Cyclotomic Research Articles

The R-transform as power map and its generalizations to higher degree

We give iterative constructions for irreducible polynomials over Fq of degree n⋅tr for all r≥0, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions xt. The R-transform introduced by Cohen is recovered as a particular case corresponding to x2, hence we obtain a generalization of Cohen's R-transform (t=2) to arbitrary degrees t≥2. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of Fq we recover and generalize a recursive construction of Panario, Reis and Wang.

A remarkable basic hypergeometric identity

We give a closed form for quotients of truncated basic hypergeometric series where the base q is evaluated at roots of unity.

The lowest discriminant ideal of a Cayley–Hamilton Hopf algebra

Discriminant ideals of noncommutative algebras A A , which are module finite over a central subalgebra C C , are key invariants that carry important information about A A , such as the sum of the squares of the dimensions of its irreducible modules with a given central character. There has been substantial research on the computation of discriminants, but very little is known about the computation of discriminant ideals. In this paper we carry out a detailed investigation of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose identity fiber algebras are basic. The lowest discriminant ideals are the most complicated ones, because they capture the most degenerate behaviour of the fibers in the exact opposite spectrum of the picture from the Azumaya locus. We provide a description of the zero sets of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in terms of maximally stable modules of Hopf algebras, irreducible modules that are stable under tensoring with the maximal possible number of irreducible modules with trivial central character. In important situations, this is shown to be governed by the actions of the winding automorphism groups. The results are illustrated with applications to the group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity.

Division quaternion algebras over some cyclotomic fields

Let p 1, p 2 be two distinct prime integers, let n be a positive integer, n ≥ 3 and let ξn be a primitive root of order n of the unity. In the 3rd section of this paper we obtain a complete characterization for a quaternion algebra H ( p 1 , p 2 ) to be a division algebra over the nth cyclotomic field Q ( ξ n ) , when n ∈ { 3 , 4 , 6 , 7 , 8 , 9 , 11 , 12 } and we also obtain a characterization for a quaternion algebra H ( p 1 , p 2 ) to be a division algebra over the nth cyclotomic field Q ( ξ n ) , when n ∈ { 5 , 10 } . In the 4th section we obtain a complete characterization for a quaternion algebra H Q ( ξ n ) ( p 1 , p 2 ) to be a division algebra, when n = l k , with l a prime integer, l ≡ 3 (mod 4) and k a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra H Q ( ξ l ) ( p 1 , p 2 ) to be a division algebra, when l is a Fermat prime number.

On the cyclotomic field [formula omitted] and Zhi-Wei Sun's conjecture on det Mp

In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of p-th cyclotomic field and the finite field Fp, we confirm this conjecture.

A Note on Bivariate bi-periodic Mersenne Polynomials

We introduce a new generalization mn(x,y), which we will call Bivariate Bi-periodic Mersenne polynomials depending on whether n is even or odd. We investigated the bivariate and biperiodic forms of Mersenne polynomials, focusing on their unique structural properties, roots, and relationships among coefficients. Key contributions include deriving the generating function and Binet formula, examining the limit behavior of the polynomials, and identifying connections between positive and negative terms. Also, we give the limits of the consecutive terms of the polynomials and some important identities such as Catalan, Cassini and D’Ocagne’s identity. We also find the corresponding binomial addition formula.

Dynamical symmetry restoration in the Heisenberg spin chain

Abstract The entanglement asymmetry is an observable independent tool to investigate the relaxation of quantum many-body systems through the restoration of an initially broken symmetry of the dynamics. In this paper we use this to investigate the effects of interactions on quantum relaxation in a paradigmatic integrable model. Specifically, we study the dynamical restoration of the U(1) symmetry corresponding to rotations about the z-axis in the XXZ model quenched from a tilted ferromagnetic state. We find two distinct patterns of behaviour depending upon the interaction regime of the model. In the gapless regime, at roots of unity, we find that the symmetry restoration is predominantly carried out by bound states of spinons of maximal length. The velocity of these bound states is suppressed as the anisotropy is decreased toward the isotropic point leading to slower symmetry restoration. By varying the initial tilt angle, one sees that symmetry restoration is slower for an initially smaller tilt angle, signifying the presence of the quantum Mpemba effect. In the gapped regime, however, spin transport for non maximally tilted states is dominated by smaller bound states with longer bound states becoming frozen. This leads to much longer time scales for restoration compared to the gapless regime. In addition, the quantum Mpemba effect is absent in the gapped regime.

Dimension-free discretizations of the uniform norm by small product sets

Let f be an analytic polynomial of degree at most K−1. A classical inequality of Bernstein compares the supremum norm of f over the unit circle to its supremum norm over the sampling set of the K-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for Lp,1≤p≤∞ norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of deg(f) and is instead governed by the maximum individual degree of f; i.e., the largest degree of f when viewed as a univariate polynomial in any coordinate. For example, we find that for n-variate analytic polynomials f of degree at most d and individual degree at most K−1, ∥f∥L∞(Dn)≤C(X)d∥f∥L∞(Xn) for any fixed X in the unit disc D with |X|=K. The dependence on d in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant O(logK)2d.

A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

Sums of distinct fifth roots of unity and the regular dodecahedron

Integer linear combinations of cube, fourth, or sixth roots of unity form lattices in the complex plane . In contrast, integer linear combinations of fifth roots of unity do not form a lattice; in fact, they are dense in . Nevertheless, the geometry for fifth roots of unity has considerable structure. Here we consider only sums of distinct fifth roots of unity, and show that 20 of these sums are orthogonal projections of the vertices of a regular dodecahedron. Pentagonal symmetry here is only to be expected, but it is a little surprising to encounter a plane projection of a polyhedron with much richer dodecahedral symmetry.

Towards quantum gravity with neural networks: solving the quantum Hamilton constraint of U(1) BF theory

Abstract In the canonical approach of loop quantum gravity, arguably the most important outstanding problem is finding and interpreting solutions to the Hamiltonian constraint. In this work, we demonstrate that methods of machine learning are in principle applicable to this problem. We consider U(1) BF theory in three dimensions, quantised with loop quantum gravity methods. In particular, we formulate a master constraint corresponding to Hamilton and Gauß constraints using loop quantum gravity methods. To make the problem amenable for numerical simulation we fix a graph and introduce a cutoff on the kinematical degrees of freedom, effectively considering U q ( 1 ) BF theory at a root of unity. We show that the neural network quantum state ansatz can be used to numerically solve the constraints efficiently and accurately. We compute expectation values and fluctuations of certain observables and compare them with exact results or exact numerical methods where possible. We also study the dependence on the cutoff.

On the category 𝒪 of a hybrid quantum group

We study the representation theory of a hybrid quantum group at root of unity ζ \zeta introduced by Gaitsgory. After discussing some basic properties of its category O \mathcal {O} , we study deformations of the category O \mathcal {O} . For subgeneric deformations, we construct the endomorphism algebra of big projective object and compute it explicitly. Our main result is an algebra isomorphism between the center of deformed category O \mathcal {O} and the equivariant cohomology of ζ \zeta -fixed locus on the affine Grassmannian attached to the Langlands dual group.

Volichenko-type metasymmetry of braided Majorana qubits

Abstract This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits and clarifies their role; in particular, "mixed-bracket" Heisenberg-Lie algebras are introduced. These algebras belong to a more general framework than the Volichenko algebras defined in 1990 by Leites-Serganova as metasymmetries which do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators.
In a previous paper braided Z2-graded Majorana qubits were first-quantized within a graded Hopf algebra framework endowed with a braided tensor product. The resulting system admits truncations at 
roots of unity and realizes, for a given integer s=2,3,4, ..., an interpolation between ordinary Majorana fermions (recovered at s=2) and bosons (recovered in the s →∞ limit); it implements a parastatistics where at most s-1 indistinguishable particles are accommodated in a multi-particle sector.
The structures discussed in this work are:
- the quantum group interpretation of the roots of unity truncations recovered from a (superselected) set of reps of the quantum superalgebra Uq(osp(1|2));
- the reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products
(in a minimal representation, the N-particle sector of the braided Majorana qubits is described by 2N x 2N;
- the introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg-Lie algebras;
- the s→∞ untruncated limit of the mixed-bracket Heisenberg-Lie algebras producing parafermionic oscillators;
- (meta)symmetries of ordinary differential equations given by matrix Schrödinger equations in 0+1 dimension induced by the braided creation/annihilation operators;
- in the special case of a third root of unity truncation, a nonminimal realization of the intertwining operators defines the system
as a ternary algebra.

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed.

Crossed Product Interpretation of the Double Shuffle Lie Algebra Attached to a Finite Abelian Group

Racinet studied a scheme associated with the double shuffle and regularization relations between multiple polylogarithm values at N th roots of unity and constructed a group scheme attached to the situation; he also showed it to be the specialization for G=\mu_{N} of a group scheme \operatorname{\mathsf{DMR}}_{0}^{G} attached to a finite abelian group G . Then Enriquez and Furusho proved that \operatorname{\mathsf{DMR}}_{0}^{G} can be essentially identified with the stabilizer of a coproduct element arising in Racinet’s theory with respect to the action of a group of automorphisms of a free Lie algebra attached to G . We reformulate Racinet’s construction in terms of crossed products. Racinet’s coproduct can then be identified with a coproduct \widehat{\Delta}^{\mathcal{M}}_{G} defined on a module \widehat{\mathcal{M}}_{G} over an algebra \widehat{\mathcal{W}}_{G} , which is equipped with its own coproduct \widehat{\Delta}^{\mathcal{W}}_{G} , and the group action on \widehat{\mathcal{M}}_{G} extends to a compatible action of \widehat{\mathcal{W}}_{G} . We then show that the stabilizer of \widehat{\Delta}^{\mathcal{M}}_{G} , hence \operatorname{\mathsf{DMR}}_{0}^{G} , is contained in the stabilizer of \widehat{\Delta}^{\mathcal{W}}_{G} thus generalizing a result of Enriquez and Furusho [Selecta Math. (N.S.) 29 (2023), article no. 3]. This yields an explicit group scheme containing \operatorname{\mathsf{DMR}}_{0}^{G} , which we also express in the Racinet formalism.

Compatible pants decomposition for $\mathrm{SL}_{2}(\mathbb{C})$ representations of surface groups

For any irreducible representation of a surface group into \mathrm{SL}_{2}(\mathbb{C}) , we show that there exists a pants decomposition where the restriction to any pair of pants is irreducible and where no curve of the decomposition is sent to a trace \pm 2 element. We prove a similar property for \mathrm{SO}_{3} -representations. We also investigate the type of pants decomposition that can occur in this setting for a given representation. This result was announced by Detcherry and Santharoubane (2022), motivated by the study of the Azumaya locus of the skein algebra of surfaces at roots of unity.

Braided Zestings of Verlinde Modular Categories and Their Modular Data

Zesting of braided fusion categories is a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. In this paper, we classify and construct all possible braided zesting data for modular categories associated with quantum groups at roots of unity. We produce closed formulas, based on the root system of the associated Lie algebra, for the modular data of these new modular categories.

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Degenerate fourfold Massey products over arbitrary fields

We prove that, for all fields F of characteristic different from 2 and all a,b,c\in F^{\times} , the mod 2 Massey product \langle a,b,c,a\rangle vanishes as soon as it is defined. For every field F_{0} , we construct a field F containing F_{0} and a,b,c,d\in F^{\times} such that \langle a,b,c\rangle and \langle b,c,d\rangle vanish but \langle a,b,c,d\rangle is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod 2 cochain differential graded algebra of the absolute Galois group is not formal.

Structures of finite fields of primes and the Euclidean square roots of unity in the finite rings

Structures of finite fields of primes and the Euclidean square roots of unity in the finite rings