Let be a differential graded coalgebra, the Adams cobar construction and the dual algebra. We prove that for a large class of coalgebras there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies and . Thi…
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48 results for “graded coalgebras”
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.
We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham coh…
Invariants of 3-manifolds using modified Hopf G-coalgebra.
problem Constructing invariants for 3-manifolds.
method Purely Hopf G-coalgebra construction with modified integral.
result New invariants for 3-manifolds.
In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction.
Given a group G, we use involutary Hopf G-coalgebras to define a scalar invariant of flat G-bundles over 3-manifolds. When G=1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.
Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.
problem Developing invariants for flat bundles over 4-manifolds.
method Utilizing Hopf -triplets and colored trisection diagrams.
result Involutory Hopf -triplets yield well-defined invariants of -colored trisection diagrams.
In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth functions with compact support on G has a natural coalgebra structure over C_c(M) wh…
On Jordan (Co)Algebrasmath.DG
We present new results about Jordan algebras and Jordan coalgebras, and we discuss about their connections with the Yang-Baxter equations.
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
The Morse complex is shown to be an infinite functor.
problem Understanding the structure of Morse complexes as infinite functors.
method Showed the Morse complex of a compact Lie monoid can be given the structure of an f-bialgebra and defined an ∞-functor.
result Obtained two other ∞-functors mapping manifolds and actions to their Morse complexes.
The study analyzes stochastic Lie systems and their applications in various models.
problem Analyzing stochastic differential equations on manifolds.
method Coalgebra method for Hamiltonian stochastic Lie systems.
result New examples of stochastic Lie systems and Hamiltonian stochastic Lie systems are analyzed.
We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provi…
This paper extends Riemannian geometry concepts to Hom--commutative algebras.
problem Extending Riemannian geometry concepts to Hom--commutative algebras.
method Recalling Hom--commutative algebras, developing metric, connection, torsion, curvature, and differential operators.
result Established differential calculus and symplectic/Poisson structures on Hom--commutative algebras.
We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.
We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative) Hopf algebra of pointed loops on a surface. In the last version I added sections…
Paper corrects and expands stochastic Lie systems theory.
problem Stochastic Lie systems and their properties.
method Corrected stochastic Lie theorem, introduced new stochastic Lie systems.
result Stochastic Lie systems can differ significantly between Stratonovich and Itô approaches.
Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.
problem Constructing and studying new invariants for 4D 2-handlebodies.
method Defining invariants for pairs , using unimodular ribbon Hopf coalgebras.
result Decomposition formulas for original invariants in terms of refined ones.
A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group , we introduce a notion of a modular crossed -category and show that such a category gives rise to a 3-dimensional HQFT with target sp…
Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.
problem Computing and enumerating gradings of Lie algebras.
method Constructive approach to torsion-free gradings, computation of maximal grading, enumeration of all gradings.
result Computation of a maximal grading and enumeration of all torsion-free gradings.
The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle co…
Introduces graded bundles as a generalization of vector bundles.
problem Lack of a unified framework for geometric structures.
method Definition and discussion of graded spaces, graded bundles, and their linearizations.
result Graded bundles provide a natural generalization of vector bundles.
The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.
problem Finding general solutions for Lie systems.
method Develops mixed superposition rules for Lie systems with imprimitive Lie algebras and semidirect sums.
result Extends coalgebra method to Lie systems of partial differential equations.
Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-…
Three definitions of graded vector bundles are shown to be equivalent.
problem Defining graded vector bundles in three different ways.
method Equivalence of categories among sheaves, graded modules, and locally trivial graded manifolds.
result All three approaches to graded vector bundles are equivalent.
The paper integrates DGLA to DGLG using HCPs and Hopf algebras.
problem Integrating DGLA to DGLG.
method Definition of DGLG and HCPs, use of graded Hopf algebras.
result Construction of DGLG from DGLA and vice versa.
Three new types of graded Lie groups are constructed and analyzed.
problem Generalizing Lie theory to -graded geometry.
method Direct geometric construction and functor-of-points perspective.
result Isomorphic Lie algebras of the new graded Lie groups.
The paper introduces -graded manifolds and their properties.
problem Defining functions and vector fields on -graded manifolds.
method Introduced -graded manifolds from local models, defined functions and vector fields, and reviewed correspondences.
result Properties and definitions of -graded manifolds.
This paper develops a theory of graded manifolds in differential geometry.
problem Defining consistent global descriptions of graded manifolds with mixed graded coordinates.
method Using sheaves of graded commutative associative algebras on topological spaces.
result Resolved known issues in the definition of graded manifolds, especially those involving mixed graded coordinates.
Graded Transformerscs.LG
Graded Transformers embed algebraic structure in neural networks through graded transformations.
problem Efficiently modeling hierarchical and structured data in neural networks.
method Introduces Linearly Graded Transformer (LGT) and Exponentially Graded Transformer (EGT) with graded scaling operators.
result Establishes rigorous guarantees and improved efficiency for structured data.
Graded Generalized Geometrymath.DG
Combines generalized and graded geometry to explore new structures.
problem Exploring new structures on generalized tangent bundles of graded manifolds.
method Introduces canonical brackets, Dirac structures, and generalized complex structures.
result Canonical bracket on a generalized tangent bundle of a graded manifold.
Duality for graded manifoldsmath.DG
The paper explores dual structures in graded manifolds.
problem Understanding dual objects in graded manifolds.
method Developed dual objects for graded bundles and applied them to double vector bundles and graded bundles of degree 2.
result Elegant characterizations of double vector bundles and graded bundles of degree 2.
Peer grading in a MOOC did not improve over simple mean grading.
problem Improving on simple mean grading in peer evaluation.
method Applied statistical and machine learning methods to aggregate peer grades.
result None of the machine learning methods improved over simple mean grading.
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
New connections adapted to graded structures defined for a class of manifolds.
problem Defining connections for graded bundles and their applications.
method Formalism of supermanifolds to describe Lie algebroids, defining weighted A-connections.
result Existence of adapted connections on graded bundles and double vector bundles.
The paper defines Z-graded hom-Lie superalgebras and explores their properties.
problem Understanding the structure and properties of Z-graded hom-Lie superalgebras.
method Definition and exploration of Z-graded hom-Lie superalgebras, invariant bilinear forms, and simplicity conditions.
result Maximal and minimal Z-graded hom-Lie superalgebras for local hom-Lie superalgebras are identified, and conditions for simplicity are checked.
In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group define…
Characterizes fundamental groups of disjointly tree-graded spaces.
problem Understanding fundamental groups of complex geometric structures.
method Defines and analyzes disjointly tree-graded spaces, characterizing their fundamental groups.
result Fundamental groups of disjointly tree-graded spaces embed into inverse limits of free products of fundamental groups of pieces.
We define a formal exponential map for graded manifolds.
problem No specific problem stated; focus on the method and results.
method Algebraic definition of a formal exponential map for -graded manifolds.
result Simple new construction of a Fedosov type resolution and extension of the Emmrich--Weinstein theorem.
The paper studies prolongations of Lie algebras associated with pseudo -type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo -type Lie algebras.
method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…
Study higher arity self-distributive operations and their cohomology.
problem Understanding cohomology groups of higher arity operations.
method Introduced mutually distributive n-ary operations and defined a cohomology theory.
result Geometric interpretation of cohomology in terms of framed links.
In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate…
Graded Jet Geometrymath.DG
Constructs graded jet bundles for Z-graded manifolds and vector bundles.
problem Generalizing jet manifolds to Z-graded structures for differential equations.
method Directly constructs the sheaf of sections of the k-th order jet bundle of a Z-graded vector bundle.
result Establishes a graded version of Atiyah Lie algebroid.
Wavelet scattering method improves glioma grade prediction accuracy.
problem Improving glioma grading accuracy for prognosis and treatment planning.
method Wavelet scattering feature extraction, dimensionality reduction (PLS), and glioma grade prediction (SVM, LR, RF).
result Glioma grade prediction AUC increased to 0.99 with multimodal features, 13% higher than traditional radiomics.
Monoidally graded manifoldsmath.DG
Extends manifold theory to -graded manifolds.
problem Generalizing manifold theory to non-integer grading.
method Introduces -graded manifolds and proves Batchelor's theorem.
result Proves Batchelor's theorem for -graded manifolds.
A new algorithm improves medical image grading accuracy.
problem Improving automatic grading of medical images for disease severity or risk score.
method Sparse range-constrained learning (SRCL) algorithm integrating sparse representation and grading.
result Improves accuracy in cup-to-disc ratio computation and cataract grading.