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.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
Extends Drinfel'd doubles to include twists and generalizes Lie bialgebroids.
The paper introduces a method to deform submanifolds in Euclidean space using Drinfel'd twists.
A family of algebras that extends the Lie algebra of the Drinfel'd double is proposed. This allows us to systematically construct the generalized frame fields which realize the proposed algebra by means of the generalized Lie derivative, i.e., $\hat{\mathcal{L}}_{E_A}E_B{}^I = - \mathcal{F}_{A…
Construct noncommutative deformations of algebraic submanifolds in R^n.
We survey physical models which capture the main concepts of double field theory on para-Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para-Kahler geometry which extends to a natural example of a Born geometry. The corresponding phase space geom…
Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of mot…
Extends Lie bialgebroids for string and M theories with new calculus framework.
Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structures for the corresponding Lie bialgebroids. Applications include Drinfel'd's classification in the case of Poisson groups and a description of leaf spaces of foliations as homogeneous spaces of pair groupoids.
The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as …
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…
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this…
The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfeld (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra as…
We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step …
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…
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We v…
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket…
A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups H is isomorphic to the category of Manin triples (d, g, h), where h is the Lie algebra of H. In this paper, we consider Dirac Lie groups, that is, Lie groups H endowed with a multiplicative Courant algebroid A and a Dirac s…
Paper discusses gliding algorithm to transform tangle diagrams into a specific form.
Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids a…
Monograph explores algebraic structures related to Yang-Baxter equation.
We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and various iterated tangent/cotangent constructions in symplectic and Poisson geometry…
New knot polynomials derived from Nichols algebras and braided Hopf algebras.
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…
New infinite families of twisted torus knots found.
The study finds hyperbolic twisted torus links for certain twists.
Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.
The paper studies the twisted Calabi flow on Kähler manifolds.
Paper proves any twisted link can be described as a unique twisted braid.
Study on singular twisted links and virtual braids, extending knot theory concepts.
In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory o…
The paper extends knot theory to twisted virtual braids and links.
Solves infinite family of cubic polynomial problems.
The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.
This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. By differential twists we will mean smooth U(1)-gerbes with connection, and w…
We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce t…
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidem…
We generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form for a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axiom…
Characterizes a specific type of spacetime using vector fields.
New families of twisted torus knots found with essential surfaces.
Formula for Alexander polynomial of twisted torus knots derived.
Study lattice paths from twist knots and double twist knots.
Paper studies involutions generating the twist subgroup of nonorientable surfaces.
The paper classifies twisted torus links that are unlinks.
Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.
Paper shows how to twist knots to make them trivial or non-trivial.
The paper uses Floer homology to study twist coefficients and their behavior after capping off.
We study the characteristic foliation of a twisted Jacobi manifold. We show that a twisted Jacobi manifold is foliated into leaves that are, according to the parity of the dimension, endowed with a twisted contact or a twisted locally conformal symplectic structure.