The purpose of this paper is to define an α-type cohomology, which we call α-type Chevalley–Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley–Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology, we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map α are deformed. Furthermore, we provide a generalization of the grand crochet and study, in a particular case, the α-type cohomology for Hom-Lie bialgebras.

Varieties of the form $G\times S_{\!\text{reg}}$, where $G$ is a complex semisimple group and $S_{\!\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’, Math. Res. Let., to appear] use a Hamiltonian $G$-action to endow $G\times S_{\!\text{reg}}$ with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian $G$-actions, we consider a holomorphic symplectic variety $X$ carrying an abstract integrable system induced by a Hamiltonian $G$-action. Under certain hypotheses, we show that there must exist a $G$-equivariant variety isomorphism $X\cong G\times S_{\!\text{reg}}$.

Lie algebroids cannot always be integrated into Lie groupoids. We introduce a new structure, ‘Weinstein groupoid’, which may be viewed as stacky groupoids. We use this structure to present a solution to the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into such a groupoid.