arXiv subjects
Publications and source records attributed to Kaichuan Qi.
We construct a family of finite-rank Lie algebroids realizing, through their anchor maps, the singular foliations of vector fields vanishing to a prescribed order at the origin. The construction is obtained from the sub-adjacent Lie algebroids of a class of left-symmetric algebroids.
We study affine deformations of the cotangent groupoid $T^*\mathcal{G} \rightrightarrows A^*$, governed by a one-form $γ\inΩ^1(\mathcal G^{(2)})$, and characterize the conditions on $γ$ under which this construction is valid. We show that these deformations arise naturally from $\mathbb{S}^1$-central extensions of Lie groupoids via symplectic reduction, and identify the reduced symplectic form as a multiplicative magnetic form. In particular, for Kac-Moody extensions, this construction yields nontrivial deformations of quotient stacks and $\mathbb{S}^1$-gerbes.
For a compact and connected Lie group $G$, we present an explicit construction of an $\mathbb{S}^1$-gerbe over the differentiable stack $[G/G]$ in the framework of $\mathbb{S}^1$-central extensions of Lie groupoids. This gives a complete proof of the construction outlined earlier by Behrend--Xu--Zhang, together with an explicit proof of the differential-form identity stated there without proof. In particular, when $G$ is compact, simple, and simply connected, the Dixmier--Douady class of the resulting gerbe is the canonical generator of ${\rm H}^3_G(G,\mathbb Z)$.
Due to its rich structure and close connection with gauge theory, hyperkähler manifolds have attracted increasing interest. Using infinite dimensional hyperkähler reduction, Kronheimer proved that certain adjoint orbits of complexified semisimple Lie algebras admits hyperkähler structures. Later on, Xu obtained a proof for the existence of hyperkähler structures on adjoint orbits of $\mathfrak{sl}(2,\mathbb{C})$ from the viewpoint of symplectic geometry. This paper aims to thoroughly investigate and elucidate the key differences as well as the underlying connections between two distinct construction methods.