Primitive Nth Root Research Articles

Unit cyclotomic multiple zeta values for μ2,μ3 and μ4

Denote by ε a primitive Nth-root of unity. In this paper, we show that the unit cyclotomic multiple zeta values for μN generate all the cyclotomic multiple zeta values for μN when N=2,3,4. Moreover, the unit cyclotomic multiple zeta values for μN can be written as Q-linear combinations of (ζ(1ε))n,(ζ(1ε−1))n and lower depth terms in each weight n when N=2,3 and 4. By detailed analysis of the motivic Galois action, we compute the coefficients of (ζ(1ε))n,(ζ(1ε−1))n in the above expressions of unit cyclotomic multiple zeta values.

On special values of generators of modular function fields

Let N be a positive integer and ζN be a primitive Nth root of unity. Let X(N) be the modular curve of level N and FX(N),Q(ζN) be its field of meromorphic functions with Fourier coefficients in Q(ζN). Let u and v be nonconstant functions in FX(N),Q(ζN) which generate a sufficiently large subfield of FX(N),Q(ζN) over Q(ζN). In this paper, we give sufficient conditions for generating the ray class field of an imaginary quadratic field K modulo N over K(ζN) by the special values of u and v.

Tensor representations for the Drinfeld double of the Taft algebra

Over an algebraically closed field k of characteristic zero, the Drinfeld double Dn of the Taft algebra that is defined using a primitive nth root of unity q∈k for n≥2 is a quasitriangular Hopf algebra. Kauffman and Radford have shown that Dn has a ribbon element if and only if n is odd, and the ribbon element is unique; however there has been no explicit description of this element. In this work, we determine the ribbon element of Dn explicitly. For any n≥2, we use the R-matrix of Dn to construct an action of the Temperley-Lieb algebra TLk(ξ) with ξ=−(q12+q−12) on the k-fold tensor power V⊗k of any two-dimensional simple Dn-module V. This action is known to be faithful for arbitrary k≥1. We show that TLk(ξ) is isomorphic to the centralizer algebra EndDn(V⊗k) for 1≤k≤2n−2.

On consecutive primitive nth roots of unity modulo q

Given n∈N, we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n. In particular, we analyze the resultant of the cyclotomic polynomial Φn(x) with Φn(x+1), and exhibit Lucas and Mersenne divisors of this quantity. For each n≠1,2,3,6, we prove the existence of a prime qn for which there is an element α∈Zqn where α and α+1 both have multiplicative order n. Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.

Resultants of minimal polynomials of maximal real cyclotomic extensions

Define the nth real cyclotomic polynomial to be the minimal polynomial over Z of ζn+ζn−1, where ζn=e2πi/n is a primitive nth root of unity. We prove that the real cyclotomic polynomials can be formed from compositions of polynomials closely related to the Chebyshev polynomials of the first kind. We use these relations to determine the resultant of two real cyclotomic polynomials.

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] G of a three dimensional representation W of G where G ⊂ SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] G . The case where W is any two dimensional representation of G is also handled.

Wavelet-type frames for an interval

We construct a sequence of rational functions, which will be called disc wavelets, parametrized by points in the unit disc λk,n(γ)=(1−γ2n)(ωn)k,n∈N, 0≤k<n, where ωn=e2πi/n is the primitive nth root of unity, and obtain a full description, in terms of the parameter γ, of the sets {λk,n(γ)} yielding frames for the space L2(−1,1). This is done using a disc version of the Bargmann transform, which maps L2(−1,1) to the classical spaces of the unit disc (Bergman, Hardy, Dirichlet) and applying Seip’s description of sampling sets in the unit disc. We also describe how to replace the points λk,n(γ) by the orbit of a Fuchsian group Γ, observing that Seip’s density of the orbit of a point contained in the fundamental region of a Fuchsian group Γ is equal to m0, where m0 is the smallest number among the weights of the automorphic forms with a zero in the fundamental region of the group Γ.

Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials

Almost all algebra texts define cyclotomic polynomials using primitive nth roots of unity. However, the elementary formula gcd(xm - 1, xn - 1) = xgcd(m,n) - 1 in ℤ[x] can be used to define the cyclotomic polynomials without reference to roots of unity. In this article, partly motivated by cyclotomic polynomials, we prove a factorization property about strong divisibility sequences in a unique factorization domain. After illustrating this property with cyclotomic polynomials and sequences of the form An − Bn, we use the main theorem to prove that the dynamical analogues of cyclotomic polynomials over any unique factorization domain R are indeed polynomials in R[x]. Furthermore, a converse to this article's main theorem provides a simple necessary and sufficient condition for a divisibility sequence to be a rigid divisibility sequence.

Algorithms for Relatively Cyclotomic Primes

We present a general method of generating primes p and q such that q divides Φnp, where n > 2 is a fixed number. In particular, we present the deterministic method of finding a primitive nth roots of unity modulo q. We estimate the computational complexity of our methods.

An analogue of Bridges and Mena’s theorem for local fields and a local-global principle

Let G be an abelian group of finite order n,K a field and R⊆K a ring. Let D=∑g∈Gagg∈R[G] such that χ(D)∈R for every character χ:G→K(ξn) (where χ(D)=∑g∈Gagχ(g) and ξn is a primitive nth root of unity). What does D look like? The case where K=Q and R=Z was settled by Bridges and Mena. Here we obtain a complete characterization for the case where K is a finite extension of the field Qp and R is its valuation ring under the condition that p does not divide n.As an application we obtain the following local-global principle for Z/q1q2Z (where q1 and q2 are distinct primes): If D∈Z[Z/q1q2Z], then χ(D)∈Z for every character χ:Z/q1q2Z→C× if and only if ψ(D)∈Zp for every prime p and every character ψ:Z/q1q2Z→Qp(ξn).

Colourings of cyclotomic integers with class number one

This paper continues the study of colourings of the sets of cyclotomic integers ℳ n = ℤ[ξ n ] (ξ n = e 2πi/n , a primitive nth root of unity) with class number one. We present results for the colour symmetry group and colour preserving group for a given ideal colouring of ℳ n , with φ(n) = 8 and 10, thus completing the characterisation of the colour preserving group for the cases φ(n) ≤ 10, where φ is Euler's totient function.

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was: Theorem (See Borwein, Choi, Yazdani, 2001.) Let f ( z ) = ± z ± z 2 ± ⋯ ± z N − 1 , and ζ a primitive Nth root of unity. If N is an odd positive integer then max i | f ( ζ i ) | ⩾ N with equality if and only if N is an odd prime. Moreover, if equality holds, they gave an explicit construction for f ( z ) . In this paper, we look at the case when N is even. In particular, we investigate the following Conjecture Let f ( z ) and ζ be as above. If N > 2 is an even positive integer then max i | f ( ζ i ) | ⩾ N + 1 with equality if and only if N + 1 is a power of an odd prime. This conjecture was made after extensive computations. Partial results towards proving this conjecture are given.

Generalization of connection based on the concept of gradedq-differential algebra

We propose a generalization of the concept of connection form by means of a graded q-differential algebra Wq, where q is a primitive Nth root of unity, and develop the concept of curvature N-form for this generalization of the connection form. The Bianchi identity for a curvature N-form is proved. We study an Wq-connection on module and prove that every projective module admits an Wq-connection. If the module is equipped with a Hermitian structure, we introduce a notion of an Wq-connection consistent with the Hermitian structure.

Symbol algebras and cyclicity of algebras after a scalar extension

For a field F and a family of central simple F-algebras we prove that there exists a regular field extension E/F preserving indices of F-algebras such that all the algebras from the family are cyclic after scalar extension by E. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n with a primitive nth root of unity ρn. We construct a quasi-affine F-variety Symb(\( \mathcal{A} \)) such that for a field extension L/F Symb(\( \mathcal{A} \)) has an L-rational point if and only if \( \mathcal{A}{ \otimes_F}L \) is a symbol algebra. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n and K/F be a cyclic field extension of degree n. We construct a quasi-affine F-variety C(\( \mathcal{A} \),K) such that, for a field extension L/F with the property [KL : L] = [K : F], the variety C(\( \mathcal{A} \),K) has an L-rational point if and only if KL is a subfield of \( \mathcal{A}{ \otimes_F}L \).

Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing

We present two deterministic polynomial time algorithms for the following problem: check whether a sparse polynomial f(x) vanishes at a given primitive nth root of unity ζ n . A priori f(ζ n ) may be nonzero and doubly exponentially small in the input size. The existence of a polynomial time procedure in the case of factored n was conjectured by D. Plaisted in 1984, but all previously known algorithms are either randomized, or do not run in polynomial time. We apply polynomial zero testing algorithms to construct a nondeterministic polynomial time algorithm for the torsion point problem (TP). The problem TP is a particular case of the feasibility problem for a system of polynomial equations in complex numbers (coefficients of polynomials are integers). In the problem TP all coordinates of a solution must be roots of unity.

Minimal vanishing sums of roots of unity with large coefficients

A vanishing sum a 0 + a 1 ζ n + … + a a − 1 ζ n n − 1 = 0, where ζn is a primitive nth root of unity and the ais are non-negative integers is called minimal if the coefficient vector (a0, … , an−1) does not properly dominate the coefficient vector of any other such non-zero sum. We show that for every c∈ℕ there is a minimal vanishing sum of nth roots of unity with its greatest coefficient equal to c, where n is of the form 3pq for odd primes p, q. This solves an open problem posed by Lenstra Jr.

Connection on module over a graded q-differential algebra 1

We study a concept of a q-connection on a left module, where q is a primitive Nth root of unity. This concept is based on a notion of a graded q-differential algebra whose differential d satisfies dN = 0. We propose a notion of a graded q-differential algebra with involution and making use of this notion we introduce and study a concept of a q-connection consistent with a Hermitian structure of a left module. Assuming module to be a finitely generated free module we define the components of q-connection and show that these components with respect to different basises are related by gauge transformation. We also derive the relation for components of a q-connection consistent with Hermitian structure of a module.

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying dN = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.

Thesl2loop algebra symmetry of the twisted transfer matrix of the six-vertex model at roots of unity

We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity, q 2N = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are also valid for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the operators generate the sl 2 loop algebra in the sector of the total spin operator S Z ≡ N/2 (mod N). The infinite-dimensional symmetry leads to exponentially-large spectral degeneracies, as shown for the periodic boundary conditions (Deguchi T, Fabricius K and McCoy B M 2001 J. Stat. Phys. 102 701). Furthermore, we derive explicitly the sl 2 loop algebra symmetry for the periodic XXZ spin chain with an odd number of sites in the sector S Z ≡ N/2 (mod N) when q is a primitive Nth root of unity with N odd. Interestingly, in the case of N = 3, various conjectures of combinatorial formulae for the XXZ spin chain with odd sites have been given by Stroganov and other authors. We also note a connection to the spectral degeneracies of the eight-vertex model.

IRREDUCIBLE REPRESENTATIONS OF A CLASS OF FINITE-DIMENSIONAL ALGEBRAS

ABSTRACT In the previous papers1-2 we constructed an -dimensional Hopf algebra which is isomorphic to Drinfeld quantum double of Taft's Hopf algebra if and is a primitive nth root of unity, and studied the irreducible representations of . In this paper, we continue our study and examine the irreducible representations of when q is any nth root of 1. We give the structures of all simple -modules, and then classify them. We show that there are only distinct simple -modules up to isomorphism. Though is not a bialgebra, we show that the tensor product of two simple -modules possesses an -module structure in away similar to the case that q is a primitive nth root of 1. Then we give a sufficient and necessary condition for the tensor product of two simple -modules to be semisimple. More generally, we describe the structure of socal of the tensor product. Quantum groups arose from the study of quantum inverse scattering method, especially the Yang-Baxter equation. Quantum groups added new aspects to representation theory. Let be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. It was shown that R can be derived from a left H-module structure and a right H-comodule structure on M for some bialgebra H over k (see[3], [5]. The module and comodule structures satisfy a natural compatibility condition. M together with this structure is called a quantum Yang-Baxter H-module, which is also called a Yetter-Drinfeld module over H (see6-7. A quantum Yang-Baxter H-module is a (left) crossed H-bimodule.[5] Let H be a finite-dimensional Hopf algebra over afield k, and let be the Drinfeld's quantum double derived from H.[8] It is well known9-10 that a vector space M possesses a crossed H-bimodule structure if and only if M possesses a left -module structure. Hence the Yetter-Drinfeld module category is the same as the left -module category Suppose that and that is a primitive nth root of unity. Then is also a primitive nth root of unity. Taft constructed an -dimensional Hopf algebra in[11]. The 's form an interesting class of pointed Hopf algebras from a combinatorial point of view. When n is odd, provides an invariant of three-manifolds.[12] Generally, the double of is of interest in connection with knot theory. Kauffman and Radford showed[13] that is a ribbon Hopf algebra if and only if n is odd. In the paper,[1] the author constructed an infinite-dimensional noncommutative and noncocommutative Hopf algebra for any with . When q is a root of the nth cyclotomic polynomial over Z, has an -dimensional quotient Hopf algebra . If q is a primitive nth root of unity, then is isomorphic to as a Hopf algebra for any . If q is just a nth root of 1, then is merely an algebra.In[2], the author examined the irreducible representations of , equivalently, of , where and q is a primitive nth root of 1. In this paper, we will consider the more general case that q is any nth root of 1, and examine the irreducible representations of the algebra . In this case, let m be the order of q, then , and is an algebra which is not necessarily a bialgebra (Hopf algebra). Let . In Sec. 2, we give the classification of the simple -modules. We show that for any and there exists a unique l-dimensional simple -module , and that for any and , there exists a unique m-dimensional simple -module . We also show that any simple -module is either isomorphic to some or isomorphic to some . Upto isomorphisms, there are only distinct simple -modules. Though is not a bialgebra, we will show that the tensor product of two simple -modules possesses an -module structures which is similar to the case[2] that is a Hopf algebra. In Sec. 3, we consider the tensor product of two simple -modules U and V, and give a sufficient and necessary condition for the tensor product to be semisimple. We first show that if U and V are simple -module with dim or dim then is also a simple -module. Then we prove that is a semisimple -module if and only if , that with is semisimple if and only if , and that is semisimple if and only if or (mod m). Generally, we describe the structure of socal of the tensor product. In particular, if we take , then we can get all the results described in[2].