This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,657 papers · 148 categories
Trend · papers per month
48 results for “affine structure groups”
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
Constructs generalized Frobenius manifolds for specific Weyl groups.
problem Creating structures for orbit spaces of Weyl groups.
method Applying a previously established construction method to specific Weyl groups.
result Generalized Frobenius manifold structures constructed for and .
The goal of this paper is to provide a method, based on the theory of extensions of left-symmetric algebras, for classifying left-invariant affine structures on a given solvable Lie group of low dimension. To better illustrate our method, we shall apply it to classify complete left-invariant affine structures on the os…
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
A special symplectic Lie group is a triple such that is a finite-dimensional real Lie group and is a left invariant symplectic form on which is parallel with respect to a left invariant affine structure . In this paper starting from a special symplectic Lie group we show how to ``defo…
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.
Affine Artin groups have a finite classifying space.
problem Proving the conjecture for affine Artin groups.
method Dual Garside structures, Euclidean isometries, and shellability of noncrossing partitions.
result Affine Artin groups have a finite classifying space.
The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…
In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of L…
A -dimensional Lie group equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on . Relatively to this affine structure we show that the left invariant Poisson tensor corresponding to $\om^+$ is po…
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…
Study the geometry of a Lie group using Hessian and flat affine structures.
problem Understanding the geometry of a specific Lie group.
method Examined using bi-invariant Hessian metric and flat affine structure, focusing on curvatures, causal structure, and developed map.
result Determined curvatures, tidal force, and Jacobi vector fields of the Hessian metric.
We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.
New complex manifolds found with flat structure.
problem Finding compact complex manifolds with flat affine structure.
method Using Lie groups with left-invariant complex structure.
result Retrieved Inoue surfaces in 2D.
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study th…
The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.
problem Exploring the relationship between 3d gravity and Chern-Simons theory.
method A variational problem of Chern-Simons type on a principal fiber bundle with general affine group structure is studied. The connection is established through a generalized notion of extension and reduction of connections.
result Established a correspondence between 3d gravity and Chern-Simons theory using affine group connections.
Complete left-invariant affine structures on solvable non-unimodular three-dimensional Lie groupsmath.DG
In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.
The study examines closed manifolds with ray nil-affine structures and their completeness.
problem Characterizing closed manifolds with ray nil-affine structures.
method Analyzing properties of nilpotent spaces and flag manifolds, applying rank conditions and automorphism group actions.
result Closed manifolds are either complete or their developing map is a cover onto the complement of a nil-affine subspace.
Let G be a Lie group with Lie algebra $ \Cal G: = T_εG$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter…
We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
This paper is a review of the twistor theory of irreducible G-structures and affine connections. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections, while the situation with holonomies of…
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental for…
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…
Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cas…
We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalis…
Constructs special Kähler structures on Lie groups.
problem Creating special Kähler structures on Lie groups.
method Introducing twisted cartesian product and double extension process.
result Characterizes left invariant flat special Kähler structures.
Builds geometric structures for algebraic groups over real closed fields.
problem Characterizing and decomposing algebraic groups over specific valued fields.
method Real algebraic geometry to construct and analyze affine buildings.
result Computed stabilizers and obtained group decompositions.
We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras , where is a commutative algebra. These affine Lie algebras are natural generalizations of and the corresponding Lie grou…
Study rigidifies torus bundles under first Betti number constraints.
problem Understanding the structure of torus fibrations under first Betti number restrictions.
method Established rigidity results and necessary/sufficient conditions for topological splitting.
result Classification of torus bundles under specific Betti number constraints.
Classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on three-dimensional Lorentzian Lie groups.
method Examined canonical and perturbed canonical connections, as well as Kobayashi-Nomizu connections and perturbed Kobayashi-Nomizu connections.
result Classified affine Ricci solitons on three-dimensional Lorentzian Lie groups with product structure.
Closed ray affine manifoldsmath.DG
Study closed manifolds with rank one ray structures, proving completeness or covering properties.
problem Characterize closed manifolds with specific affine structures.
method Analyze the developing map and automorphism group properties.
result Closed manifolds with rank one ray structures are either complete or cover the complement of an affine subspace.
We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of …
The results of the paper concern the topological structure of complete riemannian manifolds with cyclic holonomy groups and low-dimensional orientable complete flat manifolds. We also discuss related results such as the affine classification of orientable complete flat 4-manifolds, an algebraic criterion of an affine e…
This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira…
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct a…
Study of symmetries in deformed q-map spaces reveals a complex group structure.
problem Understanding the symmetries of deformed q-map spaces.
method Analyzing the group of isometries in deformed q-map spaces, focusing on the semidirect product structure.
result The group of isometries always contains a semidirect product of an affine group and a Heisenberg group.
Conelike radiant structuresmath.DG
Develops radiant structures for statistical manifolds.
problem Statistical manifolds with radiant vector fields.
method Formulates Einstein equations for special statistical structures.
result Conelike radiant structures exist and have canonical normalizations.
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…