Three-dimensional Lie Research Articles

On the Cerbo Conjecture on Lie Groups with the Left-Invariant Lorentzian Metric

Manifolds with constraints on tensor fields include Einstein manifolds, Einstein-like manifolds, conformally flat manifolds, and a number of other important classes of manifolds. The work of many mathematicians is devoted to the study of such manifolds, which is reflected in the monographs of A. Besse, M. Berger, M.-D. Cao, M. Wang.
 Ricci solitons are one of the natural generalizations of Einstein's metrics. If a Riemannian manifold is a Lie group, one speaks of invariant Ricci solitons.
 Invariant Ricci solitons were studied in most detail in the case of unimodular Lie groups with left-invariant Riemannian metrics and the case of low dimension. Thus, L. Cerbo proved that all invariant Ricci solitons are trivial on unimodular Lie groups with left-invariant Riemannian metric and Levi-Civita connection.A similar result up to dimension four was obtained by P.N. Klepikov and D.N. Oskorbin for the non-unimodular case.
 We study invariant Ricci solitons on three-dimensional unimodular Lie groups with the Lorentzian metric.The study results show that unimodular Lie groups with left-invariant Lorentzian metric admit invariant Ricci solitons other than trivial ones. In this paper, a complete classification of invariant Ricci solitons on three-dimensional unimodular Lie groups with leftinvariant Lorentzian metric is obtained.

Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups

Riemann solitons are generalized fixed points of the Riemann flow. In this note, we study left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups. We completely classify left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.

Об инвариантных солитонах Риччи на трехмерных метрических группах Ли с полусимметрической связностью

Об инвариантных солитонах Риччи на трехмерных метрических группах Ли с полусимметрической связностью

∗-Ricci tensor on almost Kenmotsu 3-manifolds

In this paper, we obtain the expressions of the ∗-Ricci operator of a three-dimensional almost Kenmotsu manifold [Formula: see text] and find that the ∗-Ricci tensor is not symmetric for [Formula: see text]. We obtain a necessary and sufficient condition for the ∗-Ricci tensor to be symmetric and proved that if the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-[Formula: see text]-manifold [Formula: see text] is symmetric, then [Formula: see text] is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Further, it is shown that the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-manifold [Formula: see text] is parallel if and only if [Formula: see text] is ∗-Ricci flat. In addition, [Formula: see text] satisfying [Formula: see text] is locally isometric to [Formula: see text]. Finally, we discuss about [Formula: see text]-parallel ∗-Ricci tensor on almost Kenmotsu 3-manifolds.

Levi flat CR structures on 3D Lie algebras

We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties.

Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension

We extend a recent classification of three-dimensional spatially isotropic homogeneous spacetimes to Chern-Simons theories as three-dimensional gravity theories on these spacetimes. By this we find gravitational theories for all carrollian, galilean, and aristotelian counterparts of the lorentzian theories. In order to define a nondegenerate bilinear form for each of the theories, we introduce (not necessarily central) extensions of the original kinematical algebras. Using the structure of so-called double extensions, this can be done systematically. For homogeneous spaces that arise as a limit of (anti-)de Sitter spacetime, we show that it is possible to take the limit on the level of the action, after an appropriate extension. We extend our systematic construction of nondegenerate bilinear forms also to all higher-dimensional kinematical algebras.

Projections of Semisimple Lie Algebras

It is proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra. Over fields with the same restrictions, we give a lattice characterization of a three-dimensional simple split Lie algebra and a direct product of a one-dimensional algebra and a three-dimensional simple nonsplit one.

Classification of three-dimensional zeropotent algebras over an algebraically closed field

ABSTRACTA nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

Quadratic Hamilton–Poisson Systems in Three Dimensions: Equivalence, Stability, and Integration

Quadratic Hamilton---Poisson systems on three-dimensional Lie---Poisson spaces are considered. The homogeneous (positive) semidefinite systems are classified up to linear isomorphism; an exhaustive and nonredundant list of 23 normal forms is exhibited. For each normal form, the stability nature of the equilibria is determined. Each normal form is explicitly integrated, with the exception of three families of systems. Based on the analysis of the normal forms, some simple invariants are identified.

A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

An integrable extension of a soliton hierarchy associated with so(3,ℝ) and its bi-Hamiltonian formulation

A hierarchy of soliton equations is constructed from an extended spectral problem associated with the three-dimensional special orthogonal real Lie algebra so(3,ℝ) by means of zero curvature equations. The bi-Hamiltonian structure of this hierarchy is also presented via the trace identity.

Colength of the variety generated by a three-dimensional simple Lie algebra

The variety generated by a three-dimensional simple Lie algebra over a field of characteristic zero was studied rather well. The study of this variety is continued in the paper and a formula for calculation of its colengths is presented.

The normal sub-Riemannian geodesic flow on E(2) generated by a left-invariant metric and a right-invariant distribution

We consider a sub-Riemannian problem on the three-dimensional solvable Lie group E(2) endowed with a left-invariant metric and a right-invariant distribution. The problem is based on construction of a Hamilton structure for the given metric by the Pontryagin maximum principle.

The bienergy of unit vector fields

The bienergy of a unit vector field on a Riemannian manifold \((M, g)\) is defined to be the bienergy of the corresponding map \((M, g)\mapsto (T_{1}M, g_{\mathrm{S}})\), where the unit tangent sphere bundle \(T_{1}M\) is equipped with the restriction of the Sasaki metric \(g_{\mathrm{S}}\). The constrained variational problem is studied, where variations are confined to unit vector fields, and the corresponding critical point condition characterizes biharmonic unit vector fields. Finally, we completely determine invariant biharmonic unit vector fields in three-dimensional unimodular Lie groups and the generalized Heisenberg groups \(H(1, r), r\ge 2\).

On the Lengths of Certain Chains of Subalgebras in Lie Algebras

In this paper we study the lengths of certain chains of subalgebras of a Lie algebra L: namely, a chief series, a maximal chain of minimal length, a chain of maximal length in which each subalgebra is modular in L, and a chain of maximal length in which each subalgebra is a quasi-ideal of L. In particular we show that, over a field F of characteristic zero, a Lie algebra L with radical R has a maximal chain of subalgebras and a chain of subalgebras all of which are modular in L of the same length if and only if L = R, or and L/R is a direct sum of isomorphic three-dimensional simple Lie algebras.

A Classification of the Symmetries of Uniform Discrete Defective Crystals

Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometrical symmetries of these structures can be computed in terms of the changes of generators of the discrete subgroup which preserve the discrete set of points. Here a classification of the symmetries for the discrete subgroups of a particular class of three-dimensional solvable Lie group is presented. It is a fact that there are only three mathematically distinct types of Lie groups which model uniform defective crystals, and the calculations given here complete the discussion of the symmetries of the corresponding discrete structures. We show that those symmetries corresponding to automorphisms of the discrete subgroups extend uniquely to symmetries of the ambient Lie group and we regard these symmetries as (restrictions of) elastic deformations of the continuous defective crystal. Other symmetries of the discrete structures are classified as ‘inelastic’ symmetries.

Group Classification of a General Bond-Option Pricing Equation of Mathematical Finance

We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra. We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.

The geodesic flow of a sub-Riemannian metric on a solvable Lie group

We consider the sub-Riemannian problem on the three-dimensional solvable Lie group SOLV+. The problem is based on constructing a Hamiltonian structure for a given metric by the Pontryagin Maximum Principle.

INVARIANT METRICS ON THE IWASAWA MANIFOLD

We compute the automorphism group of the complex three-dimensional Heisenberg Lie algebra and study its action (by isometries) on the set of inner products of R6. As a consequence, we give a description of the moduli space of invariant metrics on the Iwasawa manifold. We also show that the action mentioned above is not polar and has a minimal orbit