Root of unity ≈ Root of unity
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Depth-graded motivic multiple zeta values Open
We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp for…
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On the Q operator and the spectrum of the XXZ model at root of unity Open
The spin-1/2 Heisenberg XXZ chain is a paradigmatic quantum integrable model. Although it can be solved exactly via Bethe ansatz techniques, there are still open issues regarding the spectrum at root of unity values of the anisotropy. We c…
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Modularization of small quantum groups Open
We construct a large family of ribbon quasi-Hopf algebras related to small quantum groups, with a factorizable R-matrix. Our main purpose is to obtain non-semisimple modular tensor categories for quantum groups at even roots of unity, wher…
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Cyclotomic analogues of finite multiple zeta values Open
We study the values of finite multiple harmonic $q$ -series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic…
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Nonsemisimple quantum invariants and TQFTs from small and unrolled quantum groups Open
We show that unrolled quantum groups at odd roots of unity give rise to\nrelative modular categories. These are the main building blocks for the\nconstruction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple\nquantum invar…
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Logarithmic Hennings invariants for restricted quantum 𝔰𝔩(2) Open
We construct a Hennings-type logarithmic invariant for restricted quantum [math] at a [math] root of unity. This quantum group [math] is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a [ma…
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Asymptotics of Nahm sums at roots of unity Open
We give a formula for the radial asymptotics to all orders of the special q -hypergeometric series known as Nahm sums at complex roots of unity. This result is used in Calegari et al. (Bloch groups, algebraic K-theory, units and Nahm’s con…
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A two-variable series for knot complements Open
The physical 3d \mathcal N = 2 theory T[Y] was previously used to predict the existence of some 3 -manifold invariants \widehat{Z}_{a}(q) that take the form of power series with integer coefficients, converging in the unit disk. Their radi…
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Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity Open
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimens…
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Fusion and braiding in finite and affine Temperley-Lieb categories Open
Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the group algebra of) the famous Artin's braid group $B_N$, while the affine TL algebras arise as diagram algebras from a generalized version of the braid group. We stud…
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Entanglement in fermionic chains and bispectrality Open
Entanglement in finite and semi-infinite free Fermionic chains is studied. A parallel is drawn with the analysis of time and band limiting in signal processing. It is shown that a tridiagonal matrix commuting with the entanglement Hamilton…
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Factorizable R-Matrices for Small Quantum Groups Open
Representations of small quantum groups uq(g) at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig to endow these c…
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Orderly generation of Butson Hadamard matrices Open
In this paper Butson-type complex Hadamard matrices $\mathrm {BH}(n,q)$ of order $n$ over the complex $q$th roots of unity are classified for small parameters by computer-aided methods. The results include a classification of $\mathrm {BH}…
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Generalized Q-functions for GKM Open
Recently we explained that the classical $Q$ Schur functions stand behind\nvarious well-known properties of the cubic Kontsevich model, and the next step\nis to ask what happens in this approach to the generalized Kontsevich model\n(GKM) w…
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Discovering and Proving Infinite Pochhammer Sum Identities Open
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $π,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as sp…
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On the K-theory stable bases of the Springer resolution Open
Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent b…
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Profinite Groups with a Cyclotomic $p$-Orientation Open
Let $p$ be a prime. A continuous representation $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ of a profinite group $G$ is called a cyclotomic $p$-orientation if for all open subgroups $U\subseteq G$ and for all $k,n\geq1$ the natural maps…
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Logarithmic Link Invariants of $\overline{U}_q^H(\mathfrak{sl}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras Open
We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families o…
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Classical shadows of stated skein representations at roots of unity Open
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the\nskein algebras of closed surfaces to L^e's stated skein algebra associated to\nopen surfaces. We prove that the stated skein algebra with deforming parameter\n+1 e…
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Categorification at prime roots of unity and hopfological finiteness Open
We survey some recent results in hopfological algebra and the program of\ncategorification at prime roots of unity. A categorical Jones-Wenzl projector\nat prime roots of unity is studied, and it is shown that this projector is\nhopfologic…
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Root vectors of the composition algebra of the Kronecker algebra Open
According to the canonical isomorphism between
\n the positive part Uq⁺(g) of the Drinfeld–Jimbo quantum group
\n Uq(g) and the generic composition algebra C(∆) of Λ, where the
\n Kac–Moody Lie algebra g and the finite dimensional heredita…
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Field Extensions and Kronecker’s Construction Open
Summary This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [ X …
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New Family of Cross Z-Complementary Sequences With Large ZCZ Width Open
In this paper, we present a new family of cross $Z$-complementary pairs (CZCPs) based on generalized Boolean functions and two roots of unity. Our key idea is to consider an arbitrary partition of the set $\{1,2,\cdots, n\}$ with two subse…
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Knots, Perturbative Series and Quantum Modularity Open
We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the bounda…
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An explicit theory of $π_{1}^\mathrm{un,crys}(\mathbb{P}^{1} - \{0,μ_{N},\infty\})$ - II-3 : Sequences of multiple harmonic sums viewed as periods Open
Let $X=\text{ }\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }W(\mathbb{F}_{q})$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p$ prime to $N$ and containing a primitive $N$-th root of unity. We establ…
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Classical shadows of stated skein representations at roots of unity Open
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embe…
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An explicit theory of $\pi_1^\mathrm{un,crys}(\mathbb{P}^1 - \{0,\mu_{N},\infty\})$ - I-3 : The number of iterations of the Frobenius viewed as a variable Open
Let $X_{0}$ be a curve $\text{ }\mathbb{P}^{1} - (\{0,\infty\} \cup \mu_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p$ prime to $N$ and containing a primitive $N$-th root of …
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The Center of Small Quantum Groups I: The Principal Block in Type A Open
We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of sl(3) and sl(4) are computed explicitly. This…
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An explicit theory of $\pi_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ - II-2 : From standard algebraic relations of weighted multiple harmonic sums to those of cyclotomic $p$-adic multiple zeta values Open
Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup \mu_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit …
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A Fock space model for decomposition numbers for quantum groups at roots of unity Open
In this paper we construct an abstract Fock space for general Lie types that serves as a generalization of the infinite wedge q-Fock space familiar in type A. Specifically, for each positive integer l, we define a Z[q, q(-1)]-module F-l wi…