Homological algebra used to study local equivalence of complex rings.
problem Local equivalence of bounded complexes over polynomial rings.
method Homological algebra approach
result Results have been proved in many places in the literature.
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48 results for “algebraic equivalence”
Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of -algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship betw…
Study algebraic obstructions to knot-like complex realizability.
problem Algebraic obstructions to knot-like complex realizability.
method Classification of local equivalence classes over .
result Classification answers a question about knot-like complexes.
Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras are homotopy equivalent. We apply this result to -algebroids to show th…
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the -category of curved Lie algebras using homotopy theory of algebras over a complete operad.
result Equivalence of -categories between Lie algebroids and certain kinds of curved Lie algebras.
Alternative algebraic characterization of 3D cobordisms category.
problem Characterize the category of 3D cobordisms algebraically.
method Define and prove equivalence of Hopf algebra categories and use a functor to present new axioms.
result Existence of a functor between Hopf algebra categories and implications for .
Undecidability proved for DG algebras problems.
problem Stable isomorphism, quasi-isomorphism, and Morita equivalence problems for semifree DG algebras.
method Essentially autonomous solutions by Gemini Deep Think and Aletheia.
result Proved undecidability of problems for semifree DG algebras.
We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the -algebras governing …
Let be a simply connected Lie group with Lie algebra . We show that the following categories are naturally equivalent. The category , of sufficiently smooth modules over the DG-algebra of singular chains on . The category of representations of the DG-Lie algeb…
Study of Pascal algebra matrices and their jet bundle map for vector bundles.
problem Defining and studying Pascal algebra matrices and their map on jet bundles.
method Identifying Pascal algebra matrices, showing generator well defines Pascal map, using it for intrinsic contact definition.
result Intrinsic definition of point-wise contact between Hermitian vector bundles using unitary equivalence of Pascal maps.
An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the sm…
This paper extends a category equivalence to an A-infinity quasi-equivalence for compact Lie groups.
problem Equivalence of DG categories for smooth singular chains on Lie groups.
method Construction of A-infinity quasi-isomorphisms and use of Van Est map, De Rham theorem.
result Extension of equivalence to A-infinity quasi-equivalence for compact Lie groups.
Study shows weak homotopy equivalences for complete minimal surfaces.
problem Understanding complete minimal surfaces and their properties.
method Analyzes algebraic null immersions and conformal minimal immersions.
result Inclusion and differential mappings are weak homotopy equivalences.
Models for self-equivalences and diffeomorphisms of manifolds.
problem Classifying spaces of self-equivalences and diffeomorphisms of manifolds.
method Construct rational models using equivariant algebraic methods.
result Formula for rational cohomology of classifying spaces.
We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantiz…
The paper studies differential operator invariants and equivalence under Lie pseudogroups.
problem Understanding invariants and equivalence of differential operators under Lie pseudogroups.
method Analysis of invariants, use of n-invariants, and application of local symplectomorphisms as an example.
result Normal forms and solutions to equivalence problems for differential operators.
Study shows algebraic structure in 2-dimensional CW-complex cobordisms.
problem Characterize cobordisms of 2-dimensional CW-complexes.
method Algebraic characterisation using Hopf algebras and symmetric monoidal categories.
result Category of cobordisms is equivalent to a freely generated Hopf algebra.
We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched over the disc. Our results addresses the problem to decide equiva…
Paper defines multiplicities for quaternion eigenvalues without complex matrix concepts.
problem Defining multiplicities for quaternion eigenvalues without traditional matrix concepts.
method Introduces two definitions for algebraic and geometric multiplicities equivalent to classical definitions.
result Definitions are equivalent to classical ones and prove all properties easily.
Genus 2 Goeritz Equivalence in math.GT
Study on Goeritz equivalence in genus 2 Heegaard splitting of .
problem Understanding Goeritz equivalence of curves in genus 2 Heegaard splitting of .
method Introduce Goeritz equivalence of curves, present algebraic obstructions, and provide examples.
result Algebraic obstructions to Goeritz equivalence of simple closed curves are computed and demonstrated.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
We prove the existence of a map of spectra between connective topological K-theory and connective algebraic L-theory of a complex -algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural e…
In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras , which are not topologically equivalent to the Homflypt polynomial. We then present the algebra which is the appropriate Temperley-Lieb analogu…
A natural occurrence of shift equivalence in a purely algebraic setting constitutes the subject matter of the following short exposition.
Review and Prospect of Algebraic Research in Equivalent Framework between Statistical Mechanics and Machine Learning Theorycond-mat.stat-mech
Paper reviews algebraic research in machine learning theory.
problem Understanding phase transitions in machine learning models.
method Algebraic approaches in statistical mechanics.
result Algebraic methods are essential for analyzing machine learning models with singularities.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
Establishes equivalence between models of derived stacks.
problem Tackles the equivalence between different models of derived geometry.
method Uses Quillen equivalence to show categories of higher derived stacks are equivalent.
result Shows equivalence among models of derived manifolds, Carchedi-Roytenberg, Behrend-Liao-Xu, and Alexandrov-Kontsevich-Schwarz-Zaboronsky.
We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these equivalences.
Paper establishes equivalence between algebraic and functorial QFTs.
problem Challenges in globally hyperbolic Lorentzian bordisms and their impact on field theories.
method Introduces a new concept of bordisms and pseudo-operads to address challenges, defining FQFTs as pseudo-multifunctors.
result Equivalence theorem between globally hyperbolic Lorentzian FQFTs and AQFTs.
Equivalence proven between algebraic stability and geometric stability.
problem Equivalence of algebraic and geometric stability criteria.
method Algebraic proof of equivalence, existence and uniqueness of minimal centers.
result Existence and uniqueness of minimal optimal destabilizing centers.
The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the mom…
In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double exten…
Let be a finite group. Noncommutative geometry of unital -algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these…
The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.
New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.
problem Classifying conformally superintegrable systems in arbitrary dimensions.
method Algebraic geometric approach extended to conformally superintegrable systems.
result An algebraic equation governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems.
For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.
In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
Proves knots in handlebodies can be represented as plats of braids.
problem Representing knots in handlebodies as plats of braids.
method Established Hilden braid group and mixed braid group, proving algebraic equivalence.
result Plat closure representation for knots in handlebodies.
-Algebras from Lie Pairsmath.DG
New algebra structure derived from Lie pairs.
problem Constructing -algebras from Lie pairs.
method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
Every metric symplectic Lie algebra has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism classes of metric symplectic Lie algebras and give a complete list …
Local equivalence found between maximally symmetric rolling and flat Cartan distributions.
problem Establishing local equivalence between maximally symmetric rolling and flat Cartan distributions.
method Using complex parametrisation of su(2), a change of coordinates maps the maximally symmetric rolling -distribution to the flat Cartan distribution.
result Local equivalence between maximally symmetric rolling and flat Cartan distributions established.
New bialgebra structures for relative Poisson algebras are introduced.
problem Extending bialgebra structures from commutative differential algebras to relative Poisson algebras.
method Introducing new bialgebra structures (relative PCA bialgebras) and using commutative 2-cocycles.
result New bialgebra structures (relative PCA bialgebras) are equivalent to certain Manin triples.
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.