Ugo Bruzzo
We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie algebroids.
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We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.
Towards differentiation and integration between Hopf algebroids and Lie algebroids
本文在交换 Hopf algebroids 与 Lie-Rinehart algebras 背景下，构造了形式微分与积分的反变函子，揭示了二者的伴随关系，并利用微分函子分析了 Hopf algebroids 态射的几何可分性。
We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.
Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order $n$ deformations to order $n+1$ deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorp…
Cohomology of Lie conformal algebroids
该研究运用λ-括号和李共形代数语言探究李共形代数胚（LCAd）及其表示，提出多种构造方式，发展LCAd上同调理论并建立其与泊松顶点代数（PVA）上同调的联系，发现两者特定上同调间的同构关系。
We describe Lie-Rinehart algebras in the tensor category $\mathcal{LM}$ of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras in $\mathcal{LM}$ to Leibniz algebroids.
We prove that Hochschild cohomology of a certain class of fully group-graded algebras is a Mackey functor. We use the machinery of transfer maps between the Hochschild cohomology of symmetric algebras.
Differential calculus over double Lie algebroids
本文研究双李代数胚的微分学，给出该结构的新实例，构建相应微分演算体系，并将非交换德拉姆复形、双泊松-利希诺维茨上同调纳入其构造特例。
We introduce the category of bicomodules for a comonad in a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [11], and the characterization of coseparable coalgebras coextensions stated in [16].
Stable functors and cohomology theory in abelian categories
该研究通过预包络/预覆盖子范畴上的稳定函子构建阿贝尔范畴的相对完备上同调，并据此刻画多种有限同调维数。
Let $H$ be a Hopf algebra over a field $k$, and $A$ an $H$-comodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.
Cohomology and Extensions of $C_p$-Green Functors of Lie Type
本文提出了李型 $C_p$-Green 函子理论，通过引入等变 Chevalley-Eilenberg 上同调及杯积，建立了奇异扩张等价类与二阶上同调群的对应关系，将经典李代数扩张推广至等变情形。
We define the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures. Representations of a hom-Lie algebroid on a hom-bundle are defined and a cohomology of a regular hom-Lie algebroid with coefficients in a representation is studied. We discuss about relationship between these classes of hom-Gerstenhaber algebras and geometric structures on a vector bundle. As an application, we associate a homology to a regular hom-Lie algebroid and …
Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories
本文研究Hopf代数胚及SAYD模上的测量，证明其诱导循环同调与循环上同调态射，并给出SAYD模范畴及非Σ operad上循环单位comp模的充实结构。
We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.
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