Dimensional Lie Research Articles

Special Hermitian metrics and Lie groups

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω is ∂ ∂ ¯ -closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a 2 n-dimensional SKT Lie algebra g and using a Hermitian flat connection on g we construct a 4 n-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.

Classification of abelian complex structures on 6-dimensional Lie algebras

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 PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk, p maps S1 → ℙ1 is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ1) of holomorphic line bundles on Lℙ1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H0, 1(Lℙ1). The group G of Möbius transformations and its loop group LG act on Lℙ1. We prove that an element of pic(Lℙ1) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ1.

LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON THE SEMI-DIRECT PRODUCT Rn

We provide the set of left-invariant minimal unit vector fields on the semi-direct product R n oPR, where P is a nonsingular diagonal matrix and on the 7 classes of 4-dimensional solvable Lie groups of the form R 3 oPR which are unimodular and of type (R).

Irreducible Representations for the Baby Tits–Kantor–Koecher Algebra

The baby Tits–Kantor–Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac–Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp 4(ℂ).

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

Nilpotent-by-abelian Lie algebras of type FPm

AbstractLet L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

Examples of homogeneous vectors on the fiber space of the associated and the tangent bundles

The paper considers the associated bundle ξ = (G × KG/K, ρξ, G/K, G/K) and the tangent bundle τG/K = (TG/K, πG/K, G/K, Rm), and gives special examples of odd dimensional solvable Lie groups equipped with left invariant Riemannian metric. Some conditions about existence of homogeneous geodesic vectors on the fiber space of ξ and τG/K are proved.

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

For any complex 6-dimensional nilpotent Lie algebra \(\mathfrak{g},\) we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain \(\mathfrak{g}\) by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.

Harmonic morphisms from solvable Lie groups

AbstractIn this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.

ON SOME HYPERCOMPLEX 4-DIMENSIONAL LIE GROUPS OF CONSTANT SCALAR CURVATURE

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.

A note on Ricci signatures

We show that only two types of Ricci signatures cannot be realized by any left-invariant metric on 4-dimensional Lie groups.

Integrable magnetic geodesic flows on lie groups

Right-invariant geodesic flows on manifolds of Lie groups associated with 2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of motion for magnetic geodesic flows is considered and necessary and sufficient condition of integrability in quadratures is formulated. Canonic forms for 2-cocycles of all 4-dimensional Lie algebras are given and integrable cases among them are separated.

Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities

We prove that any totally geodesic hypersurface $N^5$ of a 6-dimensional nearly K\"ahler manifold $M^6$ is a Sasaki-Einstein manifold, and so it has a hypo structure in the sense of \cite{ConS}. We show that any Sasaki-Einstein 5-manifold defines a nearly K\"ahler structure on the sin-cone $N^5\times\mathbb R$, and a compact nearly K\"ahler structure with conical singularities on $N^5\times [0,\pi]$ when $N^5$ is compact thus providing a link between Calabi-Yau structure on the cone $N^5\times [0,\pi]$ and the nearly K\"ahler structure on the sin-cone $N^5\times [0,\pi]$. We define the notion of {\it nearly hypo} structure that leads to a general construction of nearly K\"ahler structure on $N^5\times\mathbb R$. We determine {\it double hypo} structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5-dimensional Lie algebras with non-zero first Betti number. An extension of the concept of nearly K\"ahler structure is introduced, which we refer to as {\it nearly half flat} SU(3)-structure, that leads us to generalize the construction of nearly parallel $G_2$-structures on $M^6\times\mathbb R$ given in \cite{BM}. For $N^5=S^5\subset S^6$ and for $N^5=S^2 \times S^3\subset S^3 \times S^3$, we describe explicitly a Sasaki-Einstein hypo structure as well as the corresponding nearly K\"ahler structures on $N^5\times\mathbb R$ and $N^5\times [0,\pi]$, and the nearly parallel $G_2$-structures on $N^5\times\mathbb R^2$ and $(N^5\times [0,\pi])\times [0,\pi]$.

Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-K\"ahler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-K\"ahler is given.

UNIMODULAR GROUPS OF TYPE ℝ3⋊ ℝ

There are 7 types of 4-dimensional solvable Lie groups of the form <TEX>${\mathbb{R}^3}\;{\times}_{\varphi}\;{\mathbb{R}}$</TEX> which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups <TEX>$({\mathbb{R}^4},\;Nil^3\;{\times}\;{\mathbb{R}\;and\;Nil^4)$</TEX> are well known to have lattices. All the compact forms modeled on the remaining four solvable groups <TEX>$Sol^3\;{\times}\;{\mathbb{R}},\;Sol_0^4,\;Sol_0^'4\;and\;Sol_{\lambda}^4$</TEX> are characterized: (1) <TEX>$Sol^3\;{\times}\;{\mathbb{R}}$</TEX> has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, <TEX>${\mathbb{Z}}_2\;or\;{\mathbb{Z}}_4$</TEX>. (2) Only some of <TEX>$Sol_{\lambda}^4$</TEX>, called <TEX>$Sol_{m,n}^4$</TEX>, have lattices with no non-trivial infra-solvmanifolds. (3) <TEX>$Sol_0^{'4}$</TEX> does not have a lattice nor a compact form. (4) <TEX>$Sol_0^4$</TEX> does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on <TEX>$Sol_0^4$</TEX>. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.

Formal Deformations, Contractions and Moduli Spaces of Lie Algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions.

Actions of Lie Superalgebras on Reduced Rings

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is

On the Dimensions of Commutative Subalgebras and Subgroups of Nilpotent Algebras and Lie Groups of Class 2

We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.

Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2

Nilpotent 6-dimensional Lie algebras over any field of characteristic not 2 are classified. The proof of this classification is essentially constructive: for a given 6-dimensional nilpotent Lie algebra L, following the steps of the proof, it is possible to find a Lie algebra M that occurs in the classification, and an isomorphism L → M . In the proof a method due to Skjelbred and Sund is used, along with a method based on Gröbner bases to find isomorphisms.