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…
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.
169,341 papers · 148 categories
Trend · papers per month
48 results for “Affine Lie Algebras”
Affine structures on Lie groupoids are studied, showing rich algebraic properties.
problem Understanding affine structures on Lie groupoids.
method Analyzing affine -vector fields, -forms, and -tensors, and showing their algebraic properties.
result The space of affine structures forms a 2-vector space over multiplicative structures, and affine multivector fields have a Lie 2-algebra structure.
In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.
The paper characterizes flat affine connections on manifolds and Lie groups.
problem Characterizing flat affine connections on manifolds and Lie groups.
method New characterization through affine representations of automorphisms.
result Existence of a Lie group with a flat affine bi-invariant connection.
In this note we prove that every non characteristically filiform Lie algebra is endowed with an affine structure.
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…
Real Lie groups' invariant theory matches that of their affine counterparts.
problem Matching invariant theory of real Lie groups with affine groups.
method Simple remark showing coincidence.
result Invariant theory of real Lie groups equals that of their affine counterparts.
We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space . Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtai…
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.
Left invariant affine structures in a Lie group are in one-to-one correspondence with left-symmetric algebras over its Lie algebra (``over'' means that the commutator coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the mult…
Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.
problem Characterizing contact Lie algebras and their stabilizers.
method Analyzing algebraic Lie algebras of index 1 and their orbits.
result Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.
Affine representations of Lie algebras and geometric interpretation in the case of smooth manifoldsmath.DG
In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations and 1-cohomology classes of the algebra. We classify the affine representations of…
The paper classifies vector fields on 5D nilpotent Lie groups.
problem Classifying left-invariant affine and projective vector fields on 5D nilpotent Lie groups.
method Algebraic characterization and case-by-case analysis of vector fields.
result All projective vector fields are affine, extending classical results.
We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.
The paper identifies all flat CR Lie groups and their structures.
problem Identifying flat CR Lie groups and their structures.
method Analyzing Lie algebras and structures of Lie groups.
result Only specific Lie algebras are Cartan flat non-degenerate CR Lie algebras.
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…
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
This work explores algebraic structures from curvature and torsion in affine connections.
problem Understanding algebraic structures from curvature and torsion in affine connections.
method Post-Lie algebra, D-algebra, and special polynomials.
result A particular class of geometrically special polynomials is generated by torsion and curvature.
Characterizes flat affine symplectic Lie groups and their properties.
problem Characterizing flat affine symplectic Lie groups.
method Using symplectic étale affine representations and central translations.
result Obtains nontrivial examples of flat affine symplectic Lie groups in every even dimension.
Study special affine connections on symmetric spaces and their products.
problem Characterize special affine connections on symmetric spaces.
method Analyze canonical affine connections, introduce special products, and study holonomy Lie algebras.
result Established a correspondence between special affine connections and special products on the Lie algebra.
This paper studies actions of solvable Lie groups on nilpotent Lie groups.
problem Characterizing which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Using Lie algebra properties and semisimple splitting, the paper provides methods to check for such actions.
result A full description of possibilities for actions up to dimension 4.
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.
Positive energy representations of affine algebras and Stokes matrices of the affine Toda equationsmath.DG
Constructs positive energy representations from Toda equations Stokes data.
problem Creating positive energy representations of affine algebras.
method Using Stokes data of tt*-Toda equations to construct representations.
result Illustrates construction with examples in conformal field theory.
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.
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.
Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
Study torsion's impact on 2D affine Killing vectors on homogeneous surfaces.
problem Effects of torsion on affine Killing vectors on homogeneous surfaces.
method Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
result Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
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…
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.
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 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.
An LR-structure on a Lie algebra is a bilinear product, satisfying certain commutativity relations, and which is compatible with the Lie product. LR-structures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LR-str…
Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.
problem Understanding Casimir functions for free nilpotent Lie groups of steps 3 and 4.
method Construction of Casimir functions for free nilpotent Lie groups of steps 3 and 4.
result For 3-step groups, coadjoint orbits are fully described as affine subspaces or direct products of quadrics.
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 classifies special Hessian rank 2 hypersurfaces in 4D space.
problem Classifying hypersurfaces with constant Hessian rank 2.
method Power series method of equivalence, Lie's classification spirit.
result 34 inequivalent terminal branches, each with a nonempty moduli space.
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on .
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 aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear …
Constructs proper affine actions for right-angled Coxeter groups.
problem Proper affine actions for right-angled Coxeter groups.
method Constructs proper actions of right-angled Coxeter groups on O(p,q+1) and its Lie algebra by affine transformations.
result Any virtually special group admits proper affine actions on some R^n.
We classify all of the 4-dimensional linear Poisson structures of which the corresponding Lie algebras can be considered as the extension by a derivation of 3-dimensional unimodular Lie algebras. The affine Poisson structures on R^3 are totally classified.
Notes on flat pseudo-Riemannian manifolds, focusing on their characterization and properties.
problem Characterizing flat pseudo-Riemannian manifolds and their properties.
method Survey of basic concepts in affine and Riemannian geometry, characterization of flat manifolds, and analysis of Lie groups.
result Characterization and properties of flat pseudo-Riemannian Lie groups and their metrics.
Characterizes flat affine connections on manifolds.
problem Understanding flat affine connections on manifolds.
method Characterization through natural affine representation of diffeomorphisms.
result Group of affine transformations acts on R^n with open orbit when dimension > n.
Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.
problem Constructing a Cartan calculus in tangent categories.
method Define scalar multiplication by a commutative ring object to equip tangent bundles with -module structure.
result Every object in tangent categories carries a Cartan calculus of Lie-Rinehart forms.
New classification of complex hypersurfaces in 3D.
problem Classifying simply-transitive Levi non-degenerate hypersurfaces in .
method Novel Lie algebraic approach, new coordinate-free formula for quartic tensor.
result Unique non-tubular model with geometric relations to planar equi-affine geometry.
Study of coloured invariants of torus knots using algebras.
problem Understanding coloured invariants of torus knots .
method Representation theory of principal affine algebras and asymptotic weight multiplicities.
result Limits of renormalized invariants are equal to characters of algebra modules.
Left-symmetric algebras help define affine spheres.
problem Characterizing improper affine spheres in flat affine space.
method Analyzing left-symmetric algebras and their properties.
result Conditions for left-symmetric algebras to yield proper affine spheres.
Uniform Jordan property proven for Lie groups and complex space transformations.
problem Proving Jordan property for Lie groups and complex space transformations.
method Analyzing families of Lie groups and using algebraic groups properties.
result Uniform Jordan property established for Lie groups and complex space transformations.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.