We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…
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48 results for “quasitriangular r-matrices”
The paper studies T-leaves and stabilizers in Poisson structures on flag varieties.
problem Understanding the structure of T-leaves and their stabilizers in Poisson structures.
method Developed a general theory for T-leaves and leaf stabilizers, applied to specific Poisson structures on flag varieties.
result Described T-leaf decompositions and computed leaf stabilizers and symplectic leaf dimensions.
Reformulates Fock-Rosly Poisson structure using quasi-triangular r-matrices.
problem Defining Fock-Rosly Poisson structure on moduli spaces.
method Using Lie algebra actions and quasi-triangular r-matrices.
result Shows Fock-Rosly structure as mixed product Poisson structure.
The purpose of this paper is to establish a connection between various subjects such as dynamical r-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures developed in dg-ga/9508013 and dg-ga/9611001. In particular, we give a new method of classifying dynamical r-mat…
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf -algebras and coloring Kirby diagrams.
result Invariants defined for 4-manifolds and connections.
New methods reveal colored Jones polynomials from quantum R-matrices and knot invariants.
problem Understanding colored Jones polynomials of knots.
method Two realizations: quantum R-matrices and refined quantum modularity conjecture.
result New insights into knot invariants from quantum R-matrices and matrix conjectures.
In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and -matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{…
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalismmath-ph
Direct proof of Alexander polynomial scaling for L-shaped representations.
problem Proving scaling property of Alexander polynomials for specific representations.
method Direct use of Reshetikhin-Turaev formalism to compute R-matrices.
result Normalized Alexander polynomial for one-hook representations scales with .
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equa…
We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. …
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 -matrices.
We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical -matrices and matched pairs induced by Poisson group actions
New braid group action defined on projective quantum sl(2) modules.
problem Defining a new braid group action on quantum sl(2) modules.
method Action via R-matrix on tensor powers of simple projective modules.
result The action is faithful for the extended representation.
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…
fkcompute calculates a knot invariant from a braid presentation.
problem Computing the Gukov-Manolescu invariant for complex knots and links.
method Three-phase pipeline: braid presentation search, state space encoding, R-matrix multiplication.
result fkcompute efficiently computes the invariant for knots and links up to 12 crossings.
What is a virtual link?math.GT
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L in S X I, a thickened, compact, oriented surface S, such that the link complement (S X I) - L has no essential vertical cylinder.
Linv invariant and web spacemath.GT
In this paper, we reconstruct Kuperberg's web space. We introduce a new web (a trivalent diagram) and new relations between Kuperberg's web diagrams and the new diagram. Using the webs, we define crossing formulas corresponding to R-matrices associated to some irreducible representations and calculate…
Unified 3D R-matrices from quantum cluster algebra.
problem Constructing new solutions to the tetrahedron equation.
method Symmetric butterfly quiver, quantum cluster algebra, quantum dilogarithms, q-Weyl algebra.
result Unified 3D R-matrices from various sources.
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting propertie…
For an arbitrary identity L=R between compositions of maps L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit…
In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new fi…
This paper describes a method to obtain state model parameters for an infinite series of Links-Gould link invariants LG^{m,n}, based on quantum R matrices associated with the (\dot{0}_m | \dotα_n) representations of the quantum superalgebras U_q[gl(m|n)]. Explicit details of the state models for the cases n=1 and m=1,2…
Study extends knot polynomials to links, identifying them with known invariants.
problem Extending knot polynomials to links.
method Applying Reshetikhin-Turaev functor to braided Hopf algebras with automorphisms.
result Identifies some knot polynomials with known link invariants.
Study reveals hidden structure behind Racah matrices for twisted knots.
problem Understanding non-associativity in representation products of twisted knots.
method Analysis of quantum R-matrices and their eigenvalues to decompose Racah matrices.
result Discovery of pentad structure associated with universal R-matrix.
This paper is a continuation of [KS]. We develop the results of [KS] principally in two directions. First, we generalize the main result of [KS], the connection between the solutions of the classical dynamical Yang-Baxter equation and Poisson homogeneous spaces of Poisson Lie groups. We hope that now we present this re…
Study geometric bases for A-polynomials in SU(3) using arcade formalism.
problem Understanding relations between different knots and their representations.
method Use arcade formalism and Kuberberg rule to simplify calculations.
result Equations for knot polynomials become independent of representation.
We construct a certain cross product of two copies of the braided dual of a quasitriangular Hopf algebra , which we call the elliptic double , and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371 corresponds to the case m=2, and our generalizations sheds additional light on …
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …
Classifies Lie bialgebras using Darboux families.
problem Classifying real four-dimensional indecomposable coboundary Lie bialgebras.
method Introducing Darboux families to classify Lie bialgebras geometrically.
result Classification of coboundary Lie bialgebras on real four-dimensional indecomposable Lie algebras.
New method for calculating HOMFLY polynomials in symmetric representations.
problem Calculating HOMFLY polynomials for symmetric representations.
method Planar decomposition and projection to symmetric representations.
result Restoration of planarity and new insights into HOMFLY polynomials.
Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of…
This paper explores the relationship between Leibniz algebras and Nijenhuis operators.
problem Understanding the relationship between Leibniz algebras and Nijenhuis operators.
method Investigation of Nijenhuis operators on Leibniz algebras and classification of Leibniz bialgebras.
result Leibniz algebras are closely related to Nijenhuis operators, and triangular symplectic Leibniz bialgebras possess Nijenhuis operators.
The paper explores knot invariants using quantum dilogarithm representations and compares them to compact cases.
problem Constructing non-compact knot invariants using infinite-dimensional representations.
method Using Reshetikhin-Turaev approach and modular transformations of conformal blocks.
result Explicit expressions for non-compact knot invariants can be derived from ordinary ones, differing in homogeneity.
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. O…
New algorithm learns low-rank matrices with linear number of samples.
problem Learning low-rank matrices efficiently in latent-variable applications.
method Proposed algorithm that uses linear number of samples in high dimension.
result Learning , rank-, matrices requires $Ω( rac{kr}{ε^2})$ samples.
Analyzes solutions of stratified Lie systems and their geometric structures.
problem Nonautonomous systems of differential equations on manifolds.
method Analyzes particular solutions and properties of stratified Lie systems.
result Generalizes properties of Lie systems to stratified Lie systems.
Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete matrix is finitely rank- completable if there are at …
Lie algebras with -action are studied.
problem Lie algebra structures on symplectic Lie groups.
method Definition of -quasi-Frobenius Lie algebras.
result Induced -action on -quasi-Frobenius Lie algebras.
We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum…
Quantizes Chern-Simons invariant for tangle exteriors.
problem Geometric quantization of Chern-Simons invariant for tangles.
method Defining a sequence of invariants using modules over quantum and holonomy -matrices.
result Directly recovers Chern-Simons invariant when .
Quantum link invariants derived from skein algebras.
problem Defining invariants for framed links with SL2 local systems.
method Theory of representations of stated skein algebras, quantum coadjoint action, Drinfeld double, Bonahon-Wong quantum trace.
result Explicit formulas for link invariants and alternative proof of Murakami-Murakami relation.
The paper generalizes knot invariants and their connections to quivers and ideals.
problem Understanding knot complements and their invariants.
method Generalizing invariants, knots-quivers correspondence, and -polynomials; associating to branch of -polynomial; quiver generating series; -matrices; quantum -deformed -polynomial; 3d-5d theory.
result Explicit expressions for invariants and their quiver representations for several simple knots.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Abstract: New invariants for links and three-manifolds from finite groups.
problem Creating invariants for links and three-manifolds.
method Using finite groups to define matrices and extended matrices, constructing invariants through colored tangle categories.
result Invariants for links and three-manifolds are group invariants.
In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe only single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological p…
Bayesian neural networks can be simplified by parameterizing weights as rank- matrices, reducing parameter count and improving performance.
problem High parameter count in standard Bayesian neural networks.
method Parameterize weights as with , , inducing a singular posterior.
result PAC-Bayes generalization bounds and loss bounds show improved performance with fewer parameters.