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Register a new userAn Atiyah sequence for noncommutative principal bundles
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We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given *-automorphism on the quantum base space lifts to a *-automorphism on the quantum total space that commutes with the underlying structure group.
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We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.
In this paper we construct Cech cohomology groups that form a Gysin-type long exact sequence for principal torus bundles. This sequence is modeled on a de Rham cohomology sequence published in earlier work by Bouwknegt, Hannabuss and Mathai, which was developed to compute the global properties of T-duality in the presence of NS H-Flux.
Given a holomorphic principal bundle $Q, longrightarrow, X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for $text{ad} Q otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When $X,=, G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L, subset, P$ is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kahler metric on $T^*(G/P)$, recovering Biquards description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.
In this paper, a notion of a principal $2$-bundle over a Lie groupoid has been introduced. For such principal $2$-bundles, we produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal $2$-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We constructed a class of principal $mathbb{G}=[G_1rightrightarrows G_0]$-bundles and connections from a given principal $G_0$-bundle $E_0rightarrow X_0$ over $[X_1rightrightarrows X_0]$ with connection. An existence criterion for the connections on a principal $2$-bundle over a proper, etale Lie groupoid is proposed. The action of the $2$-group of gauge transformations on the category of strict and semi-strict connections has been studied. Finally we noted an extended symmetry of the category of semi-strict connections.
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