Generalized Yetter–Drinfeld (quasi)modules and Yetter–Drinfeld–Long bi(quasi)modules for Hopf quasigroups

2019 ◽  
Vol 18 (02) ◽  
pp. 1950034
Author(s):  
Gui-Qi Shi ◽  
Xiao-Li Fang ◽  
Blas Torrecillas
Keyword(s):  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Hopf Quasigroup

As generalizations of Yetter–Drinfeld module over a Hopf quasigroup, we introduce the notions of Yetter–Drinfeld–Long bimodule and generalize the Yetter–Drinfeld module over a Hopf quasigroup in this paper, and show that the category of Yetter–Drinfeld–Long bimodules [Formula: see text] over Hopf quasigroups is braided, which generalizes the results in Alonso Álvarez et al. [Projections and Yetter–Drinfeld modules over Hopf (co)quasigroups, J. Algebra 443 (2015) 153–199]. We also prove that the category of [Formula: see text] having all the categories of generalized Yetter–Drinfeld modules [Formula: see text], [Formula: see text] as components is a crossed [Formula: see text]-category.

scholarly journals Galois criterion for torsion points of Drinfeld modules

2021 ◽  
pp. 1-12
Author(s):  
Chien-Hua Chen
Keyword(s):  
Maximal Ideal ◽  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Drinfeld Modules ◽  
Torsion Points ◽  
Ideal Formula ◽  
Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

Torsion Points of Drinfeld Modules

1995 ◽  
Vol 38 (1) ◽  
pp. 3-10
Author(s):  
Sunghan Bae ◽  
Jakyung Koo
Keyword(s):  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Locally Compact ◽  
Torsion Points ◽  
Complete Field ◽  
Rank 2

AbstractThe finiteness of K-rational torsion points of a Drinfeld module of rank 2 over a locally compact complete field K with a discrete valuation is proved.

Characteristic p Galois Representations That Arise from Drinfeld Modules

2000 ◽  
Vol 43 (3) ◽  
pp. 282-293 ◽  
Author(s):  
Nigel Boston ◽  
David T. Ose
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Galois Representations ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Absolute Galois Group ◽  
Characteristic P ◽  
Finite Characteristic ◽  
The Absolute

AbstractWe examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group’s action on the division points of an appropriate Drinfeld module.

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

2021 ◽  
Vol 28 (02) ◽  
pp. 213-242
Author(s):  
Tao Zhang ◽  
Yue Gu ◽  
Shuanhong Wang
Keyword(s):  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category ◽  
Symmetric Monoidal Category ◽  
Category Equivalence ◽  
Hopf Modules ◽  
Hopf Quasigroup

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

scholarly journals Towards the Full Mordell–Lang Conjecture for Drinfeld Modules

2010 ◽  
Vol 53 (1) ◽  
pp. 95-101
Author(s):  
Dragos Ghioca
Keyword(s):  
Finite Rank ◽  
Drinfeld Module ◽  
Drinfeld Modules

AbstractLet ϕ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of . We show that the intersection of X with a finite rank ϕ-submodule of is finite.

scholarly journals Two-cocycles and cleft extensions in left braided categories

2020 ◽  
pp. 2140013
Author(s):  
István Heckenberger ◽  
Kevin Wolf
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Cleft Extensions ◽  
Braided Categories

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra [Formula: see text] a Yetter–Drinfeld module braids from the left with [Formula: see text]-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and [Formula: see text]-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques.

On Tate-Drinfeld Modules

1992 ◽  
Vol 35 (2) ◽  
pp. 145-151
Author(s):  
Sunghan Bae ◽  
Pyung-Lyun Kang
Keyword(s):  
Elliptic Curves ◽  
Drinfeld Module ◽  
Drinfeld Modules

AbstractThe Tate-Drinfeld module is defined by Gekeler. We define the Tate- Drinfeld map and show the analogous properties concerning Tate elliptic curves and Tate map.

Symmetry and pseudosymmetry of v-Yetter–Drinfeld categories for Hom–Hopf algebras

2017 ◽  
Vol 14 (09) ◽  
pp. 1750129 ◽  
Author(s):  
Xiao-Li Fang ◽  
Tae-Hwa Kim ◽  
Xiao-Hui Zhang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Tensor Category ◽  
Baxter Equation ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Sufficient And Necessary Conditions

The purpose of this paper is to introduce the category of [Formula: see text]-Yetter–Drinfeld modules ([Formula: see text]) over a Hom–Hopf algebra. We first prove that every category of [Formula: see text]-Yetter–Drinfeld modules over a Hom–Hopf algebra with a bijective antipode [Formula: see text] is a braided tensor category and that every [Formula: see text]-Yetter–Drinfeld module can provide the solution of the Hom–Yang–Baxter equation. Secondly, we find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Finally, we construct examples of [Formula: see text]-Yetter–Drinfeld modules by a quasitriangular Hom–Hopf algebra and study their relationship.

Endomorphism algebras over weak (α, β)-Yetter–Drinfeld modules

2015 ◽  
Vol 14 (07) ◽  
pp. 1550097
Author(s):  
Ling Liu ◽  
Bingliang Shen
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Azumaya Algebra ◽  
Finite Dimensional ◽  
Endomorphism Algebras

Let H be a weak Hopf algebra with a bijective antipode, α, β ∈ Aut weak Hopf (H) and M a finite-dimensional weak (α, β)-Yetter–Drinfeld module. Then in this paper we prove that the endomorphism algebras End Hs(M) and End Ht(M) op endowed with certain structures become algebras in H𝒲𝒴𝒟H and we also study the isomorphic relations between different endomorphism algebras. We prove that End Hs(M) endowed with certain structures becomes an H-Azumaya algebra.

Cryptanalysis of a cryptosystem based on Drinfeld modules

2006 ◽  
Vol 153 (1) ◽  
pp. 12
Author(s):  
S.R. Blackburn ◽  
C.F.A. Cid ◽  
S.D. Galbraith
Keyword(s):  
Drinfeld Modules
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