ such that the square root $\mathcal{L}$ of the line bundle $\wedge^{\mathrm{top}}A^\ast \otimes \wedge^{\mathrm{top}}T^\ast M$ exists, the Clifford bundle associated to the split pseudo-Euclidean vector bundle $(E = A \oplus A^\ast, \langle \cdot, \cdot \rangle)$, admits a spinor bundle $\wedge^\bullet A \otimes \mathcal{L}$, whose section space can be thought of as that of Berezinian half-densities of the graded manifold $A^\ast[1]$. We give an explicit construction of Dirac generating operators of split Courant algebroid (or proto-bialgebroid) structures on $A \oplus A^\ast$ introduced by Alekseev and Xu. We also prove that the square of the Dirac generating operator gives rise to an invariant of the split Courant algebroid.

math.DG↗

arXiv2021

Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras

In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras play important roles in the whole study. Finally we generalize some results to the Lie group context. In particular, we show that there is a one-to-one correspondence between factorizable Poisson Lie groups and quadratic Rota-Baxter Lie groups.

math-ph↗

arXiv2021

Double Principal Bundles

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.

math.DG↗

arXiv2021

A review of Lie 2-algebras

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.

math.RA↗

arXiv2020

Integration and geometrization of Rota-Baxter Lie algebras

This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight $1$ on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.

math.QA↗

arXiv2020

Linearization of the higher analogue of Courant algebroids

In this paper, we show that the spaces of sections of the $n$-th differential operator bundle $\dev^n E$ and the $n$-th skew-symmetric jet bundle $\jet_n E$ of a vector bundle $E$ are isomorphic to the spaces of linear $n$-vector fields and linear $n$-forms on $E^*$ respectively. Consequently, the $n$-omni-Lie algebroid $\dev E\oplus\jet_n E$ introduced by Bi-Vitagliago-Zhang can be explained as certain linearization, which we call pseudo-linearization of the higher analogue of Courant algebroids $TE^*\oplus \wedge^nT^*E^*$. On the other hand, we show that the omni $n$-Lie algebroid $\dev E\oplus \wedge^n\jet E$ can also be explained as certain linearization, which we call Weinstein-linearization of the higher analogue of Courant algebroids $TE^*\oplus \wedge^nT^*E^*$. We also show that $n$-Lie algebroids, local $n$-Lie algebras and Nambu-Jacobi structures can be characterized as integrable subbundles of omni $n$-Lie algebroids.

math-ph↗

arXiv2019

On Lie Bialgebroid Crossed Modules

We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples $(K,P,Q)$ is established, where $K$ is a co-quadratic Lie algebroid and $(P,Q)$ is a pair of transverse Dirac structures in $K$.

math.QA↗

arXiv2019

Atiyah classes of strongly homotopy Lie pairs

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[α^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class $[α^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and the Atiyah class $[α^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,E)$. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

math.QA↗

arXiv2019

Affine structures on Lie groupoids

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.

math.DG↗

arXiv2018

Coadjoint orbits of Lie groupoids

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.

math.DG↗

arXiv2017

VB-Courant algebroids, E-Courant algebroids and generalized geometry

In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\gl(V)\oplus V$ correspond to complex Lie algebra structures on V.

math.DG↗