arXiv research

New model for rational tropical points using $\mathfrak{sp}_4$-webs and measures.

Study of $\mathfrak{sp}_4$-webs on surfaces, proving cluster algebra structure.

Cluster algebras match for specific Lie algebras and surfaces.

Quantum cluster algebra constructed from web skein relations on surfaces.

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transi…

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

Researchers compute $\mathfrak{gl}_2$-skein modules for lens spaces.

Researchers determined the second homology group of a specific symplectic derivation Lie algebra.

Quantum traces map skein algebras to Fock-Goncharov spaces.

If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero, it is well known that $\dim H^{\ast }(\mathfrak{n})\geq L(p)$ where $p$ is the polynomial associated to the grading and $L(p)$ is the sum of the absolute values of the coefficients of $p$. From …

Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.

Study of webs in quantum type C, proving equivalence to quantum representations.

Paper constructs super integrable systems on color Lie algebra.

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…

Let $M$ be an analytic complete finite volume pseudo-Riemannian manifold and $\widetilde{Sp}(n,\mathbb{R})\times\widetilde{Sp}(1,\mathbb{R})$ a connected semisimple Lie group such that its Lie algebra is $\mathfrak{sp}(n,\mathbb{R})\oplus\mathfrak{sp}(1,\mathbb{R})$. We characterize the structure of the manifold $M$ as…

We study the structure of the symplectic invariant part $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ of the Lie algebra $\mathfrak{h}_{g,1}$ consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface $Σ_{g}$ of genus $g$. First we describe the orthogonal dir…

We construct a new grading on the Goldman Lie algebra of a closed oriented surface by the winding number. This grading induces a grading on the HOMFLY-PT skein algebra and related algebras. Our work supports the conjectures of B. Cooper and P. Samuelson

We show that the Kauffman bracket skein algebra of any oriented surface F (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rel…

We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…

New proof shows Lie algebras are rigid under certain conditions.

Relates two types of skein algebras using explicit correspondences.

We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tan…

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' graded cellular basis and has two major application…

In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we cate…

We present a holomorphic representation of the Jacobi algebra $\mathfrak{h}_n\rtimes \mathfrak{sp}(n,\R)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}^n\times \mathcal{D}_n$. We construct the Hilbert space of holomorphic functions on which these differential operators a…

Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by studying a natural non-associative algebra $\mathfrak{m}$ comming from the affine struc…

A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly s…

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g…

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of thos…

We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that…

The study examines geodesics and tight geodesics in surface curve complexes.

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincaré superalge…

The paper finds formulas for flat models of certain Lie algebras.

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core …

We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms introduced by V.Hinich.

The paper studies deformations of Lie ideals in Lie algebras.

The paper calculates a specific weight system for chord diagrams with a particular graph structure.

Study Type $C$ skein modules using $Sp(2n)$ webs and construct transparent elements.

This research extends Lie algebra actions to singular foliations.

To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $π$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\hbox{Ann}({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_π\subset{…

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructe…

The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $τ_2\equiv 0$, while the second family has $τ_1\equivτ_2\equiv 0$. The families are differen…

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

Modified Hennings invariant defined using quantum groups and integrals.

Develops higher representation theory for odd Khovanov homology and rewriting theory.

Defines a new 2+1-G-HQFT using graded skein modules.