Dimensional Lie Research Articles

INFRA-SOLVMANIFOLDS OF Sol14

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group Sol1 4 , and more gener- ally, the crystallographic groups of Sol1 4 . The maximal compact subgroup of Isom(Sol1 4 ) is D4 = Z4 ⋊ Z2. We shall exhibit an infra-solvmanifold of Sol1 4 whose holonomy is D4. This implies that all possible holonomy groups do occur; the trivial group, Z2 (5 families), Z4, Z2×Z2 (5 families), and Z4 ⋊ Z2 (2 families).

Commutative n-ary superalgebras with an invariant skew-symmetric form

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras with an invariant form, and a classification of real simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for $L_{\infty}$-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric $n$-ary algebras.

A Class of Finite Dimensional Modular Lie Superalgebras of Special Type

This paper is concerned with the Lie superalgebra S(n, m) of special type over a field of prime characteristic. We construct the modular Lie superalgebra S(n, m) and discuss some properties of this algebra. Then the derivation superalgebra of S(n, m) is determined.

Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras

The purpose of this paper is to obtain some inequalities and certain bounds for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras. Also, we give an inequality for the dimension of the c-nilpotent multiplier of L connected with dimension of the Lie algebras γd (L) and L / Zd−1 (L) . Finally, we compare our results with the previously known result.

Characterization of Nilpotent Lie Algebras Pair by Their Schur Multipliers

Let (N, L) be a pair of finite dimensional nilpotent Lie algebras, in which N is an ideal in L and ℳ(N, L) is the Schur multiplier of the pair (N, L). In 2011, the second author and colleagues proved that if N admits a complement K say, in L with dim N = n and dim K = m, then , where t(N, L) is a non-negative integer. In the present article, we characterize the pairs (N, L) for which t(N, L) = 0, 1, 2, 3, and 4. Also, in a special case when L is a filiform Lie algebra, we determine all pairs (N, L), when t(N, L) = 0, 1,…, 10.

Lorentz geometry of 4-dimensional nilpotent Lie groups

In the present work we classify, up to automorphism, left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $$H_3\times \mathbb {R}$$ and $$G_4$$ . We investigate their geometry, especially holonomy groups and decomposability of these metrics. Finally, we find projective classes of the metrics and prove all of these metrics are also left invariant.

Harmonic morphisms from four-dimensional Lie groups

We consider 4-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.

On generating (2 + 1)-dimensional hierarchies of evolution equations

Two isospectral problems are constructed with the help of a 6-dimensional Lie algebra. By using the Tu scheme, a (1+1)-dimensional expanding integrable couplings of the KdV hierarchy is obtained and the corresponding Hamiltonian structure is established. In addition, the 2-order matrix operators proposed by Tuguizhang are extended to the case where some 4-order matrices are given. Based on the extension, a new hierarchy of 2+1 dimensions is obtained by the Hamiltonian operator of the above (1+1)-dimensional case and the TAH scheme. The new hierarchy of 2+1 dimensions can be reduced to a coupled (2+1)-dimensional nonlinear equation and furthermore it can be reduced to the (2+1)-dimensional KdV equation which has important physics applications. The Hamiltonian structure for the (2+1)-dimensional hierarchy is derived with the aid of an extended trace identity. To the best of our knowledge, generating the (2+1)-dimensional equation hierarchies by virtue of the TAH scheme has not been studied in detail except to previous little work by Tu et al.

Two Expanding Integrable Models of the Geng-Cao Hierarchy

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

SU(3)-holonomy metrics from nilpotent Lie groups

One way of producing explicit Riemannian 6-manifolds with holonomy SU(3) is by integrating a flow of SU(2)-structures on a 5-manifold, called the hypo evolution flow. In this paper we classify invariant hypo SU(2)-structures on nilpotent 5-dimensional Lie groups. We characterize the hypo evolution flow in terms of gauge transformations, and study the flow induced on the variety of frames on a Lie algebra taken up to automorphisms. We classify the orbits of this flow for all hypo nilpotent structures, obtaining several families of cohomogeneity one metrics with holonomy contained in SU(3). We prove that these metrics cannot be extended to a complete metric, unless they are flat.

The centralizer of a 3-dimensional simple subalgebra in the universal enveloping algebra of a 7-dimensional simple Malcev algebra

Under study are the centralizers of 3-dimensional simple Lie subalgebras in the universal enveloping algebra of a 7-dimensional simple Malcev algebra. We find some sets of generators for these centralizers in characteristic not 2 nor 3 and for the subalgebra generated by the centralizer in the central closure of the universal enveloping algebra in characteristic 3. As a corollary of the main theorem we obtain the available description of the center of universal enveloping algebra of a 7-dimensional simple Malcev algebra.

Homogeneous 4-Dimensional Kähler–Weyl Structures

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kahler–Weyl structure; this structure is locally conformally Kahler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kahler–Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.

About Leibniz cohomology and deformations of Lie algebras

We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding whether it has more Leibniz deformations than just the Lie ones. We also give the complete description of the Leibniz (and Lie) universal infinitesimal deformation of the 4-dimensional diamond Lie algebra, used in the WZW model, and study the case of its 5-dimensional analogue by computing Massey products.

Complete Classification of BKM Lie Superalgebras Possessing Special Imaginary Roots

In this article, we give a complete classification of BKM Lie superalgebras possessing special imaginary roots. In this classification, we have also included infinite dimensional BKM Lie superalgebras with finite or infinite number of real or imaginary simple roots.

The Natural Filtration of Finite Dimensional Modular Lie Superalgebras of Special Type

This paper is concerned with the natural filtration of Lie superalgebraS(n,m)of special type over a field of prime characteristic. We first construct the modular Lie superalgebraS(n,m). Then we prove that the natural filtration ofS(n,m)is invariant under its automorphisms.

A note on the isometry group of the hyperbolic plane

This note concerns \(G\), the connected component of the identity of the isometries of the hyperbolic plane. \(G\) operates on the closed unit disk. Here we characterize \(G\) both as a Lie group among the centerless, connected, real semisimple Lie groups of noncompact type as having no two dimensional abelian Lie subgroups, then as a transformation group: any two nontrivial isometries of \(G\) commute if and only if they have the same fixed point set, this property being particular to \(G\) among all isometry groups of irreducible symmetric spaces of noncompact type. From this it follows that any subgroup of \(G\) with a nontrivial center is abelian. In particular, each nilpotent subgroup of \(G\) is abelian. Finally, we uniquely characterize SO\((2,1)\) among the centerless, connected, real simple Lie groups of noncompact type with this property.

Almost Kähler structures on four dimensional unimodular Lie algebras

Let J be an almost complex structure on a 4-dimensional and unimodular Lie algebra g. We show that there exists a symplectic form taming J if and only if there is a symplectic form compatible with J. We also introduce groups HJ+(g) and HJ−(g) as the subgroups of the Chevalley–Eilenberg cohomology classes which can be represented by J-invariant, respectively J-anti-invariant, 2-forms on g. and we prove a cohomological J-decomposition theorem following Draˇghici et al. (2010) [12]: H2(g)=HJ+(g)⊕HJ−(g). We discover that tameness of J can be characterized in terms of the dimension of HJ±(g), just as in the complex surface case. We also describe the tamed and compatible symplectic cones. Finally, two applications to homogeneous J on 4-manifolds are obtained.

A New Integrable Coupling of a Soliton Hierarchy

With the help of the isospectral problem given by Li and Geng,a soliton hierarchy is obtained by use of the Tu Scheme.Finally,a type of new integrable coupling is produced by taking the 5-dimensional Lie algebra.

About Zero Torsion Linear Maps on Lie Algebras

We prove that any zero torsion linear map on a nonsolvable real Lie algebra is an extension of some CR-structure. We then study the cases of (2, ) and the 3-dimensional Heisenberg Lie algebra . In both cases, we compute up to equivalence all zero torsion linear maps on , and deduce an explicit description of the equivalence classes of integrable complex structures on .

Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(ℋ) and the holomorph L of finite dimensional Heisenberg Lie color algebra ℋ graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(ℋ) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.