arXiv research

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

Classifies complex structures on specific nilpotent Lie algebras.

New method constructs nilpotent Lie algebras from quivers.

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

Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.

Criterion for nilpotent Lie groups to have nilsolitons.

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(…

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…

In math.DG/0312243 we developed a general classification scheme for metric Lie algebras, i.e. for finite-dimensional Lie algebras equipped with a non-degenerate invariant inner product. Here we determine all nilpotent Lie algebras l with dim l'=2 which are used in this scheme. Furthermore, we classify all nilpotent met…

The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.

New examples of Lie algebras with ad-invariant metrics found.

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph. We then use properties of the Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie algebra. As an application, if we start with pairs of non-isomorphic Schreier graphs co…

The study characterizes G₂-structures on 2-step nilpotent Lie groups.

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …

Study abelian factors in Lie algebras from graph edge labels.

In this work we study the problem of existence of symplectic structures on free nilpotent Lie algebras. Necessary and sufficient conditions are given for even dimensional ones. The one dimensional central extension for odd dimensional free nilpotent Lie algebras is also considered.

We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the graphs from which they arise are isomorphic.

Every symplectic Lie algebra with degenerate (including non-abelian nilpotent symplectic Lie algebras) has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism …

We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $n\leq9$. On every nilpotent Lie algebra of dimension $\leq 7$, we determine the number of inequivalent nice bases, which can be $0$, $1$, o…

A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected …

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals…

New methods find Ricci-flat metrics on specific Lie groups.

We present the classification of real nilpotent quasi-filiform Lie algebras endowed with a complex structure. A nilpotent Lie algebra g is called quasi-filiform is the nilindex is equal to dim(n)-2. We recall that the filiform case (nilindex =dim(g)-1) has already been studied.

Paper generalizes results from nilpotent Lie algebras to broader types.

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\math…

Study derivations for nilpotent Lie algebras with negative Ricci curvature.

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial pr…

Classifies nilpotent Lie groups with specific $G_2$-structures.

In this note we consider 2-step nilpotent Lie algebras associated with graphs. We prove that 2-step nilpotent Lie algebras $\n$ and $\n'$ associated with graphs $(S, E)$ and $(S', E')$ respectively are isomorphic if and only if $(S, E)$ and $(S', E')$ are isomorphic.

The paper studies algebraic relations of first integrals on specific Lie groups.

A Lie algebra is called nonsoliton if it does not admit a soliton inner product. We demonstrate that the subset of nonsoliton Lie algebras in the moduli space of indecomposable n-dimensional N-graded nilpotent Lie algebras is discrete if and only if n <= 7.

This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.

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

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, …

In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.

Study on nilpotent Lie algebras with specific metrics.

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…

Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.

We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie gr…

Constructs Lie algebras from labeled directed graphs and identifies properties of these algebras.

Defines and extends flat pseudo-Riemannian F-Lie algebras.

This paper studies actions of solvable Lie groups on nilpotent Lie groups.

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplec…

The paper classifies vector fields on 5D nilpotent Lie groups.

The paper studies nice bases for Lie algebras and their properties.

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…