arXiv research

Introduces semi-abelian generalized complex structures.

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.

Study of complex and Hermitian structures on specific Lie groups.

The paper classifies invariant structures on complex almost Abelian groups.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

New theory connects non-abelian bundle gerbes to abelian ones.

The paper defines angle structures on 3-manifolds using abelian groups.

Let $M = Γ\backslash G$ be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result of C. Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon for 2-step nilmanifolds. We characterize small def…

Characterizes almost abelian Lie algebras with integrable complex structure

Unique complex structures on specific Lie algebras.

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting gro…

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct p…

Study transcendence of abelian differential periods from bi-algebraic perspective.

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are uno…

Let $Λ$ be a finite abelian group. A dynamical system with transformation group $Λ$ is a triple $(A,Λ,α)$, consisting of a unital locally convex algebra $A$, the finite abelian group $Λ$ and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of $Λ$ on $A$. In this paper we present a new, geometricall…

Characterizes hypercomplex Lie groups and their solvmanifolds.

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebra…

Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.

Study non-Abelian gauge theories using Poisson bracket structures.

Galois action on manifold structures of complex varieties is abelian.

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.

The paper constructs toric vector bundles using spectral networks and non-abelianization.

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

Study locally conformally balanced metrics on specific Lie algebras.

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilp…

We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in …

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie grou…

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the…

In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure group is abelian.

Defines non-Abelian gerbes with connections for physical applications.

Researchers extend geometric quantization to complex Abelian Lie supergroups.

Study of trigonal curves in abelian differentials with specific divisor properties.

Generalizes quandle constructions and defines a multiplication that results in an abelian group.

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.

Frobenius manifold structures on the spaces of abelian integrals were constructed by I. Krichever. We use D-modules, deformation theory, and homological algebra to give a coordinate-free description of these structures. It turns out that the tangent sheaf multiplication has a cohomological origin, while the Levi-Civita…

If $A$ is an abelian quandle and $Q$ is a quandle, the hom set $\mathrm{Hom}(Q,A)$ of quandle homomorphisms from $Q$ to $A$ has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom …

The paper studies the structure of a specific homology group related to mapping class groups.

Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.

Characterizes almost Abelian Lie algebras with special $H$-structures.

The paper constructs a free abelian group from Anosov representations on bundles.

Classifies two-step solvable Lie groups with SKT structures.

The paper finds small exotic 4-manifolds with free abelian groups.