Narrow Results

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras.

Given a representation $(\rho ;A)$ of a 3-Lie algebra $B$, we construct first-order cohomology classes by using derivations of $A$, $B$ and obtain a Lie algebra $G_{\rho }$ with a representation $\mathrm{\Phi }$ on $H^1(B;A)$. In the case that $\rho$ is given by an abelian extension $0\to A\hookrightarrow L\to B\to 0$ of 3-Lie algebras with $[A,A,L]=0$, we obtain obstruction classes for extensibility of derivations of $A$ and $B$ to those of $L$. An application of the representation $\mathrm{\Phi }$ to derivations is also discussed.

The aim of this paper is to introduce 3-Hom-ρ-Lie algebra structures generalizing the algebras of 3-Hom-Lie algebra. Also, we investigate the representations and deformations theory of this type of Hom-Lie algebras. Moreover, we introduce the definition of extensions and abelian extensions of 3-Hom-ρ-Lie algebras and show that associated to any abelian extension, there is a representation and a 2-cocycle.

Let ϕ be a Drinfel'd module defined over a finite extension K of 𝔽_q(T); we establish a uniform lower bound for the canonical height of a point of ϕ, rational over the maximal abelian extension of K, and thus solve the so-called abelian version of the Lehmer problem in this situation. The classical original problem (a non torsion point in 𝔾_m(ℚ^ab)) was solved by Amoroso and Dvornicich [1].

Soit ϕ un module de Drinfel'd défini sur une extension finie K de 𝔽_q(T); nous démontrons une minoration uniforme pour la hauteur canonique d'un point de ϕ, rationnel sur l'extension abélienne maximale de K, et résolvons ainsi la version dite abélienne du problème de Lehmer dans cette situation. Dans le cadre classique (un point d'ordre infini de 𝔾_m(ℚ^ab)), cette question a été résolue par Amoroso et Dvornicich dans [1].

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pⁿ-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.