New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
Examines algebraic conditions for positive sectional curvature in 4D and higher.
problem Determining when the sectional curvature of a Riemannian manifold is positive.
method Analyzes algebraic conditions for sectional positivity in 4D and higher dimensions.
result Characterizes a dense open subset of operators in 4D for sectional positivity.
The duality principle connects algebraic curvature tensors in pseudo-Euclidean spaces.
problem Understanding algebraic curvature tensors in pseudo-Euclidean spaces.
method Proving equivalence between the Jordan-Osserman condition and the Rakić duality principle.
result The Osserman condition and the duality principle are equivalent in the diagonalisable case.
The paper explores structures on 3-Lie algebras, including product, complex, and symplectic.
problem Exploring structures on 3-Lie algebras.
method Introducing phase spaces, product structures, complex structures, and compatibility conditions.
result Four types of special integrability conditions for product, complex, and symplectic structures on 3-Lie algebras.
Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Conditions for exponentiating Lie algebras on complete locally convex spaces are established.
problem Conditions for exponentiating Lie algebras of linear operators on complete locally convex spaces.
method Focus on equicontinuous case, establishing necessary conditions for exponentiation to compact Lie groups.
result Necessary conditions for exponentiation to compact Lie groups are established.
We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integra…
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.
This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.
problem Characterizing LCSKT structures on almost abelian Lie algebras.
method Analyzing the LCSKT condition and its compatibility with other Hermitian structures.
result Classification of LCSKT almost abelian Lie algebras in dimension 6.
Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.
problem Determining when degree-2 Azumaya algebras with C2-actions extend to entire curves.
method Criteria for extension of algebra and new condition for extension with action, testable by computer algebra systems.
result New conditions for extending degree-2 Azumaya algebras with C2-actions over curves.
In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra…
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
Paper describes stability conditions on contraction algebra derived categories.
problem Stability conditions on contraction algebra derived categories.
method Description of full space of stability conditions on derived category.
result Stability manifold is universal cover of hyperplane arrangement.
H-type Lie algebras were introduced by Kaplan as a class of real Lie algebras generalizing the familiar Heisenberg Lie algebra h3. The H-type property depends on a choice of inner product on the Lie algebra g. Among the H-type Lie algebras are the complex Heisenberg Lie algebras $\mathfrak{h}…
The study examines the rigidity of 2-step Carnot groups and their Lie algebra structures.
problem The rigidity of 2-step Carnot groups and their Lie algebra structures.
method Analysis of bi-dimensions and Lie algebra structures to determine rigidity.
result Explicit criteria for rigidity of pseudo H- and J-type algebras are given, and the relation of the J2-condition to rigidity is explored. Extends machine learning models for analytic boundary conditions in differential equations.
problem Inclusion of data in differential equations using symbolic algorithms.
method Combines computer algebra with Gaussian processes and extends to analytic boundary conditions using Gröbner and Janet bases of Weyl algebras.
result Describes divergence-free flow in domains bounded by analytic functions.
We prove that any connected proper Dupin hypersurface in Rn is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in Rn that satisfies a certain finiteness condition. Hence any taut submanifo…
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann th…
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a Lie algebra of symmetries has invariant solutions with respect to this Lie alge…
Study on affine surfaces with specific algebraic properties.
problem Characterize homogeneous affine surfaces with Hessian rank 2.
method Investigate algebra of differential invariants under affine transformation group.
result Organize homogeneous models into inequivalent branches.
Study algebraic concordance groups for non-trivial links.
problem Invariance of signature, Fox-Milnor condition, and Blanchfield pairing under concordance.
method Defined algebraic concordance groups using generalized Seifert matrices.
result Recovery of invariance of signature, Fox-Milnor condition, and Blanchfield pairing for concordant links.
Study characterizes G2-structures on specific Lie groups and identifies harmonic conditions.
problem Characterizing and identifying harmonic G2-structures on almost Abelian Lie groups. method Analyzing left-invariant G2-structures, characterizing torsion forms, and using algebraic conditions. result Established algebraic conditions for harmonic G2-structures and identified admissible torsion classes. Quandle 2-cocycles yield invariant values for knots under certain algebraic conditions.
problem Defining and understanding invariants of knots using quandle 2-cocycles.
method Analyzing algebraic properties of quandle extensions and their impact on knot invariants.
result The invariant values are constant or follow a restricted form for classical knots under specific conditions.
Characterizes non-degenerate cyclic metric Lie algebras.
problem Understanding the structure of non-solvable cyclic metric Lie algebras.
method Using sufficient conditions, cyclic quadruples, and double extension method.
result Complete characterization of non-degenerate cyclic metric Lie algebras.
This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution σ of the adjoint bundle $\mc{A}$ gives rise, given certain algebraic conditions, to a unique preferred affine connection ∇…
Study pseudo-bundles of exterior algebras and their Clifford modules, addressing compatibility issues.
problem Compatibility issues in diffeological pseudo-bundles and their duals.
method Analysis of pseudo-bundles of exterior algebras, Clifford actions, and gluing conditions.
result A natural map ensuring commutativity of duals under gluing, also an isometry.
We provide necessary conditions for the Alexander polynomials of algebraically split component-preservingly amphicheiral links. We raise a conjecture that the Alexander polynomial of an algebraically split component-preservingly amphicheiral link with even components is zero. Our necessary conditions and some examples …
We give a homological interpretation of the coefficients of the Hilbert series for an algebra associated with a directed graph and its dual algebra. This allows us to obtain necessary conditions for Koszulity of such algebras in terms of homological properties of the graphs. We use our results to construct algebras wit…
Lie algebras of quotient groups defined under specific conditions.
problem Conditions for Lie differentiation of quotient groups.
method Diffeological group theory, tangent structure, Lie functor instantiation.
result Lie algebra structure on quotient groups derived from Lie algebras of parent groups.
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.
The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We int…
We study holonomy algebras generated by an algebraic element of the Clifford algebra, or equivalently, the holonomy algebras of certain spin connections in flat space. We provide series of examples in arbitrary dimensions and establish general properties of the holonomy algebras under some mild conditions on the genera…
This work is devoted to the study of a class of Poisson-Lie groups endowed with left invariant metrics. The triples (G,π,<,>) are considered, where G is a simply connected Lie group, ?π is a multiplicative Poisson tensor and <,> is a left invariant riemannian metric such that Hawkins conditions are satisfied. H…
Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.
problem Approximation and interpolation for regular immersions directed by algebraically elliptic cones.
method Uses homotopy-theoretic necessary and sufficient conditions for approximation and interpolation.
result Homotopy-theoretic conditions for approximation and interpolation are satisfied in many cases of interest.
In this paper we introduce the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yok…
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Study of Riemann-Poisson Lie groups with compatibility conditions.
problem Characterizing and constructing Riemann-Poisson Lie groups.
method Left-invariant metrics and Poisson tensors compatible in Lie groups.
result Characterization and construction of Lie algebras up to dimension 5.
A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is stu…
Holomorphic Poisson cohomology on nilmanifolds identified and characterized.
problem Characterizing the cohomology of holomorphic Poisson structures on nilmanifolds.
method Construction of non-trivial holomorphic Poisson structures and identification of conditions for cohomology isomorphisms.
result Conditions for the cohomology of non-trivial holomorphic Poisson structures to be isomorphic to trivial ones.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.
problem Defining operators with varying orders on filtered manifolds.
method Using generalized fixed point algebras and nilpotent Lie groups, they construct a new calculus.
result They establish a new calculus that reflects the behavior of differential operators on filtered manifolds.
We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced functio…
Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.
problem Characterizing abelian structures on odd-dimensional Lie algebras.
method Introducing and analyzing abelian almost contact and almost 3-contact structures, and their compatibility conditions.
result Classification of 5-dimensional Sasakian Lie algebras and 7-dimensional abelian almost 3-contact Lie algebras.
New criteria for effective prolongations of graded Lie algebras.
problem Effective criteria for the finiteness of prolongations of graded Lie algebras.
method Explicit construction of matrices to check the rank for effective criteria.
result Results apply to geometries defined by structure algebras on contact distributions.
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…
In this work we study the problem of existence of symplectic structures on free nilpotent Lie algebras. Necessary and sufficient conditions are given for even dimensional ones. The one dimensional central extension for odd dimensional free nilpotent Lie algebras is also considered.
Develops methods for conditional symmetries of higher-order PDEs, removing unnecessary assumptions.
problem Formulating conditional symmetries for higher-order PDEs with unnecessary assumptions.
method Geometrical formulation and Lie systems approach to derive Lie algebras of conditional symmetries.
result New insights and methods for solving higher-order PDEs.