scholarly journals On the Tits–Weiss conjecture and the Kneser–Tits conjecture for and (With an Appendix by R. M. Weiss)

2021 ◽  
Vol 9 ◽  
Author(s):  
Seidon Alsaody ◽  
Vladimir Chernousov ◽  
Arturo Pianzola
Keyword(s):  
Algebraic Groups ◽  
Equivalence Classes ◽  
Structure Group ◽  
Arbitrary Field ◽  
Norm Principle

Abstract We prove that the structure group of any Albert algebra over an arbitrary field is R-trivial. This implies the Tits–Weiss conjecture for Albert algebras and the Kneser–Tits conjecture for isotropic groups of type $\mathrm {E}_{7,1}^{78}, \mathrm {E}_{8,2}^{78}$ . As a further corollary, we show that some standard conjectures on the groups of R-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type $^1\mathrm {E}_6$ .

scholarly journals Monothetic algebraic groups

2007 ◽  
Vol 82 (3) ◽  
pp. 315-324 ◽  
Author(s):  
Giovanni Falcone ◽  
Peter Plaumann ◽  
Karl Strambach
Keyword(s):  
Algebraic Group ◽  
Cyclic Subgroup ◽  
Algebraic Groups ◽  
Arbitrary Field

AbstractWe call an algebraic group monothetic if it possesses a dense cyclic subgroup. For an arbitrary field k we describe the structure of all, not necessarily affine, monothetic k-groups G and determine in which cases G has a k-rational generator.

scholarly journals Finite symmetric tensor categories with the Chevalley property in characteristic 2

2020 ◽  
pp. 2140010
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Function Algebra ◽  
Equivalence Classes ◽  
Closed Field ◽  
Arbitrary Field ◽  
Tensor Categories ◽  
Characteristic 2 ◽  
Chevalley Property

We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. Namely, we prove that every such category [Formula: see text] admits a symmetric fiber functor to the symmetric tensor category [Formula: see text] of representations of the triangular Hopf algebra [Formula: see text]. Equivalently, we prove that there exists a unique finite group scheme [Formula: see text] in [Formula: see text] such that [Formula: see text] is symmetric tensor equivalent to [Formula: see text]. Finally, we compute the group [Formula: see text] of equivalence classes of twists for the group algebra [Formula: see text] of a finite abelian [Formula: see text]-group [Formula: see text] over an arbitrary field [Formula: see text] of characteristic [Formula: see text], and the Sweedler cohomology groups [Formula: see text], [Formula: see text], of the function algebra [Formula: see text] of [Formula: see text].

Local-Global Norm Principle for Algebraic Groups

2004 ◽  
Vol 32 (3) ◽  
pp. 829-838 ◽  
Author(s):  
Pedro B. Barquero
Keyword(s):  
Algebraic Groups ◽  
Norm Principle

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

Cohomology of Arithmetic Subgroups of Algebraic Groups: II

1968 ◽  
Vol 87 (2) ◽  
pp. 279 ◽  
Author(s):  
M. S. Raghunathan
Keyword(s):  
Algebraic Groups

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

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