2 research outputs found
From Atiyah Classes to Homotopy Leibniz Algebras
A celebrated theorem of Kapranov states that the Atiyah class of the tangent
bundle of a complex manifold makes into a Lie algebra object in
, the bounded below derived category of coherent sheaves on .
Furthermore Kapranov proved that, for a K\"ahler manifold , the Dolbeault
resolution of is an
algebra. In this paper, we prove that Kapranov's theorem holds in much wider
generality for vector bundles over Lie pairs. Given a Lie pair , i.e. a
Lie algebroid together with a Lie subalgebroid , we define the Atiyah
class of an -module (relative to ) as the obstruction to
the existence of an -compatible -connection on . We prove that the
Atiyah classes and respectively make and
into a Lie algebra and a Lie algebra module in the bounded below
derived category , where is the abelian
category of left -modules and is the universal
enveloping algebra of . Moreover, we produce a homotopy Leibniz algebra and
a homotopy Leibniz module stemming from the Atiyah classes of and ,
and inducing the aforesaid Lie structures in .Comment: 36 page
Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes
Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie
algebras (with space isotropy) have recently been classified in all dimensions.
In this paper, we continue the study of these "maximally symmetric" spacetimes
by investigating their local geometry. For each such spacetime and relative to
exponential coordinates, we calculate the (infinitesimal) action of the
kinematical symmetries, paying particular attention to the action of the
boosts, showing in almost all cases that they act with generic non-compact
orbits. We also calculate the soldering form, the associated vielbein and any
invariant aristotelian, galilean or carrollian structures. The (conformal)
symmetries of the galilean and carrollian structures we determine are typically
infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the
space of invariant affine connections on each homogeneous spacetime and work
out their torsion and curvature.Comment: 62 pages, 3 figures, 4 tables, v2: Matches published version, mistake
corrected in Section 4.1.3., 10.2, 10.3, other minor improvements, added
reference