arXiv research

Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.

We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry.

This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra co…

Paper introduces a new symbol map for differential symmetry breaking operators.

Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.

Study of symmetry distributions in Lorentzian naturally reductive nilmanifolds.

Generalizes pseudo-product structures with abnormal extremals.

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overde…

The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a b…

New CR hypersurfaces in complex space with specific properties.

Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.

This work reviews left-invariant optimal control problems on Lie groups.

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…

Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.

Study of symmetries in 4D Lie groups.

We show certain symmetry of the dimensions of cohomologies of the funda- mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basic cohomology. As applications, we show that the polycyclic fundamental groups of compact Sasakian manifolds are virtually nilpotent and Sasakian solvmanifolds are…

We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symm…

A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. An automorphism of a dessin is a permutation of the edges of the underlying graph which preserves the colouring of the vertices and extends to an orientation-preserving self-homeomorphism of the supporting surface. A dessin…

Study of sub-Riemannian problem on specific Lie groups, revealing symmetries and bounds.

The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex a…

We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we prove it here for nilpotent models. We also prove that in any dimension the trans…

Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.

We consider the octonionic self-duality equations on eight-dimensional manifolds of the form $M_8=M_4\times \R^4$, where $M_4$ is a hyper-Kähler four-manifold. We construct explicit solutions to these equations and their symmetry reductions to the non-abelian Seiberg-Witten equations on $M_4$ in the case when the gauge…

An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($\dim M=5$), and for $\dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolo…

This paper reinterprets Khovanov-Sano symmetries using BV formalism.

We demonstrate the agreement between the Higgs branches of two N=2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E_6 flavor symmetry. In mathematical terms, we demonstrate…

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extre…

Nilpotent quandles have simple characterizations and are closely related to nilpotency.

Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.

New method constructs nilpotent Lie algebras from quivers.

Study knot invariants using automorphism groups of free nilpotent groups.

The paper generalizes cyclic metrics in homogeneous Finsler geometry.

We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact $G_2$ holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive ex…

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.

Classifies complex structures on specific nilpotent Lie algebras.

We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…

$Γ$-structures are weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting $Γ$-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of $Γ$-structures on manifolds which are nilpotent…

This paper classifies Ricci soliton subgroups in a specific type of nilpotent group.

Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.

Completes classification of G2-structures on specific nilpotent Lie groups.

New findings on complex structures in nilpotent Lie algebras and pseudo-Kähler geometry.

The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.

This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.

Criterion for nilpotent Lie groups to have nilsolitons.

We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex pro…

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.