arXiv subjects

Semeon Arthamonov

Publications and source records attributed to Semeon Arthamonov.

7 recordsLinked to original sources

arXiv2026

Categorification of the genus two DAHA

We construct three derived endofunctors on the derived category of graded integrable representations of the current algebra of a flat degeneration of $D(2|1,\alpha)$, and prove that their classes in the Grothendieck group are the genus two Macdonald operators. Computing the graded $\operatorname{Ext}$ pairing explicitly, we deduce the self-adjointness of the genus two Macdonald operators and the orthogonality of genus $2$ Macdonald polynomials with respect to the certain skew-bilinear form.

math.RT↗

arXiv2026

Quasi-Poisson varieties from double quasi-Poisson algebras in types $B,C,D$

Double (quasi-)Poisson brackets were introduced on associative algebras by Van den Bergh to induce a (quasi-)Poisson structure on their representation spaces naturally equipped with a $\mathrm{GL}$-action (type $\mathtt{A}$). If there exists a compatible involutive anti-automorphism on the underlying associative algebras, Olshanski and Safonkin proved that this construction can be upgraded to induce a Poisson structure on twisted representation spaces (types $\mathtt{B},\mathtt{C},\mathtt{D}$). We provide an analogous result for double quasi-Poisson brackets, and over an arbitrary semisimple base. We also apply our theory to quivers in order to understand the Poisson structure on twisted (localised multiplicative) quiver varieties. The formalism permits that different vertices are assigned different types. As a first application, we recover the framework of Massuyeau and Turaev for Hopf algebras with a Fox pairing, which induces in particular the Poisson structure of character varieties for the orthogonal or symplectic groups. As a second application, we introduce a modified Kontsevich system.

math.RT↗

arXiv2026

Finite group actions on genus two $SL(2, \mathbb{C})$-character variety and applications to SCFTs

We investigate irreducible components of the fixed point sets of $ SL(2,\mathbb{C}) $-character variety of the genus two surface group under orientation preserving actions of the finite groups of the $ Mod(\Sigma_{2}) $. We work in the $ \mathcal{O} $-generator presentation of the genus two DAHA and its classical limit $ \mathcal{A}_{q=1,t} $, where we observe nontrivial coincidences between fixed loci attached to different subgroups and establish genus/irregularity transitions. The subvarieties obtained in this way provide novel geometric candidates for symmetry-reduced moduli spaces relevant to $ 4d $ $ \mathcal{N} = 2 $ SCFTs.

math-ph↗

arXiv2024

Cluster structure on genus 2 spherical DAHA: seven-colored flower

We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type $A_1$ as sum over lattice points in a convex polyhedron in 7-dimensional space.

math.RT↗

arXiv2016

Modified Double Poisson Brackets

We propose a non skew-symmetric generalization of the original definition of double Poisson Bracket by M. Van den Bergh. It allows one to explicitly construct more general class of H0-Poisson structures on finitely generated associative algebras. We show that modified double Poisson brackets inherit certain major properties of the double Poisson brackets.

math.QA↗

arXiv2015

Refined Chern-Simons Theory in Genus Two

Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. Our construction is built upon a q,t-deformation of the square of q-6j symbol of U_q(sl_2), which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.

hep-th↗

arXiv2014

Noncommutative Inverse Scattering Method for the Kontsevich system

We formulate an analog of Inverse Scattering Method for integrable systems on noncommutative associative algebras. In particular we define Hamilton flows, Casimir elements and noncommutative analog of the Lax matrix. The noncommutative Lax element generates infinite family of commuting Hamilton flows on an associative algebra. The proposed approach to integrable systems on associative algebras satisfy certain universal property, in particular it incorporates both classical and quantum integrable systems as well as provides a basis for further generalization. We motivate our definition by explicit construction of noncommutative analog of Lax matrix for a system of differential equations on associative algebra recently proposed by Kontsevich. First we present these equations in the Hamilton form by defining a bracket of Loday type on the group algebra of the free group with two generators. To make the definition more constructive we utilize (with certain generalizations) the Van den Bergh approach to Loday brackets via double Poisson brackets. We show that there exists an infinite family of commuting flows generated by the noncommutative Lax element.

math-ph↗