A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Reformulates Fock-Rosly Poisson structure using quasi-triangular r-matrices.
Paper extends quantum invariant to colored ideal triangulations.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.
In this paper we consider the Poisson algebraic structure associated with a classical -matrix, i.e. with a solution of the modified classical Yang--Baxter equation. In Section 1 we recall the concept and basic facts of the -matrix type Poisson orbits. Then we describe the -matrix Poisson pencil (i.e the pair o…
New combinatorial type helps distinguish plane curve topologies.
The paper defines and computes a knot complement invariant for simple links.
We show that for any coboundary Poisson Lie group G, the Poisson structure on G^* is linearizable at the group unit. This strengthens a result of Enriquez-Etingof-Marshall, who had established formal linearizability of G^* for quasi-triangular Poisson Lie groups G. We also prove linearizability properties for the group…
We introduce a construction of the differential calculus on the quantum supergroup GL. We obtain two differential calculi, respectively, associated with the left and right Cartan-Maurer one-forms. We also obtain the quantum superalgebra of GL. Although all of the structures we obtain are der…
Paper reviews Viro's definition of Khovanov homology for tangles.
Abstract: New invariants for links and three-manifolds from finite groups.
A general scheme for construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to WDVV associativity equations is formulated. The advantage is taken from the Rota-Baxter identity and some relation being counterpart of the modified Yang-Baxter identity from the classical …
Extends knot invariant computation to symmetrically colored sl_N.
New braid group action defined on projective quantum sl(2) modules.
New proof for knot state-sum formula using bijection between states.
The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We u…
New knot invariants derived using quantum cluster algebras.
Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.
Paper proves a conjecture about Links-Gould invariants using specialized quantum algebras.
A non-commutative differential calculus on the -superplane is presented via a contraction of the -superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the dimensional classical phase space (the super-Heisenberg algebra) is introduced.
Study Lie bialgebra structures on flat metric Lie algebras, leading to explicit Poisson-Lie groups.
Any classical r-matrix on the Lie algebra of linear operators on a real vector space V gives rise to a quadratic Poisson structure on V which admits a deformation quantization stemming from the construction of V. Drinfel'd. We exhibit in this article an example of quadratic Poisson structure which does not arise this w…
Invariants for long knots defined using algebraic and categorical methods.
Unified 3D R-matrices from quantum cluster algebra.
We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-mat…
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…
The differential calculus on the quantum supergroup GL was introduced by Schmidke {\it et al}. (1990 {\it Z. Phys. C} {\bf 48} 249). We construct a differential calculus on the quantum supergroup GL in a different way and we obtain its quantum superalgebra. The main structures are derived without an…
If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the …
AGD converges in polynomial iterations to optimal matrix factorization.
Defect of knot polynomials remains invariant under certain braid substitutions.
A systematic description of the Wess-Zumino-Witten model is presented. The symplectic method plays the major role in this paper and also gives the relationship between the WZW model and the Chern-Simons model. The quantum theory is obtained to give the projective representation of the Loop group. The Gauss constraints …
This paper updates knot invariants using Hopf algebras and categorifies their structure.
Researchers create a new invariant for knot theory.
We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are -series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of th…
New approach solves non-square matrix sensing without local minima issues.
Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are …
New algorithm speeds up knot polynomial calculations.
We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from quantum field theory based on the Chern-Simon Lagrangian. From our approach we ca…
Develops quantum cluster algebra approach to solve tetrahedron equation.
Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.
The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwi…
The study investigates linearizability of Poisson structures on groupoids.
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space , the space which is covariant under the action of the quantum group . For each of the two covariant differential calculi over based on the -matrix formalism, we…
Skein theory aids in computing YB homology and cohomology groups.
New knot polynomials reveal patterns and mutations.
This paper describes a method for the automatic evaluation of the Links-Gould two-variable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are present…
New algorithm reduces sample and computational complexities for matrix completion.
We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results …