Link invariants explained using HOMFLYPT polynomials and Yokonuma-Hecke algebras.
problem Link invariants constructed using Markov traces on Yokonuma-Hecke algebras.
method Description of link invariants in terms of HOMFLYPT polynomials and linking matrix.
result Link invariants explained completely using HOMFLYPT polynomials and Yokonuma-Hecke algebras.
New invariants for links derived from algebraic structures.
problem Creating invariants for links using algebraic connections.
method Using isomorphisms between affine and classical Yokonuma--Hecke algebras to construct invariants.
result Reduction in the number of invariants needed to study a link.
In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can…
In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the p--adic framed braids and the p--adic Yokonuma--Hecke algebras in…
New link invariants from algebraic framizations.
problem Developing new link invariants not topologically equivalent to existing ones.
method Construction of new algebras mYd,n(q) and mFTLd,n(q), and proving isomorphisms. result Introduction of new 2-variable and 1-variable link invariants.
Survey on new link invariants derived from quotients of Yokonuma-Hecke algebras.
problem Construction and properties of quotients of Yokonuma-Hecke algebras.
method Quotient construction and analysis of dimension, linear bases, representation theory.
result New one-variable link invariants derived from quotients, stronger than Jones polynomial.
We develop several applications of the fact that the Yokonuma--Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori--Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma--Hecke algebras of…
New polynomial invariants for links identified from Yokonuma-Hecke algebras.
problem Defining new polynomial invariants for classical links.
method Using a Markov trace on Yokonuma-Hecke algebra of type A.
result New invariants distinguish knots from links and are stronger than Homflypt polynomial.
The paper studies link invariants from Yokonuma-Hecke algebras.
problem Developing link invariants from algebraic structures.
method Investigating properties of traces on Yokonuma-Hecke algebras and defining link invariants.
result The invariants are topologically equivalent to the Homflypt polynomial on knots but not on links.
In this paper we introduce the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yok…
Extends Framization to Coxeter system of type B.
problem Extending Framization to Coxeter systems of type B.
method Define a natural extension of the classical Temperley-Lieb algebra, prove the existence of a unique linear Markov trace function, introduce Framization as a quotient of the Yokonuma-Hecke algebra, and provide conditions for the Markov trace to pass to the quotient.
result Construct invariants for framed and classical links inside the solid torus.
The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007), we explore further the structures of the p-adic framed braids and the p-adic Yokonuma-Hecke algebras constructed in Juyumaya and Lambropoulo…
We compare the invariant for classical knots and links defined using the Juyumaya trace on the Yokonuma-Hecke algebras with the HOMFLYPT polynomial. We show that the two invariants, as maps on the set L of oriented link types in S3, do not coincide except in a few trivial cases.
Study framizations of algebras using Schur--Weyl duality and tied braids.
problem Understanding framizations of algebras and their connections to quantum groups.
method Developing a general setting for framizations of algebras, including Yokonuma--Hecke and tied braids.
result Obtained Schur--Weyl duality for various algebras, including new framizations.
We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the…
We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra Yd,n(q), based on the study of the spectrum of its Jucys-Murphy elements which are defined here. We give explicit formulas for the irreducible representations of Yd,n(q) in terms of standard d-tableaux; w…
In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma--Hecke algebras Yd,n(u) and the theory of singular braids. The Yokonuma--Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphis…
In this paper we define the p-adic framed braid group F∞,n, arising as the inverse limit of the modular framed braids and we give topological generators for F∞,n. We also give geometric interpretations for the p-adic framed braids. We then construct a p-adic Yokonuma-Hec…
This paper presents results on the framization of some knot algebras, defined by the authors. We explain the motivations of the concept of framization, coming from the Yokonuma--Hecke algebras, as well as recent results on the framization of the Temperley--Lieb algebra. Finally, we propose framizations for other knot a…
In this work we introduce the concept of Modular Framization or simply Framization. We construct a framization Fd,n of the Birman--Wenzl--Murakami algebra, also known as BMW algebra, and start a systematic study of this framization. We show that Fd,n is finite dimensional and the \lq braid generators\rq\ of t…
The thesis explores centralisers and Hecke algebras in representation theory with applications to knots and physics.
problem Understanding centralisers and Hecke algebras in representation theory.
method Review of classical and quantum Schur-Weyl duality, discussion of quantum groups and their centralisers, and application to knot theory and physics.
result New insights into centralisers and Hecke algebras, leading to solutions for the Yang-Baxter equation and link invariants.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending L∞-algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms. result Braided L∞-algebra is derived from the process. Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
Study on Lie algebras from Clifford modules, focusing on specific types.
problem Characterizing Lie algebras from Clifford modules and their graded structures.
method Analysis of pseudo H-type Lie algebras and their representations through Clifford algebras. result Different types of Lie algebras have varying possibilities of containing pseudo H-type Lie algebras in their negative part. 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 …
The paper studies prolongations of Lie algebras associated with pseudo H-type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo H-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.
The paper extends cellular algebras with relative orderings and explores their properties.
problem Generalizing cellular algebras with different partial orderings.
method Classification and construction of simple modules, characterizations, and examples.
result Examples of relative cellular algebras that are not cellular.
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
problem Classifying Hsiang algebras and understanding their properties.
method Introducing quasicomposition and tripling constructions to study Hsiang algebras.
result The triple of a quasicomposition algebra is an exceptional Hsiang algebra.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
New grading on algebras of curves by winding number.
problem Understanding the structure of algebras of curves.
method Constructing a new grading on the Goldman Lie algebra and related algebras by winding number.
result Induces a new grading on the HOMFLY-PT skein algebra and related algebras.
Study biderivations in complete Leibniz algebras, extending Lie algebra results.
problem Defining and studying biderivations in complete Leibniz algebras.
method Analyze biderivations according to two definitions, provide conditions for biderivations, and compare symmetric and skew-symmetric biderivations.
result Necessary and sufficient conditions for biderivations in Leibniz algebras are provided.
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.
The paper classifies Lie algebras with special operators.
problem Classifying 3D Lie algebras with regular semisimple algebraic Nijenhuis operators.
method Described all Nijenhuis eigenbases for each 3D Lie algebra.
result Different answers in real and complex cases, some Lie algebras admit operators, others do not.
New theory disassembles Lie algebras into dyons and triadons.
problem Understanding the structure of Lie algebras through disassembly.
method Modular disassembly of Lie algebras into dyons and triadons.
result Any Lie algebra can be assembled from dyons and triadons.
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
Abstract defines and identifies a planar algebra with spin properties.
problem Identifying a specific planar algebra.
method Generators and relations defined, structure studied, identified with Jones' spin planar algebra.
result Identified a specific planar algebra with spin properties.
The paper investigates gradings of complex simple Lie algebras, focusing on ∣3∣-gradings and their algebraic structures.
problem Investigating the algebraic structure of ∣3∣-gradings of complex simple Lie algebras. method Completely determining the possible reductive algebras n0 and proving the uniqueness of a specific free nilpotent Lie algebra. result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a ∣3∣-grading is the usual ∣3∣-grading of the exceptional Lie algebra g2. Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
Study resolves conjecture linking two algebraic structures on surfaces.
problem Compatibility of skein and cluster algebra structures on surfaces.
method Established compatibility between skein and cluster algebras of surfaces.
result Cluster algebra of positive genus surfaces is not finitely generated.
Cluster C*-algebras' K-theory matches their corresponding cluster algebras.
problem Understanding the K-theory of cluster C*-algebras.
method Proved isomorphism between K_0-group of cluster C*-algebras and cluster algebras.
result Positivity conjecture for cluster algebras is proven more succinctly.
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with ε-estimates. result Similarity structures converge to classical algebraic objects as εightarrow0.