arXiv subjects
Publications and source records attributed to Maosong Xiang.
Motivated by the extended deformation theory of complex manifolds, we give a combinatorial description of holomorphic polyvector fields on a smooth compact toric variety via Klyachko's filtrations. This yields dimension and formal equivariant-character formulas and recovers the descriptions of Demazure roots and anticanonical sections.
Consider a Lie subalgebra $\mathfrak{l} \subset \mathfrak{g}$ and an $\mathfrak{l}$-invariant open submanifold $V \subset \mathfrak{l}^{\ast}$. We demonstrate that any smooth dynamical twist on $V$, valued in $U(\mathfrak{g}) \otimes U(\mathfrak{g})\llbracket \hbar \rrbracket$, establishes a twistor on the associated quantum groupoid when combined with the Gutt star product on the cotangent bundle $T^\ast L$ of a Lie group $L$ that integrates $\mathfrak{l}$. This result provides a framework for constructing equivariant star products from smooth dynamical twists on those Poisson homogeneous spaces arising from nondegenerate polarized Lie algebras, leveraging the structure of twistors of quantum groupoids.
We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian algebra. Given a Lie algebroid pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we investigate isomorphism classes of infinitesimal deformations of $(L,A)$ modulo automorphisms from exponentials of derivations of $L$ and those from the exponentials of inner derivations of $L$, respectively. For the associated two deformation functors, we find the associated governing $L_\infty$-algebras in the sense of extended deformation theory. Furthermore, when $(L,A)$ is a matched Lie pair, i.e. the quotient $L/A$ is also a Lie subalgebroid of $L$, we investigate isomorphism classes of infinitesimal deformations modulo automorphisms from exponentials of derivations along the normal direction $L/A$. The extended deformation theory of the associated deformation functor recovers the formal deformation theory of complex structures and that of transversely holomorphic foliations.
A Loday-Pirashvili module over a Lie algebra $\mathfrak{g}$ is a Lie algebra object $\bigl(G\xrightarrow{X} \mathfrak{g} \bigr)$ in the category of linear maps, or equivalently, a $\mathfrak{g}$-module $G$ which admits a $\mathfrak{g}$-equivariant linear map $X:G\to \mathfrak{g}$. We study dg Loday-Pirashvili modules over Lie algebras, which is a generalization of Loday-Pirashvili modules in a natural way, and establish several equivalent characterizations of dg Loday-Pirashvili modules. To provide a concise characterization, a dg Loday-Pirashvili module is a non-negative and bounded dg $\mathfrak{g}$-module $V$ paired with a weak morphism of dg $\mathfrak{g}$-modules $α\colon V\rightsquigarrow \mathfrak{g}$. Such a dg Loday-Pirashvili module resolves an arbitrarily specified classical Loday-Pirashvili module in the sense that it exists and is unique (up to homotopy). Dg Loday-Pirashvili modules can be characterized through dg derivations. This perspective allows the calculation of the corresponding twisted Atiyah classes. By leveraging the Kapranov functor on the dg derivation arising from a dg Loday-Pirashvili module $(V,α)$, a Leibniz$_\infty[1]$ algebra structure can be derived on $\wedge^\bullet \mathfrak{g}^\vee\otimes V[1]$. The binary bracket of this structure corresponds to the twisted Atiyah cocycle. To exemplify these intricate algebraic structures through specific cases, we utilize this machinery to a particular type of dg Loday-Pirashvili modules stemming from Lie algebra pairs.
Filippov n-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov n-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the n-ary bracket of any Filippov n-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov n-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank n vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.
For any regular Courant algebroid $E$ over a smooth manifold $M$ with characteristic distribution $F$ and ample Lie algebroid $A_E$, we prove that there exists a canonical homological vector field on the graded manifold $A_E[1] \oplus (TM/F)^\ast[2]$ such that the resulting dg manifold $\mathcal{M}_E$, which we call the minimal model of the Courant algebroid $E$, encodes all cohomological information of $E$. Indeed, the standard cohomology of $E$ can be identified with the cohomology of the function space on $\mathcal{M}_E$, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.
Motivated by recent developments of $\infty$-categorical theories related to differential graded (dg for short) Lie algebras, we develop a general framework for locally finite $\infty$-$\mathfrak{g}$-modules over a dg Lie algebra $\mathfrak{g}$. We show that the category of such locally finite $\infty$-$\mathfrak{g}$-modules is almost a model category in the sense of Vallette. As a homotopy theoretical generalization of Loday and Pirashvili's Lie algebra objects in the tensor category of linear maps, we further study weak Loday-Pirashvili modules consisting of $\infty$-morphisms from locally finite $\infty$-$\mathfrak{g}$-modules to the adjoint module $\mathfrak{g}$. From the category of such weak Loday-Pirashvili modules over $\mathfrak{g}$, we find a functor that maps to the category of Leibniz$_\infty$ algebras enriched over the Chevalley-Eilenberg dg algebra of $\mathfrak{g}$. This functor can be regarded as the homotopy lifting of Loday and Pirashvili's original method to realize Leibniz algebras from Lie algebra objects in the category of linear maps.
For any regular Lie algebroid $A$, the kernel $K$ and the image $F$ of its anchor map $ρ_A$, together with $A$ itself fit into a short exact sequence, called Atiyah sequence, of Lie algebroids. We prove that Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence. That is, the Atiyah and Todd classes of $A$ restrict to the Atiyah and Todd classes of the bundle $K$ of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution $F \subseteq T_M$ on the other hand.
For the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F \subseteq T_M \otimes_{\mathbb{R}} \mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ and establish a canonical isomorphism with the Hochschild cohomology of the algebra of functions on leaf space in terms of transversal polydifferential operators of $F$. In particular, for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ associated with a complex manifold $X$, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomology $HH^{\bullet}(X)$ of the complex manifold $X$. As an application, we show that the Duflo-Kontsevich type theorem for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ implies the Duflo-Kontsevich theorem for complex manifolds.
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.
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 $\mathcal{A}$.
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 $L$.
In this paper, we study the Atiyah class and Todd class of the DG manifold $(F[1],d_F)$ corresponding to an integrable distribution $F \subset T_{\mathbb{K}} M = TM \otimes_{\mathbb{R}} \mathbb{K}$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that these two classes are canonically identical to those of the Lie pair $(T_{\mathbb{K}} M, F)$. As a consequence, the Atiyah class of a complex manifold $X$ is isomorphic to the Atiyah class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$. Moreover, if $X$ is a compact Kähler manifold, then the Todd class of $X$ is also isomorphic to the Todd class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$.