Explicit formulae for τ ′ r(L(p, q)) and τ(L(p, q)) are obtained for all L(p, q). There are three systems of invariants of Witten-type for closed oriented 3manifolds: 1. {τr(M), r ≥ 2; τ ′ r(M), r odd ≥ 3}, where τr was defined by Reshetikhin and Turaev [1], and τ ′ r was defined by Kirby-Melvin [2] 2. {Θr(M,A), r ≥ 1,whereA is a 2r-th primitive root of unity} defined by Blanchet, Habegger, Mas...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

On the 15th of September 2008, a power-sharing agreement was brokered between the major political parties in Zimbabwe, namely, the Tsvangirai-led Movement for Democratic Change (MDC-T), the Mutambara-led Movement for Democratic Change (MDC-M) and the then ruling Zimbabwe African National Union–Patriotic Front (ZANU-PF). This Global Political Agreement (GPA) paved the way for the establishment o...

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F . Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H(U, Fp) as an Fp[GF /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H(U, Fp) was determined onl...

We introduce the notion of circular words with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word abaababaabaab . . .. Second, we present an expression of the sequence A004146 of [S] in terms of a product of expressions involving roo...

A sequence of functions F = {fn(q)}∞ n=1 satisfies the functional equation for multiplication of quantum integers if fmn(q) = fm(q)fn(qm) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with coefficients in Q that satisfy this functional equation. 1. The functional equation for multiplication of quantum integers Let N = {1, 2, 3, . . ...

Recently, Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor p = t + 5t + I5t + 25/ + 25 as translates of Gaussian periods. Later, Schoof and Washington showed that these units were fundamental units. In this note, we observe that Lehmer's family comes from the covering of modular curves X",(25) —► ,Y0(25). This gives a conceptual explanation for the ...

Let G be a finite, quaternion-free group with exponent e, let F be a field of characteristic zero and let x be an absolutely irreducible character of G. Suppose that a Sylow 2-subgroup of the Galois group of F(J\) over F is cyclic. It is shown that if x is not real valued, then the Schur index of x over F is odd. Let G be a finite group with exponent e and let F be a field of characteristic zer...

We present a definition for free quantum groups. The idea is that these must satisfy S n ⊂ G ⊂ U + n , along with a technical representation theory condition. We work out in detail the case of quantum analogues of the hyperoctahedral group Hn. We first consider the hypercube in R , and show that its quantum symmetry group is in fact a q-deformation of On at q = −1. Then we consider the space fo...

We introduce the notion of circular words with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word abaababaabaab . . .. Second, we present an expression of the sequence A004146 of [S] in terms of a product of expressions involving roo...