On splitting of the normalizer of a maximal torus in linear groups

2015 ◽  
Vol 14 (07) ◽  
pp. 1550114 ◽  
Author(s):  
Alexey Galt
Keyword(s):  
Finite Field ◽  
Maximal Torus ◽  
Linear Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Maximal Tori

We describe linear groups over an algebraically closed field in which the normalizer of a maximal torus splits over the torus. We describe linear groups over a finite field and their maximal tori in which the normalizer of the maximal torus splits over the torus.

Solvable Subgroups and their Lie Algebras in Characteristic p

1979 ◽  
Vol 31 (2) ◽  
pp. 308-311
Author(s):  
David J. Winter
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Torus ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Connected Component ◽  
Characteristic P ◽  
Maximal Tori ◽  
Commutative Subalgebras

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .

CARTAN SUBMONOIDS OF ALGEBRAIC MONOIDS

1995 ◽  
Vol 05 (03) ◽  
pp. 367-377 ◽  
Author(s):  
WENXUE HUANG
Keyword(s):  
Maximal Torus ◽  
Unit Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Completely Regular ◽  
Algebraic Monoid ◽  
Algebraic Monoids

Let M be an irreducible linear algebraic monoid defined over an algebraically closed field K with idempotent set E(M), T a maximal torus of the unit group G of M. We call CM(T)c a Cartan submonoid of M. The following are proved: (1) If M is reductive with zero or completely regular, then CM(T) is irreducible and regular and [Formula: see text]; (2) If M is regular, then M is solvable iff NM(CM(T))=CM(T), in which case, CM(T) is irreducible and regular; (3) If M is regular, then [Formula: see text].

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

2021 ◽  
Vol 21 (4) ◽  
pp. 451-461
Author(s):  
Massimo Giulietti ◽  
Motoko Kawakita ◽  
Stefano Lia ◽  
Maria Montanucci
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Finite Field ◽  
Symmetric Group ◽  
Closed Field ◽  
Complex Field ◽  
Hermitian Curve ◽  
Full Automorphism Group ◽  
Algebraically Closed Field

Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)−12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

scholarly journals Abelian varieties over finite fields as basic abelian varieties

2017 ◽  
Vol 29 (2) ◽  
pp. 489-500 ◽  
Author(s):  
Chia-Fu Yu
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Finite Fields ◽  
Mass Formula ◽  
Abelian Varieties ◽  
Closed Field ◽  
Abelian Surfaces ◽  
Algebraically Closed Field

AbstractIn this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic ${p>0}$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

Rings which admit elimination of quantifiers

1978 ◽  
Vol 43 (1) ◽  
pp. 92-112 ◽  
Author(s):  
Bruce I. Rose
Keyword(s):  
Finite Field ◽  
Prime Ring ◽  
Chain Condition ◽  
Closed Field ◽  
Matrix Rings ◽  
Algebraically Closed Field ◽  
Finite Set ◽  
Descending Chain Condition ◽  
Infinite Center ◽  
Ordered Pairs

AbstractWe say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers.Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field.A ring is prime if it satisfies the sentence: ∀x∀y∃z (x =0 ∨ y = 0∨ xzy ≠ 0).Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn)⊕GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let be the set of ordered pairs (f, Q) where Q is a finite set of primes and such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q ∈ Qf(q), for some (f, Q) in .Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to.Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to.In contrast to Theorems 2 and 4, we haveTheorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition.We also generalize Theorems 1, 2 and 4 to alternative rings.

scholarly journals Reidemeister spectrum of special and general linear groups over some fields contains 1

2019 ◽  
Vol 18 (08) ◽  
pp. 1950153 ◽  
Author(s):  
Timur Nasybullov
Keyword(s):  
Transcendence Degree ◽  
General Linear ◽  
Linear Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
General Linear Groups ◽  
Zero Characteristic ◽  
Field Automorphism

We prove that if [Formula: see text] is an algebraically closed field of zero characteristic which has infinite transcendence degree over [Formula: see text], then there exists a field automorphism [Formula: see text] of [Formula: see text] and [Formula: see text] such that [Formula: see text]. This fact implies that [Formula: see text] and [Formula: see text] do not possess the [Formula: see text]-property. However, if the transcendece degree of [Formula: see text] over [Formula: see text] is finite, then [Formula: see text] and [Formula: see text] are known to possess the [Formula: see text]-property [13].

scholarly journals On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

2021 ◽  
Vol 14 (5) ◽  
pp. 547-553
Author(s):  
Peter Danchev ◽  
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Closed Field ◽  
Multilinear Algebra ◽  
Infinite Field ◽  
Algebraically Closed Field ◽  
Periodic Matrix ◽  
Nilpotent Matrix ◽  
Square Matrix

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

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