arXiv research

Computes Lie algebra structure constants using a graphical calculus.

New parameterization of Teichmüller spaces for complex Lie groups.

Study shows Steinberg representation's multiplicity in cohomology of congruence subgroups.

Formula for colored invariants of torus knots linked to $\mathcal{W}_r$ algebras.

Extends knot invariant computation to symmetrically colored sl_N.

A new invariant for links generalizes Alexander polynomial for sl_3.

Study unbounded $\mathfrak{sl}_3$-laminations around punctures.

Paper proves zero stability for one-row colored sl₃-Jones polynomials.

Generalizes reductive homogeneous spaces to arbitrary Lie groups using gauge theory.

The paper calculates colored Jones polynomials for specific link configurations.

Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.

Study reveals $\mathfrak{sl}_2$ action on link homology of specific torus links.

Geometric model of unbounded sl3 laminations with tropical coordinates.

Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.

Quantum cluster algebra constructed from web skein relations on surfaces.

New $\mathfrak{sl}_2$ action on link homologies discovered.

Researchers create a new invariant for knot theory.

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…

Study on quantum $\mathfrak{sl}_3$ invariant for positive links.

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra o…

The paper extends Khovanov homology results to $\mathfrak{sl}(n)$ homologies and provides bounds on knot properties.

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivarian…

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve …

We study holomorphic integrable systems on the hyperkähler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2(\mathbb{C})$-triple. Our main result is that this manifold carries a canonical \textit{abstract int…

A well known result of Drinfeld classifies Poisson Lie groups $(H,Π)$ in terms of Lie algebraic data in the form of Manin triples $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$; he also classified compatible Poisson structures on $H$-homogeneous spaces $H/K$ in terms of Lagrangian subalgebras $\mathfrak{l}\subset\mathfrak{…

Motivated by a possible connection between the $\mathrm{SU}(N)$ instanton knot Floer homology of Kronheimer and Mrowka and $\mathfrak{sl}(N)$ Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yam…

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…

Extends Lawrence's representations to integral $U_q \mathfrak{sl}(2)$ Verma-modules and braid groups.

This article gives matrix factorizations for the trivalent diagrams and double line appearing in $\mathfrak{sl}_n$ quantum link invariant. These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimens…

Lifts an $\mathfrak{sl}_2$ action to annular Khovanov homology's stable refinement.

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…

The paper connects isomonodromic and isospectral deformations for $\mathfrak{sl}_2(\mathbb{C})$ connections.

Khovanov-Rozansky homology shows periodic links have group actions.

Quantum invariants from $U_h\mathfrak{sl}(2|1)$ are q-holonomic.

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd …

Formula for $\mathfrak{sl}_2$ weight system on complete bipartite graphs.

We define and study the category of symmetric $\mathfrak{sl}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum $\mathfrak{sl}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric $\mathfrak{sl}_2$-spider is (braided monoidally) equivalent to …

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…

New systems of linear PDEs discovered in 3D contact manifolds.

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

We suggest an index-free formalism allowing to simplify many computations in Riemann geometry. The main ingredients are forms with values in a Clifford algebra and an action of the group $\mathfrak{sl}_2\times \mathfrak{sl}_2$ on such forms.

Second part of proving linearization theorem for sl2(C).

Study Poisson cohomology and linearize Lie algebra structures.

Study shows link polynomial evaluations from Heegaard Floer theory.

Study Higgs bundles and their reductions to prove stability and cohomology properties.

Quantum invariants derived from $\mathcal{U}_q(\mathfrak{sl}_2)$ link holonomy.

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.