The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.
problem Understanding universal enveloping algebras of Lie-Rinehart algebras.
method Extending a theorem to left Hopf algebroids and applying it to universal enveloping algebras of Lie-Rinehart algebras.
result Provides a crossed product decomposition of universal enveloping algebras for curved and flat connections.
Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is e…
New Boolean algebra method shows knot unknotting number is (c+1)/2.
problem Finding the minimum number of region crossing changes to unknot a knot.
method Boolean algebra applied to region crossing changes.
result Region unknotting number is (c+1)/2 for any knot with crossing number c.
Investigates adjustments on Lie group crossed modules for gauge theory.
problem Existence and classification of adjustments on crossed modules of Lie groups.
method Differentiation/integration correspondence with infinitesimal adjustments; Lie algebra techniques.
result Infinitesimal adjustments exist if and only if the Kassel-Loday class lies in the image of the Chern-Weil homomorphism.
Study shows crossing numbers for algebraic knots can differ by arbitrarily large amounts.
problem Comparing two crossing number definitions for algebraic knots.
method Analyzed Hopf fibration and complex singularities to compare crossing numbers.
result Difference between crossing numbers can be arbitrarily large.
Isomorphism found between filtered calculus and crossed products.
problem Tackles isomorphism in filtered calculus and crossed products.
method Uses natural R-action and structure result for C*-algebra of graded nilpotent Lie groups.
result Found isomorphism between kernel of tangent groupoid and crossed product.
Integrable dynamics explained via geometric maps and cluster algebras.
problem Integrable dynamics in projective geometry.
method Twisted triple crossing diagram maps and cluster integrable systems.
result Cross-ratio dynamics described by geometric R-matrices. Study relative commutants in von Neumann algebras using contraction notions.
problem Understanding relative commutants in group and tracial crossed product von Neumann algebras.
method Introducing contraction notions to study relative commutants.
result Results applied to negatively curved groups and SL(d, Z).
Research decouples Lie algebroids using bicocycle double cross product theory.
problem Understanding decoupling and coupling phenomena in Lie algebroids.
method Bicocycle double cross product realization method.
result Unified product, double cross product, semi-direct product, and cocycle extension frameworks are instances of the general method.
Paper constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
In \cite{PrzytyskiTraczyk} J.H.Przytyski and P.Traczyk introduced an algebraic structure, called {\it a Conway algebra,} and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in \cite{KauffmanLambropoulou} L. H. Kauffman and S. Lambropoulou in…
Abstract sketches historical development of Lie brackets, crossed modules, and Lie-Rinehart algebras.
problem Characterizing and understanding the relationships between Lie brackets, crossed modules, and Lie-Rinehart algebras.
method Historical review and combinatorial group theory considerations.
result The mutual relationship between Lie-Rinehart algebras and Lie brackets, and the historical development of these concepts.
Researchers create spectral triples for twisted crossed products using Kasparov's external product.
problem Constructing spectral triples for twisted crossed products.
method Using Kasparov's external product, the construction of spectral triples for twisted crossed products is achieved.
result The construction of spectral triples for twisted crossed products is possible under suitable assumptions.
Let G be a finite group. Noncommutative geometry of unital G-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…
The concordance orders of many algebraic order two knots of ten or fewer crossings have been heretofore unknown. We use Casson-Gordon invariants and twisted Alexander polynomials to find that, in all but one case, these knots do not have concordance order two. We also find that a certain family of algebraic order two t…
We investigate a relationship between Ozsváth and Szabó's bordered theory and the algebras and bimodules constructed by Khovanov-Seidel. Specifically, we show that (a variant of) a special case of Ozsváth-Szabó's algebras has a quotient which is isomorphic to the Khovanov-Seidel quiver algebra with coefficients in $\ma…
New algebra counts components of arborescent knots and links.
problem Counting components of arborescent knots and links.
method Developed a new algebra called the crossing algebra.
result The crossing algebra counts the number of components for arborescent knots and links.
Classifies prime algebraic tangles up to 14 crossings.
problem Classifying prime algebraic tangles systematically.
method Developed a novel canonical representation to distinguish mutant tangles.
result Increased classification of prime tangles up to 14 crossings.
We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product.…
New method classifies C-boundaries up to 6 crossings.
problem Classifying C-boundaries with up to 6 crossings. method Proposed a new construction method.
result Extended classification of C-boundaries up to 6 crossings. The paper explores rigidity and proximality in dynamical systems, proving new results about C∗-algebras.
problem Understanding rigidity and proximality in dynamical systems and their algebraic counterparts.
method Analyzing crossed products of dynamical systems and their C∗-algebras, focusing on uniform rigidity and proximality. result Uniformly rigid systems are almost reflecting, and certain crossed products are reflecting.
F. Labourie [arXiv:1212.5015] characterized the Hitchin components for PSL(n,R) for any n>1 by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank n swapping algebra, which is the quotient of the swap…
The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C^* algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set c…
We resolve parts (A) and (B) of Problem 1.100 from Kirby's list by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontriv…
Two crossing homomorphisms on braid groups are shown to be equivalent.
problem Comparing two definitions of crossing homomorphisms on braid groups.
method Diagrammatic and algebraic definitions of crossing homomorphisms compared and computed for simple braids.
result Diagrammatic and algebraic crossing homomorphisms are equivalent.
The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.
problem Characterizing flat manifolds that have unique cusp cross-sections in arithmetic hyperbolic manifolds.
method Algebraic characterization of cusp cross-sections in arithmetic hyperbolic manifolds.
result Construction of flat manifolds with unique cusp cross-sections and proof of their existence in all dimensions n≥32. We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with triv…
The paper describes topological properties of arcs and crossings in knot theory.
problem Understanding the topological nature of arcs and crossings in knot theory.
method Topological description of arcs and crossings as isotopy classes of probes, homotopy classes of diagram elements.
result Sets of arcs and crossings are fundamental for algebraic objects like quandles, partial ternary quasigroups, biquandloids, and crossoids.
This work merges 3-anchored bundles into 3-Lie algebroids.
problem Combining two 3-anchored bundles into a unified structure.
method Develops algebraic framework for merging bundles with mutual actions and cocycle terms.
result Unified setting for 3-Lie algebroids and special cases.
Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings…
The study refines algebraic domains with specific boundary conditions.
problem Understanding shapes of regions bounded by real algebraic curves.
method Inductive definition of regions, respecting characteristic finite sets.
result Generalized Poincar'e-Reeb Graphs for new types of regions.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots,…
The paper classifies algebraic curves in 4-balls and their boundaries.
problem Understanding algebraic curves in 4-dimensional balls and their boundaries.
method Analyzing algebraic curves in complex 2-space and their intersections with 4-balls.
result Classification of algebraic curves with up to 5 crossings.
Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a…
We study the dg-Lie algebra f_n generated by the coefficients of the universal translation invariant flat dg-connection on the n-dimensional affine space. We describe its "semiabelianization" (in particular, the universal quotient which is a crossed module of Lie algebras) in terms of closed differential forms of arbit…
Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.
problem Analyzing how the number of trajectories of quadratic differentials changes with variation.
method Proves an analytic wall-crossing formula using Fock-Goncharov coordinates and characterizes birational automorphisms.
result Characterizes certain birational automorphisms and computes Stokes automorphisms.
We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…
We consider diagrams of links in S2 obtained by projection from S3 with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic an…
The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…
In this paper, we investigate the relationship between algebraic soliton metrics and soliton metrics for geometric evolution equations on Lie groups. After discussing the general relationship between algebraic soliton metrics and soliton metrics, we investigate the cross curvature flow and the second order renormalizat…
Classifies flat knots up to 8 crossings using Lyndon words.
problem Classifying flat knots up to a certain number of crossings.
method Using matchings on Lyndon words and various knot invariants.
result Distinguished all flat knots up to 7 crossings except for five pairs.
When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on R3×S1 are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these…
We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial constructi…
The wall-crossing formula for Donaldson invariants of smooth, simply connected four manifolds with b+=1 is shown to be a topological invariant of the manifold for reducible connections with two or fewer singular points. The explicit formulas derived agree with those of Ellingsrud and Gottische and Friedman and Qin f…
We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in $\C^k$ with normal crossings. We construct two potential functions (of first and second kind) of variables labeled by hyperplanes of the arrangement and prove that the matrix coefficient…
In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any…
New invariant from quantum algebra for 3-manifold bundles.
problem Quantum invariants of flat 2-bundles over 3-manifolds.
method From an involutory Hopf algebra graded by a crossed module, constructing a homotopy invariant via χ-colored Heegaard diagrams. result Reduces to Kuperberg invariant when bundle is trivializable.