Searcharxiv⌕ Search

arXiv subjects

Muze Ren

Publications and source records attributed to Muze Ren.

10 recordsLinked to original sources
arXiv2026

Double shuffle relations and double antipodes

In this note, we prove that for every Lie series $\psi$ with no terms of degree less than 3, the relation $$[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$$ is equivalent to $S_*(\psi_*)=-\psi_*$, where $\psi_*$ denotes the regularization of $\psi$ and $S_*$ is the harmonic antipode. The proof relies on the calculations of the harmonic antipode $S_*$ and the shuffle antipode $S$. As a consequence, we prove that every $\psi$ in Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ satisfies the relation $[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$. We further prove that $\mathfrak{dmr}_0$ injects into the symmetric Kashiwara--Vergne Lie algebra $\mathfrak{krv}^{\mathrm{sym}}_2$ of Alekseev and Torossian.

math.QA↗
arXiv2026

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

It is proved by L.~Schneps that the double shuffle Lie algebra $\mathfrak{dmr}_0$ injects to the Kashiwara-Vergne Lie algebra $\mathfrak{krv}_2$ in \cite{Schneps2012,Schneps2025}. We show that $\mathfrak{dmr}_0$ with the infinitesimal hexagon equation $[x,\varphi(-x,-y,x)]+[y,\varphi(-x-y,y)]=0$ injects to the symmetric Kashiwara-Vergne Lie algebra $\mathfrak{krv}^{\mathrm{sym}}_2$. The proof is based on the inclusion of brunnian braids group on different genus 0 surfaces which is different from the method of mould calculus in \cite{Schneps2012,Schneps2025}. We generalize the inclusion in two directions, one using lower central series of brunnian Lie algebras and the other is to establish explicit links between the pentagon equation map, the stuffle coproduct, the divergence map and the necklace cobracket.

math.QA↗
arXiv2026

The kernel of formal polylogarithms

Polylogarithmic functions (polylogs) in $n$ variables can be viewed as elements of $(U\mathfrak{p}_{m})^*$, the dual of the universal enveloping algebra of the Lie algebra $\mathfrak{p}_{m}$ of infinitesimal spherical pure braids with $m=n+3$ strands. Polylogs with $m=4,5$ are used in the theory relating double shuffle relations and Drinfeld associators \cite{furusho_double_2011}. We give explicit formulas for elements of $(U\mathfrak{p}_{m})^*$ representing polylogs, and compute the left ideal $J_{m} \subset U\mathfrak{p}_{m}$ given by their joint kernel. We introduce Lie subalgebras $\mathfrak{k}_{m}=\mathfrak{p}_{m} \cap J_{m}$, and we compute them for $m=4, 5$.

math.QA↗
arXiv2025

Reduced coaction Lie algebra, double shuffle Lie algebra and noncommutative krv2 equation

We study the reduced coaction Lie algebra $\mathfrak{rc}_0$, which is defined by an algebraic equation satisfied by the reduced coaction (an upgraded version of the necklace cobracket) and the skew-symmetric condition. We prove that the double shuffle Lie algebra $\mathfrak{dmr}_0$ together with the skew-symmetric condition injects to $\mathfrak{rc}_0$, and that $\mathfrak{rc}_0$ together with the krv1 equation injects to the Kashiwara-Vergne Lie algebra $\mathfrak{krv}_2$.

math.QA↗
arXiv2024

Poisson brackets and coaction maps of regularized holonomies of the KZ equation

We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra ${\rm gl}(N, \mathbb{C})$. Our main technical tool is a certain projection of the generalized pentagon equation of \cite{AFR2024}.

math.QA↗
arXiv2024

Generalized Pentagon Equations

Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $\Phi_{\rm KZ}$ by considering the regularized holonomy of the KZ connection along the {\em droit chemin} $[0,1]$. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation. In this paper, we consider paths on $\mathbb{C}\backslash \{ z_1, \dots, z_n\}$ which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy $H$ of the KZ connection associated to such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter $H$, $\Phi_{\rm KZ}$, and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path.

math.QA↗
arXiv2023

Groupoid intertwiner and twist for dynamical Yang--Baxter equation: part I

Intertwiner is a homomorphism between two existing dynamical R matrices, first introduced by Baxter in eight vertex-SOS correspondence, we develop certain equivalence relations among R matrices using intertwiners. Twist is a homomorphism that twist a dynamical R matrix to get a new dynamical R matrix, we introduce a kind of notion of twist that generalize classical Drinfeld twist in quasi-triangular Hopf algebra and some dynamical twist. As applications, we obtain some examples of twists from Ocneanu cell calculus and Fendley--Ginsparg orbifold constructions. The relations between intertwiner and twist are also discussed, the groupoid structures are emphasized.

math.RT↗
arXiv2023

Baxterization for the dynamical Yang-Baxter equation

The Baxterization process for the dynamical Yang-Baxter equation is studied. We introduce the local dynamical Hecke ,Temperley-Lieb and Birman-Murakami-Wenzl operators, then by inserting spectral parameters, from each representation of these operators, we get dynamical R matrix under some conditions. As applications, we reformulate trigonometric degeneration of elliptic quantum group representations and also get dynamical R matrix for critical ADE integrable lattice models. Through Baxterization, we construct some one dimensional integrable systems that are dynamical version of the Heisenberg spin chain.

math.RT↗
arXiv2020

Quantum Groups for Restricted SOS Models

We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of $\pi$-graded vector spaces for a groupoid $\pi$.

math.RT↗