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…
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Knot invariants from XC-structures on Sweedler algebra are trivially determined.
problem Defining and characterizing knot invariants from XC-structures.
method Examining XC-structures on the Sweedler algebra and their relation to knot invariants.
result Knot invariants from XC-structures on Sweedler algebra are completely determined by the framing of the knot.
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with -estimates.
result Similarity structures converge to classical algebraic objects as .
We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. …
The abstract defines -structures and connects them to octonion algebras.
problem Classifying -structures and understanding their geometric properties.
method Established an isomorphism between -structures and octonion algebras over .
result The classification of -structures agrees with a parametrisation of octonion algebras with isometric norm.
The paper models and deforms A-infinity structures for bordered knot algebras.
problem Understanding A-infinity structures for bordered knot algebras.
method Combinatorial model and weighted deformation of A-infinity structures.
result Explicit combinatorial model for bordered knot algebras' A-infinity structure.
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).
Classifies complex structures on specific nilpotent Lie algebras.
problem Classifying complex structures on nilpotent Lie algebras with minimal center.
method Classification through algebraic and geometric methods.
result Space of complex structures on specific Lie algebras up to isomorphism.
Geometric deformations preserve post-Lie algebra structure in regularity structures.
problem Deriving geometric deformations of post-Lie algebras.
method Extending geometrical notions of torsion and curvature, deriving compatibility conditions.
result Derives a pre-Lie structure for regularity structures, isomorphic to a post-Lie algebra.
We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure tha…
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
On the algebra generated by math.DG
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.
problem Characterize coKähler structures on Lie algebras.
method Use Fino-Vezzoni correspondence to relate coKähler Lie algebras to Kähler Lie algebras.
result Complete the flat case for odd-dimensional Lie algebras, proving coKähler structures exist.
Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…
It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend o…
Study SKT and Kähler structures on specific Lie algebras.
problem Characterize SKT and Kähler structures on solvable Lie algebras with codimension two nilradical.
method Classify and construct new examples of SKT solvable Lie algebras.
result Provide a classification of SKT Lie algebras in dimension six and extend SKT nilpotent Lie algebras to higher dimensions.
AIDN uses deep learning to represent algebraic structures.
problem Building learning systems to uncover algebraic laws from data.
method AIDN is a deep learning algorithm that represents algebraic objects using neural networks.
result AIDN can robustly compute representations of various algebraic structures.
The pong algebramath.GT
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure…
Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.
problem Characterizing solvable symplectic Lie algebras.
method Symplectic double extension process.
result Classifies Lie algebras of dimensions up to 6 and proves structural theorems.
A Levi-Malcev type decomposition for -step solvable Lie algebras with a complex structure
problem Decomposition of -step solvable Lie algebras with a complex structure
method Proving a Levi-Malcev type decomposition
result Fino-Vezzoni conjecture holds for -step solvable unimodular Lie algebras
One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functori…
A new algebraic structure emerges from reductive homogeneous spaces.
problem Understanding the algebraic properties of tangent bundles.
method Defined a new algebraic structure based on connections and torsion.
result Post-Lie-Yamaguti algebra is a new algebraic structure related to Lie-Yamaguti algebras.
Study on a specific type of Lie algebras with Kähler and contact properties.
problem Characterizing and classifying transversely Kähler almost contact metric Lie algebras.
method Analyzing properties of Lie algebras with contact forms and Kähler structures, considering center dimensions and quotient properties.
result Classification of 5-dimensional η-Einstein transversely Kähler almost contact metric Lie algebras.
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…
The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.
problem Characterizing and understanding nilpotent complex structures on stratified Lie algebras.
method Introduced a new descending series to prove a new characterization of nilpotent complex structures and examined whether these structures preserve the strata.
result Found that there exists a -invariant stratification on a step 2 nilpotent Lie algebra with a complex structure.
The study examines extensions of Lie algebras with specific geometric structures.
problem Conditions for preserving geometric structures in Lie algebra extensions.
method Analyzes extensions of Sasakian and Frobenius-Kähler Lie algebras.
result Conditions for maintaining Sasakian or Frobenius-Kähler structures after extensions.
Twilled L(ie)-R(inehart) algebas generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an almost twilled pre-LR algebra, which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR-structures in terms of c…
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.
Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.
problem Investigate the existence of complex curves in hypercomplex nilmanifolds.
method Analyze quaternionic-solvable hypercomplex structures on nilpotent Lie algebras and prove the absence of complex curves in complex manifolds.
result Prove the non-existence of complex curves in complex manifolds associated with quaternionic-solvable hypercomplex structures.
New algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an algebra is associated with each Dirac-Jacobi structure.
result There is a one-to-one correspondence between MC elements of the algebra and small deformations of the Dirac-Jacobi structure.
We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex pro…
Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise t…
Real algebraic structures help classify overtwisted contact 3-spheres.
problem Classifying overtwisted contact structures on 3-spheres.
method Using real algebraic functions and open book decompositions.
result Most overtwisted contact structures are real algebraic.
We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sa…
On the classification of left-invariant para-Kähler structures on four-dimensional Lie groupsmath.DG
The paper classifies para-Kähler structures on Lie groups.
problem Classifying para-Kähler structures on Lie groups.
method Classification based on symplectic Lie algebras, finding compatible para-complex structures and pseudo-Riemannian metrics.
result Explicit forms of para-complex structures and pseudo-Riemannian metrics are found.
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…
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…
Study -Einstein Sasakian structures on Lie algebras, dividing cases based on center dimension.
problem Investigate -Einstein Sasakian structures on Lie algebras.
method Divide cases based on center dimension, use theory of normal -algebras and modifications of Hermitian Lie algebras to construct examples.
result Construct new examples of -Einstein Sasakian Lie algebras and solvmanifolds.
On Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structuresmath.DG
We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by…
Left-invariant Kähler and semi-para-Kähler structures on some six-dimensional unsolvable Lie groupsmath.DG
Research explores Kähler and semi-para-Kähler structures on specific Lie groups.
problem Existence of Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups.
method Examines four specific Lie algebras and their structures.
result One Lie algebra admits Kähler metrics, others admit semi-para-Kähler and semi-Kähler structures.
The paper generalizes the number of complex structures on metric Lie algebras.
problem How many orthogonal bi-invariant complex structures exist on metric Lie algebras?
method Developed a unique orthogonal decomposition into irreducible factors for metric Lie algebras.
result There are either 0 or 2^k such complex structures, with k the number of irreducible factors.
In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac struct…