Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric bilinear form on the Lie algebra g and the corresponding central extension of Lg, we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form on G arises as an obstruction. This involves the construction of a 2-form on PG with differential the pull-back of the Cartan form. In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the Cartan form, and for their G-equivariant extensions.
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.
This short note gives a geometric interpretation of the Atiyah class of a Lie pair. It proves that it vanishes if the subalgebroid is the kernel of a fibration of Lie algebroids. In other words, the Atiyah class of a Lie pair vanishes if the subalgebroid is the fiber of an ideal system in the Lie algebroid. In order to prove this, a new characterisation of the ideal condition for an infinitesimal ideal system is found.
Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics $h_{p,q}$ depending on two parameters with $p\in\mathbb{R}$ and $q\geq0$. This family has been introduced recently and possesses interesting geometric properties. If $p=q=0$ we recover the Sasaki metric and when $p=q=1$ we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with an Euclidean prod…
We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal bundles for the (complexified) string group and their morphisms. Our main results are a classification of string algebroids in terms of Cech cohomology, and the construction of a locally complete famil…
In this thesis we study geometric structures from Poisson and generalized complex geometry with mild singular behavior using Lie algebroids. The process of lifting such structures to their Lie algebroid version makes them less singular, as their singular behavior is incorporated in the anchor of the Lie algebroid. We develop a framework for this using the concept of a divisor, which encodes the singularities, and show when structures exhibiting such singularities can be lifted to a Lie algebroid built out of the divisor. Once one has successfully lifted the structure, it becomes possible to st…
Transitive Lie algebroids and \textbf{Q}-manifolds
Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米翁·斯托伊洛数学研究所）；Unité Mixte de Recherche 5127 CNRS, Université Savoie Mont Blanc, Laboratoire de Mathématiques (LAMA)（CNRS联合研究单位5127，萨瓦-蒙布朗大学，数学实验室）
We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid $A$ over a compact manifold $M$ vanishes unless $A=TM$, and prove a general Künneth formula. As applications we give a short proof of a vanishing result for the Euler characteristic of a principal bundle calculated using invariant differential forms, and show that the cohomology of certain Lie algebroids are exterior algebras. The latter result can be seen as a generalization o…
Let $A \Rightarrow M$ be a Lie algebroid. In this short note, we prove that a pull-back of $A$ along a fibration with homologically $k$-connected fibers, shares the same deformation cohomology of $A$ up to degree $k$.
To a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as the obstruction to the existence of $\mathbf{L}$-invariant fibrewise affine connections on the homogeneous space $\mathbf{L}/\mathbf{A}$. For Lie groupoid pairs with vanishing Atiyah class, we show that the left $\mathbf{A}$-action on the quotient space $\mathbf{L}/\mathbf{A}$ can be linearized. In addition to giving an alternative proof of a result of Calaque about the Poincare-Birkhoff-Witt map for Lie algebroid pairs with vanishing Atiyah…
Connections, metrics and Higgs fields on complex fiber bundles
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Tsinghua University（清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type corresponding to the open stratum and also in an iterated sense, with connecting equivariant fibrations over the boundary hypersurfaces covering the resolutions of the other strata. This structure descends to a resolution of the quotient as a stratified space. For an Abelian group action the equivariant K-theory can then be described in terms of bundles over the bases `dressed' by…
