Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.17482

Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group

Chandan Singh
View original source ↗Read source document ↗Export metadata ↓

Abstract

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group $\mathsf{GRT}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that $\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.

Explore related subjects

math.QAmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chandan Singh. 2026-10-12. Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group. https://arxiv.org/abs/2610.17482

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Non-Equivalence of Smooth and Nodal Conformal Block Functors in Logarithmic CFT

Let $\mathbb V$ be an $\mathbb N$-graded, $C_2$-cofinite vertex operator algebra (VOA) admitting a non-lowest generated module in $\mathrm{Mod}(\mathbb V)$ (e.g., the triplet algebras $\mathcal{W}_p$ for $p\in \mathbb{Z}_{\geq 2}$ or the even symplectic fermion VOAs $SF_d^+$ for $d\in \mathbb{Z}_+$). We prove that, unlike in the rational case, for $N\geq 4$ there is no vector bundle on $\overline{\mathcal{M}}_{0,N}$ whose fibers are the spaces of conformal blocks associated to certain $\mathbb{V}$-modules, since the dimensions of these conformal block spaces differ between smooth and nodal curves. Consequently, the sheaf of coinvariants associated to these $\mathbb V$-modules on $\overline{\mathcal{M}}_{0,N}$ is not locally free for $N\geq 4$. It also follows that, unlike in the rational case, the mode transition algebra $\mathfrak A$ introduced by Damiolini-Gibney-Krashen is not isomorphic to the end $\mathbb E=\int_{\mathbb X\in \mathrm{Mod}(\mathbb X)}\mathbb X\otimes \mathbb{X}'$ as an object of $\mathrm{Mod}(\mathbb{V}^{\otimes 2})$.

math.QA↗
arXiv2026

Evaluation-type deformed modules over the quantum affine vertex algebras of type $A$

Let $\mathcal{V}^c(\mathfrak{gl}_N)$ be Etingof-Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $ϕ$-coordinated $\mathcal{V}^c(\mathfrak{gl}_N)$-modules and the representations of quantized enveloping algebra $U_h(\mathfrak{gl}_N)$ and reflection equation algebra $\mathcal{O}_h(Mat_N)$. As an application, we demonstrate how the elements of the center of $\mathcal{V}^c(\mathfrak{gl}_N)$ at the critical level $c=-N$ give rise to the $q$-analogues of quantum immanants for $U_h(\mathfrak{gl}_N)$, which were recently found by Jing, Liu and Molev. Finally, we derive the analogues of these results for the quantum affine vertex algebra associated with the normalized Yang $R$-matrix.

math.QA↗
arXiv2026

Exact sequences of representation categories of weak Hopf algebras

We study exact sequences of representation categories of weak Hopf algebras over an arbitrary field. Given a sequence $A\overset{k}{\to} B\oversetπ{\to} H$, where $A$ and $B$ are weak Hopf algebras and $H$ is a Hopf algebra, we develop verifiable algebraic conditions on $k$ and $π$ under which there is an exact sequence $\mathrm{Rep}(H)\to\mathrm{Rep}(B)\to\mathrm{Rep}(A)$ of tensor categories in the sense of Bruguières and Natale (2011). Along the way, we develop a generalization of the restriction of scalars functor for maps $π:B\to H$ between associative algebras satisfying a weakened multiplicativity constraint depending on a relatively separable subalgebra $B_r\subseteq B$, as well as a theory of kernels and cokernels for weak Hopf algebras. We, in particular, find that the cokernel of a weak Hopf algebra homomorphism $k:A\to B$ always exists, is a Hopf algebra, and the cokernel map is surjective when $A$ is connected. We conclude by studying examples of such exact sequences built from groupoids, formal ribbon extensions of quasitriangular weak Hopf algebras, and cocycled crossed products of a Hopf algebra acting weakly on a weak Hopf algebra.

math.QA↗