This paper classifies all left-invariant Hermitian and hyper-Hermitian structures on Lie groups whose commutator groups are one-dimensional. Furthermore, for each natural number n = 2k and n = 4k (k > 1 for the case of hyper-Hermitian structures), we construct an n-dimensional Lie group with a one-dimensional commutator group that possesses left-invariant Hermitian and hyper-Hermitian structures, respectively.

We study the Hitchin maps on the moduli spaces of Hitchin sheaves and parabolic Hitchin sheaves on a nodal integral curve [Formula: see text]. We study their fibres, the BNR correspondences and the relation of the restriction of the Hitchin map with very stable bundles. As an application, in the ”coprime” case we prove that the parabolic Poincaré sheaf on [Formula: see text] is parabolic stable where [Formula: see text] is a fixed determinant moduli space of parabolic sheaves on [Formula: see text].

Let [Formula: see text] be an immersed hypersurface without umbilical points, one can define the Möbius metric [Formula: see text] and the Möbius form [Formula: see text], which are invariant under the Möbius transformation group of [Formula: see text]. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of [Formula: see text] with respect to the Möbius metric [Formula: see text]. In this paper, we classify completely the Willmore hypersurfaces with vanishing Möbius form (i.e. [Formula: see text]) in [Formula: see text] up to a Möbius transformation of [Formula: see text].

Braided algebras are associative algebras endowed with a Yang–Baxter operator that satisfies certain compatibility conditions involving the multiplication. Along with Hochschild cohomology of algebras, there is also a notion of Yang–Baxter cohomology, which is associated to any Yang–Baxter operator. In this paper, we introduce and study a cohomology theory for braided algebras in dimensions 2 and 3, that unifies Hochschild and Yang–Baxter cohomology theories. We show that its second cohomology group classifies infinitesimal deformations of braided algebras. We provide infinite families of examples of braided algebras, including Hopf algebras, tensorized multiple conjugation quandles, and braided Frobenius algebras. Moreover, we derive the obstructions to higher deformations, which lie in the third cohomology group. Relations to Hopf algebra cohomology are also discussed.

We show that pseudoconvex hypersurfaces of finite Catlin multitype with holomorphically nondegenerate models in [Formula: see text] do not admit exotic symmetries. As a consequence, this leads to a sharp 2-jet determination result for such manifolds. The results provide a necessary step toward understanding local invariants and equivalence of weakly pseudoconvex hypersurfaces.

Subfactors of the hyperfinite II 1 factor with “exotic” properties can be constructed from non-degenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones’ graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley–Lieb–Jones standard invariant and index [Formula: see text], i.e. the index of the Haagerup subfactor. We are led to a conjecture pertaining to Jones indices of irreducible, hyperfinite subfactors.

We prove that a normal hyperplane section of the Segre variety [Formula: see text] is K-unstable with respect to any polarization if [Formula: see text] or it is not smooth.

Let [Formula: see text] be a smooth projective rational variety carrying a regular action of a finite abelian group [Formula: see text]. We give examples of effective computation of the Brauer group of the quotient stack [Formula: see text] in dimensions [Formula: see text] and [Formula: see text] using residues in Galois cohomology and the geometry of fixed loci. In particular, we compute [Formula: see text] for all [Formula: see text]-minimal del Pezzo surfaces.

The concepts of the twisted knot theory and the virtual singular knot theory inspire the introduction of the singular twisted knot theory. This research presents comparable results for singular twisted links, such as the Alexander and Markov theorems derived from virtual knot theory. However, it differentiates itself from virtual singular knot theory by demonstrating that the number of Markov moves can be minimized within this framework. Furthermore, this paper introduces singular twisted virtual braids and explores their monoid structure. We also offer a monoid presentation and a reduced monoid presentation for singular twisted virtual braids.

In this paper, we characterize the existence of nontrivial invariant spinors on maximal flag manifolds associated to complex simple Lie algebras. This characterization is based on the combinatorial properties of their set of positive roots. We also give some bounds for the dimension of the space of invariant spinors in each case.