Survey on algebraic fibers of group extensions and their finiteness properties.
problem Existence and finiteness of algebraic fibers in group extensions.
method Analysis of abstract and pro-p groups. result Finiteness properties of algebraic fibers in group extensions.
Study Lie algebras with specific bilinear forms, finding exceptions.
problem Characterizing Lie algebras with symmetric, invariant, and nondegenerate bilinear forms.
method Analyzing structural properties and exceptions of Lie algebras.
result Rare exceptions to the properties of Lie algebras with these bilinear forms.
Graphs define Lie algebras with geometric properties studied.
problem Classifying geometric properties of Lie algebras defined by graphs.
method Constructing Lie algebras from directed graphs and studying their geometric properties.
result Conditions for dense sets of smoothly closed geodesics in nilmanifolds.
Expands on graded Poisson algebras, their properties, and applications.
problem None explicitly stated; focuses on overview and properties.
method Overview and discussion of properties and applications.
result Provides detailed overview of graded Poisson algebras and their contexts.
The paper studies linearisation and splitting properties for vector fields and algebras.
problem Linearisation and splitting properties for vector fields and algebras.
method Formal linearisation problem in the framework of graded coalgebras, explicit recursive construction, and characterisation of linearisable algebras.
result A formal vector field is linearisable if and only if it satisfies a splitting property, providing a streamlined proof of Basto-Gonçalves' theorem.
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
Paper explores algebraic, topological, and combinatorial properties of singular virtual braids.
problem Understanding singular virtual braids and their properties.
method Algebraic relations, topological and combinatorial bijections, presentations.
result A bijection between singular abstract braids and singular virtual braids, leading to a presentation of the singular pure virtual braid monoid.
Projectors defined in virtual Temperley-Lieb algebra with properties similar to Jones-Wenzl projectors.
problem Defining projectors in the virtual Temperley-Lieb algebra.
method Generalizing Jones-Wenzl projectors and defining new projectors with specific properties.
result Uniqueness of the projector fn and explicit formula for fn. Relates two types of skein algebras using explicit correspondences.
problem Defining and relating stated and internal skein algebras.
method Explicit correspondence between stated and internal skein algebras, distinguishing between left and right boundary edges, proving excision properties.
result Agrees with excision properties of stated skein algebras under specific conditions.
Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.
problem Understanding the structure of convolution algebras on Lie groupoids.
method Analyzing smooth functions and invariant subsets to prove H-unitality.
result H-unitality of groupoid algebras and their quotients, leading to excision properties.
Survey on foliations and diffeomorphism groups.
problem Relationship between algebraic and homotopical properties.
method Survey and analysis of existing literature.
result Explains the connection between diffeomorphism groups and foliations.
Born Lie algebras classified up to 6D, with integrable metrics studied.
problem Classifying and understanding Born Lie algebras.
method Bicross product construction from pseudo-Riemannian Lie algebras.
result Classification of Lie algebras up to 6D with integrable Born structures.
Proves properties of Torelli Lie algebra for surfaces.
problem Properties of Torelli Lie algebra for surfaces.
method Proves two theorems about Malcev Lie algebra associated to Torelli group.
result Stable Koszul property and trivial Johnson homomorphism kernel.
Study biderivations in complete Leibniz algebras, extending Lie algebra results.
problem Defining and studying biderivations in complete Leibniz algebras.
method Analyze biderivations according to two definitions, provide conditions for biderivations, and compare symmetric and skew-symmetric biderivations.
result Necessary and sufficient conditions for biderivations in Leibniz algebras are provided.
New insights link algebraic and geometric properties of connections.
problem Understanding numerical integration on manifolds.
method Relating invariant connections to Lie algebra actions.
result Generalized classical results for invariant connections on algebroids.
We study CR quadrics satisfying a symmetry property (S~) which is slightly weaker than the symmetry property (S), recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics s…
Study indefinite bi-invariant metrics and their conformal properties.
problem Characterize indefinite bi-invariant metrics and their conformal relationships.
method Analyzes Lie algebras under different cases and structures.
result Conformally Einstein properties for various Lie algebras are determined.
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …
Abstract defines and identifies a planar algebra with spin properties.
problem Identifying a specific planar algebra.
method Generators and relations defined, structure studied, identified with Jones' spin planar algebra.
result Identified a specific planar algebra with spin properties.
Study solvable Lie algebras linked to graph structures.
problem Understanding solvable Lie algebras through graph properties.
method Define and study solvable extensions of graph Lie algebras, proving isomorphism conditions.
result Two solvable Lie algebras are isomorphic if and only if their corresponding graphs are isomorphic.
Study topological properties of integrable case on Lie algebra so(4).
problem Topological analysis of integrable case for Euler's equations on so(4).
method Construction of bifurcation diagrams, determination of critical points, description of Liouville tori bifurcations, computation of loop molecules.
result Some topological properties of Kovalevskaya case can be derived from the case on so(4).
New homological results for bordered Floer algebras derived from hypertoric categories.
problem Homological properties of bordered Floer algebras.
method Affine quasi hereditary property of equivariant hypertoric convolution algebras and computation of Ext groups.
result Existence of standard modules and isomorphism of Ext groups to bordered strands dg algebras.
New geometric structures on surfaces generalize complex and real Lie algebra properties.
problem Generalizing geometric structures associated with Lie algebras.
method Define and analyze generalizations of punctual Hilbert schemes for complex and real Lie algebras.
result Construct geometric structures homeomorphic to Hitchin components.
This paper gives an overview of some basic properties of Leibniz algebras. Some of the results were known earlier, but in the article they are accompanied by new simple proofs. Some of the results are new. The article can be viewed as a digest or a mini-manual for the basic theory of Leibniz algebras
Survey on groups acting on non-positive curvature spaces.
problem Understanding groups acting on spaces with non-positive curvature.
method Presentation of examples and properties, linking algebraic/analytic and geometric properties.
result Discussion of links between group properties and space geometry.
We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and D…
The paper introduces pseudo-quotients for algebraic actions and applies them to character varieties.
problem Characterizing algebraic actions and their quotients.
method Introducing pseudo-quotients as a weak version of quotients for algebraic actions, focusing on purely topological properties.
result Pseudo-quotients are unique up to virtual class in characteristic zero and can be used to compute character varieties.
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…
Paper defines nearly Frobenius algebras and explores their properties.
problem Defining and studying algebras without traces.
method Survey of foundational results and applications.
result Characterization of nearly Frobenius algebras and their properties.
Study geometric properties of symmetric matrices with repeated eigenvalues.
problem Investigate geometric properties of symmetric matrices with repeated eigenvalues.
method Explicitly compute the volume of the intersection with the sphere and prove an Eckart-Young-Mirsky-type theorem.
result Prove connections to Real Algebraic Geometry and Random Matrix Theory.
We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form).…
The aim of this note is to introduce the notion of a D-Lie algebra and to prove some elementary properties of D-Lie algebras, the category of D-Lie algebras, the category of modules on a D-Lie algebra and extensions of D-Lie algebras. …
Explains a property of algebras related to quantum field theories.
problem Explains a property of algebras encoding line defects in quantum field theories.
method Physical explanation of a property of quantized algebras using dualities and field theories.
result Physical explanation of a large center in quantized algebras when the deformation parameter is a root of unity.
New groups algebraically fibre with high-dimensional hyperbolic groups.
problem Finding new quasi-isometry classes of hyperbolic groups.
method Constructing infinitely many hyperbolic groups as finite-index subgroups of right-angled Coxeter groups.
result Groups algebraically fibre with finitely presented kernels, expanding finiteness properties.
Proves properties of complex algebraic varieties and local systems.
problem Properties of complex algebraic varieties and local systems.
method Analytic Zariski open subsets and algebraic maps.
result Trivializing covering spaces of complex algebraic varieties.
The paper extends cellular algebras with relative orderings and explores their properties.
problem Generalizing cellular algebras with different partial orderings.
method Classification and construction of simple modules, characterizations, and examples.
result Examples of relative cellular algebras that are not cellular.
New symplectic invariants linked to odd sphere bundles.
problem Understanding symplectic manifolds through Aoo-algebras.
method Equivalence of symplectic Aoo-algebras to de Rham algebras on sphere bundles.
result Symplectic Aoo-algebras can define intersection theory.
We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how thes…
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…
Affine structures on Lie groupoids are studied, showing rich algebraic properties.
problem Understanding affine structures on Lie groupoids.
method Analyzing affine k-vector fields, k-forms, and (p,q)-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.
We study nilmanifolds admitting Anosov automorphisms by applying elementary properties of algebraic units in number fields to the associated Anosov Lie algebras. We identify obstructions to the existence of Anosov Lie algebras. The case of 13-dimensional Anosov Lie algebras is worked out as an illustration of the techn…
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.
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…
The paper studies Laplacians on smooth distributions and proves they are multipliers in C∗-algebras.
problem Understanding spectral properties of Laplacians on generalized smooth distributions.
method Survey of generalized smooth distributions, proof of Laplacian as a multiplier in foliation C∗-algebra. result Laplacians on smooth distributions define unbounded multipliers in foliation C∗-algebras. We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, C∞-algebraic) symplectic geometry and calibrated geometr…
We introduce two K-theories, one for vector bundles whose fibers are modules of vertex operator algebras, another for vector bundles whose fibers are modules of associative algebras. We verify the cohomological properties of these K-theories, and construct a natural homomorphism from the VOA K-theory to the associa…
Holomorphic quantum modular forms linked to knot volumes.
problem Understanding algebraic properties of quantum modular forms.
method Analyzing descendant state integrals for specific knots.
result Illustrated algebraic properties for the (-2,3,7)-pretzel knot.