Five-dimensional K-contact Lie algebras

We introduce a general approach to the study of left-invariant K-contact structures on Lie groups and we obtain a full classification in dimension five. We show that Sasakian structures on five-dimensional Lie algebras with non-trivial center are a relatively rare phenomenon with respect to K-contact structures. We also prove that a five-dimensional solvmanifold with a left-invariant K-contact (not Sasakian) structure is a ${\mathbb S^1}$ -bundle over a symplectic solvmanifold. Rigidity results are then obtained for five-dimensional K-contact Lie algebras with trivial center and for K-contact ??-Einstein structures. Moreover, five-dimensional Sasakian ??-symmetric Lie algebras are completely classified, and some explicit examples of five-dimensional Sasakian pseudo-metric Lie algebras are provided.     

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H _{2n + 1}. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H ₅ or a semidirect product ? ? (H ₃ × ?). In particular, the compact quotient is an S ¹-bundle over a four-dimensional Kähler solvmanifold.     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

Cotangent bundles of 4-dimensional hypercomplex Lie groups

 We study geometrical structures on the cotangent bundle T ^* G of a Lie group G which are left-invariant with respect to the Lie group structure on T ^* G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-K?hler metrics and hyper-K?hler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying In?nü-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures. Received: 4 February 2002 / Revised version: 20 August 2002 Research partially supported by MURST and GNSAGA (Indam) of Italy Mathematics Subject Classification (2000): 53C26, 22E25     

Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian manifolds.     

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting \mathfrakg^* = V₁ ?V₂ {\mathfrak{g}^*} = {V_1} \oplus {V_2} , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.     

Einstein–Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ^± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.      

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S ²ⁿ⁺¹ is the Reeb vector field of a natural Sasakian structure on S ²ⁿ⁺¹. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction. Supported by funds of the University of Lecce and M.I.U.R.(PRIN).     

Half-flat structures on decomposable lie groups

Half-at SU(3)-structures are the natural initial values for Hitchin’s evolution equations whose solutions define parallel G₂-structures. Together with the results of [SH], the results of this article completely solve the existence problem of left-invariant half-at SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras, which refines and clarifies the existing classification of five-dimensional Lie algebras.     

Complex Surfaces with Cat(0) Metrics

We study complex surfaces with locally CAT(0) polyhedral K?hler metrics and construct such metrics on \mathbbCP²{\mathbb{C}P^{2}} with various orbifold structures. In particular, in relation to questions of Gromov and Davis–Moussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of \mathbbCP²{\mathbb{C}P^{2}} of sufficiently high degree their desingularizations are of type K(π, 1).     

Simple and Nearly Simple Deep Matrix Algebras

Deep matrix algebras were originally created by Cuntz (Comm. Math. Phys. 57:173–185, 1977) and McCrimmon (2006). Further study of the associative case was done by the author in Kennedy (2004) and Kennedy (Algebr. Represent. Theory 9:525–537, 2006). In this paper, the associative algebra DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) based on a set X over a field \mathbbK{\mathbb{K}} and various of its subalgebras are studied for the purpose of determining the structure of the associated Lie algebra \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}) and its subalgebras. Several key examples of deep matrix Lie algebras are constructed. These are shown to be either simple or nearly simple depending on the cardinality of the set X. Cartan subalgebras are constructed and two of the key Lie algebras are then decomposed with respect to the adjoint action of these subalgebras. In the process, an infinite dimensional analogue to \mathfraksl₂(\mathbbK)\mathfrak{sl}_2({\mathbb{K}}) is naturally realized as a key subalgebra in deep matrix Lie algebras.     

Odd-dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd-dimensional real Lie algebra $\mathfrak{g}$. We investigate correspondences with even-dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak{g}$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$, obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3-contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.     

Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

Let W _n ( \mathbb K {\mathbb K} ) be the Lie algebra of derivations of the polynomial algebra \mathbb K {\mathbb K} [X] := \mathbb K {\mathbb K} [x ₁,…,x _n ]over an algebraically closed field \mathbb K {\mathbb K} of characteristic zero. A subalgebra L í W_n(\mathbbK) L \subseteq {W_n}(\mathbb{K}) is called polynomial if it is a submodule of the \mathbb K {\mathbb K} [X]-module W _n ( \mathbb K {\mathbb K} ). We prove that the centralizer of every nonzero element in L is abelian, provided that L is of rank one. This fact allows one to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.     

Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksl_n{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl₂\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl₂{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.     

Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products ${\mathbb{C}^{n}\ltimes_{\phi}N}$ of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product ${\mathbb{C}^{n}\times N}$ . As a corollary of this computation, we can compute the Dolbeault cohomology H ^p,q (G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.     

Segal–Bargmann and Weyl transforms on compact Lie groups

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as ólafsson and ?rsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and K_{\mathbb C}{K_{\mathbb C}} . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on K_{\mathbb C}{K_{\mathbb C}} as operators on the Hilbert space L ²(K).     

Nondegenerate graded lie algebras with degenerate transitive subalgebras

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler is analyzed. Transitive irreducible graded Lie algebras over an algebraically closed field of characteristic p > 2 with classical reductive component L ₀ are considered. We show that if a nondegenerate Lie algebra L containes a transitive degenerate subalgebra L′such that dim L′₁ > 1, then L is an infinite-dimensional Lie algebra. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Jacobi Structures on Affine Bundles

Abstract We study affine Jacobi structures （brackets） on an affine bundle π ： A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A^＋ = ∪p∈M Aff（Ap, R） of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A^＋. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited （strongly-affine or affine-homogeneous Jacobi structures） over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.