arXiv research

Finite presentations for skein algebras linked to gauge field theory.

A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…

The paper presents a new algebraic structure for planar surfaces.

Paper presents skein algebras for spheres with punctures.

Presented an algebra structure for a specific geometric surface.

New algebraic presentations for 3D cobordisms and 4D 2-handlebodies.

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogu…

Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure…

Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras…

Researchers present and compare different representations of dissipative Hamiltonian DAE systems.

The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.

Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.

The method of direct calculation of the group of $\mathbb R$-algebra automorphisms of a Weil algebra is presented in detail. The paper is focused on the case of a one-componental group and presents two cases of values of the determinant of its linear part.

We show how the fundamental cocycles on current Lie algebras and the Lie algebra of symmetries for the sigma model are obtained via the current algebra functors. We present current group extensions integrating some of these current Lie algebra extensions.

Found a basis and presentation for a specific algebra.

The notion of different kind of algebraic Casorati curvatures are introduced. Some results expressing basic Casorati inequalities for algebraic Casorati curvatures are presented. Equality cases are also discussed. As a simple application, basic Casorati inequalities for different $δ$-Casorati curvatures for Riemannian …

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…

AIDN uses deep learning to represent algebraic structures.

The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.

Paper defines a jellyfish algorithm for a specific subfactor planar algebra.

The study describes the free Lie-Yamaguti algebra.

We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the inva…

We provide a presentation of the Roger and Yang's Kauffman bracket arc algebra for the once-punctured torus and punctured spheres with three or fewer punctures.

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…

This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel presentation of several topics of the differential geometry of (smooth) manifolds of …

Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.

We define a certain abstract planar algebra by generators and relations, study various aspects of its structure, and then identify it with Jones' spin planar algebra.

Characterizes almost abelian Lie algebras with integrable complex structure

We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels' theory of multiamal…

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for co…

Generalizes uniformization to algebraic correspondences.

We present new results about Jordan algebras and Jordan coalgebras, and we discuss about their connections with the Yang-Baxter equations.

Infinitesimal calculations link fundamental groups to Lie algebras.

In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type $A$ as a quotient algebra of the Yokonuma-Hecke algebra of type $A$. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimensi…

New Poisson structures on algebras linked to derivatives.

The construction of a C*-algebra of a differential groupoid is presented. It is shown that it defines a covariant functor from the category of differential groupoids in a sense of S. Zakrzewski to the category of C*-algebras.

Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…

In this study, we classify some soliton nilpotent Lie algebras and possible candidates in dimension 8 and 9 up to isomorphy. We focus on 1 < 2 < ::: < n type of derivations where n is the dimension of the Lie algebras. We present algorithms to generate possible algebra structures.

Paper constructs observables using multisymplectic geometry and algebraic methods.

The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…

This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Marko…

The thesis explores centralisers and Hecke algebras in representation theory with applications to knots and physics.

We characterize H-like Lie algebras in terms of subspaces of cones over conjugacy classes in $\mathfrak{so}(\mathbb{R}^q)$, translating the classification problem for H-like Lie algebras to an equivalent problem in linear algebra. We study properties of H-like Lie algebras, present new methods for constructing them, in…

In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.