On the generalization of the Darboux theorem

We consider Lie algebroids over a topological ringed space as quasicoherent sheaves of Lie-Rinehart algebras. We express hypercohomology for a locally free Lie algebroid (not necessarily of finite rank) as a derived functor, and simplify it via Čech cohomology. Furthermore, we define the Hochschild hypercohomology of a sheaf of generalized bialgebras (using a derived functor) and study the cases of the universal enveloping algebroid and of the jet algebroid of a Lie algebroid. In the sequel, we present a version of Hochschild-Kostant-Rosenberg theorem for a locally free Lie algebroid, as well as its dual version.

math.DG↗

arXiv2026

Family index for Fredholm extensions of semi-Fredholm operators

This paper is devoted to an abstract analogue of elliptic boundary value problems, namely, Fredholm realizations of semi-Fredholm operators in a Hilbert space. Such a realization is determined by an abstract boundary condition, which is a subspace in the space of abstract boundary values. We find the $K^0$ index of a family of such abstract boundary value problems, or the $K^1$ index in the self-adjoint case, in terms of the corresponding family of abstract boundary conditions. Our approach is based on passing from a Fredholm operator to its graph. The graph forms a Fredholm pair with the horizontal subspace, and we prove the index formula by deforming the horizontal subspace instead of the operator.

math.DG↗

arXiv2026

Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space

We study the equivalence classes of slice-regular functions $f:Ω\to\mathbb{H}$ on a symmetric slice domain $Ω$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice--regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.

math.DG↗