arXiv research

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

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

The paper identifies all flat CR Lie groups and their structures.

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…

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

Study Poisson cohomology and linearize Lie algebra structures.

The purpose of this paper is to provide an octonionic description of the Lie group $SL(2,{\mathbb O})$. The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from $\mathfrak{h}_2({\mathbb O})$ onto itself. An interesting characterization is giv…

Researchers create a new invariant for knot theory.

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…

Quantizes Chern-Simons invariant for tangle exteriors.

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand pa…

It is known that complex constant mean curvature ({\sc CMC} for short) immersions in $\mathbb C^3$ are natural complexifications of {\sc CMC}-immersions in $\mathbb R^3$. In this paper, conversely we consider {\it real form surfaces} of a complex {\sc CMC}-immersion, which are defined from real forms of the twisted $\m…

Study of adjoint orbits in simplest non-trivial Lie algebra case.

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

Khovanov-Rozansky homology shows periodic links have group actions.

We study the monodromy of meromorphic cyclic $\mathrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the wo…

Study cohomology spaces of sl(2) acting on n-ary differential operators.

Characterizes flag geometries for Hitchin representations in SL3(R).

Proofs centerless unimodular contact Lie algebras.

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 studies geometric structures on SL(n,R) induced by the Killing form.

New coordinates for SL3-web graphs on surfaces defined by Fock-Goncharov.

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 show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

The paper proves a quantum modularity conjecture for 3-manifolds.

Extends quantum trace map to SL3(C) for 3D surfaces.

In this paper, we study the quantum $\mathfrak{sl}(n)$ representation category using the web space. Specially, we extend $\mathfrak{sl}(n)$ web space for $n\ge 4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial …

The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.

The paper constructs solutions for Higgs fields on a 4-punctured sphere.

We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $π: Σ\to X$. The proof is by constructing a pair of inverse functors $π^{\text{ab}}, π…

Geometrically describes hyperbolic structures on link complements using quantum groups.

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

Study compares two pseudo-Kähler structures on a specific mathematical component.

New invariants from quantum group theory for hyperbolic 3-manifolds.

Flat hypercomplex nilmanifolds have a specific solvability property.

Extends knot invariant computation to symmetrically colored sl_N.

New method calculates Chern-Simons volume for 3-manifolds with surgery diagrams.

A new invariant for links generalizes Alexander polynomial for sl_3.

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

We study the Chabauty compactification of two families of closed subgroups of $SL(n,\mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,\mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Le…

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

We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the ad…

The paper calculates colored Jones polynomials for specific link configurations.

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

Computes Lie algebra structure constants using a graphical calculus.

Link homology compared with geometric link invariants using Bott-Samelson varieties.

Study on holonomy of Obata connection on specific nilmanifolds.