Low dimensional Lie groups admitting left invariant flat projective or affine structures

We prove that any real Lie group of dimension ?5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension ?5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.     

On Post-Lie Algebras, Lie–Butcher Series and Moving Frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on Euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with Euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. Lie–Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie–Butcher series are related to invariants of curves described by moving frames.     

Classification of compact lorentzian 2-orbifolds with noncompact full isometry groups

Among closed Lorentzian surfaces, only flat tori can admit noncompact full isometry groups. Moreover, for every n ≥ 3 the standard n-dimensional flat torus equipped with canonical metric has a noncompact full isometry Lie group. We show that this fails for n = 2 and classify the flat Lorentzian metrics on the torus with a noncompact full isometry Lie group. We also prove that every two-dimensional Lorentzian orbifold is very good. This implies the existence of a unique smooth compact 2-orbifold, called the pillow, admitting Lorentzian metrics with a noncompact full isometry group. We classify the metrics of this type and make some examples.     

Metric properties on Lie groups of polynomial volume growth

It is well known that we have an algebraic characterization of connected Lie groups of polynomial volume growth: a Lie group G has polynomial volume growth if and only if it is of type R. In this paper, we shall give a geometric characterization of connected Lie groups of polynomial volume growth in terms of the distance distortion of the subgroups. More precisely, we prove that a connected Lie group G has polynomial volume growth if and only if every closed subgroup has a polynomial distortion in G.     

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.     

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

A Lie algebra L is called 2-step nilpotent if L is not abelian and [L,L] lies in the center of L. 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.     

Partial Dirac cohomology and tempered representations

The tempered representations of a real reductive Lie group G are naturally partitioned into series associated with conjugacy classes of Cartan subgroups H of G. We define partial Dirac cohomology, apply it for geometric construction of various models of these H–series representations, and show how this construction fits into the framework of geometric quantization and symplectic reduction.     

The dirac operator and the principal series for complex semisimple lie groups

The Dirac operator plays a fundamental role in the geometric construction of the discrete series for semisimple Lie groups. We show that, at the level of K-theory, the Dirac operator also plays a central role in connection with the principal series for complex connected semisimple Lie groups. This proves the Connes-Kasparov conjecture for such groups.     

Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group

We show that the kernel of an irreducible unitary representation π of the group algebra L¹(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.     

New HKT manifolds arising from quaternionic representations

We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K?hler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in ${SL(n,\mathbb{H})}$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K?hler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K?hler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.     

On Non-Abelian Coverings Over the Liouville Equation

Treating the hyperbolic Liouville equation as the flat connections equation on the semisimple Lie algebra A ₁, we investigate relationships between zero-curvature representations of the Liouville equation and its Bäcklund transformations provided by a special one-dimensional coverings. Formal deformations of these Bäcklund transformations and integration in nonlocal variables are studied.     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

On the de Rham cohomology of differential and algebraic stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids.Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence.We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E₁-term consists entirely of vector bundle valued cohomology groups.Our theory works for differentiable, holomorphic and algebraic stacks.     

Simple Lie algebras having extremal elements

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type.     

Geometric quantization, complex structures and the coherent state transform

It is shown that the heat operator in the Hall coherent state transform for a compact Lie group K (J. Funct. Anal. 122 (1994) 103-151) is related with a Hermitian connection associated to a natural one-parameter family of complex structures on T^*K. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of T^*K for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin (Comm. Math. Phys. 131 (1990) 347-380) and Axelrod et al. (J. Differential Geom. 33 (1991) 787-902).     

Categorifying the Knizhnik–Zamolodchikov connection

In the context of higher gauge theory, we construct a flat and fake flat 2-connection, in the configuration space of n particles in the complex plane, categorifying the Knizhnik–Zamolodchikov connection. To this end, we define the differential crossed module of horizontal 2-chord diagrams, categorifying the Lie algebra of horizontal chord diagrams in a set of n parallel copies of the interval. This therefore yields a categorification of the 4-term relation. We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, which makes it possible to formulate the notion of an infinitesimal 2-R matrix in a differential crossed module.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Flat connections on configuration spaces and braid groups of surfaces

We construct an explicit bundle with flat connection on the configuration space of n points on a complex curve. This enables one to recover the ‘1-formality’ isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n   points on a surface and an explicitly presented Lie algebra, and to extend it to a morphism from the full braid group of the surface to the semidirect product of the associated group with the symmetric group _Sn$S_n$.     

Geometric quantization for proper moment maps

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we give a solution to a conjecture of Michèle Vergne. To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Paley–Wiener theorem for certain nilpotent Lie groupsJean

We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra 𝔤^* are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)