On central extensions of algebraic groups

1999 ◽  
Vol 64 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Tuna Altinel ◽  
Gregory Cherlin
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Morley Rank ◽  
Central Extensions ◽  
Perfect Group ◽  
Theoretic Result ◽  
Abstract Group ◽  
Finite Morley Rank

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem 1. Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0 = 1.Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary 1. Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4], Section 27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of tame simple K*-groups of finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of tameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from K-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.

scholarly journals Zero-Separating Invariants for Linear Algebraic Groups

2015 ◽  
Vol 59 (4) ◽  
pp. 911-924 ◽  
Author(s):  
Jonathan Elmer ◽  
Martin Kohls
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Elementary Proof ◽  
Algebraic Groups ◽  
Linear Algebraic Group ◽  
Null Cone ◽  
Closed Field ◽  
Unipotent Group ◽  
Linear Algebraic Groups ◽  
Separating Invariants

AbstractAbstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

Classes of unipotent elements in simple algebraic groups. I

1976 ◽  
Vol 79 (3) ◽  
pp. 401-425 ◽  
Author(s):  
P. Bala ◽  
R. W. Carter
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Adjoint Action ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Elements

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

On solvable centerless groups of Morley rank 3

1993 ◽  
Vol 58 (2) ◽  
pp. 546-556
Author(s):  
Mark Kelly Davis ◽  
Ali Nesin
Keyword(s):  
Nilpotent Group ◽  
General Structure ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Closed Field ◽  
Morley Rank ◽  
Nonnilpotent Group ◽  
Finite Morley Rank ◽  
Torsion Free Subgroup ◽  
Torsion Free

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

scholarly journals On Cocharacters Associated to Nilpotent Elements of Reductive Groups

2008 ◽  
Vol 190 ◽  
pp. 105-128 ◽  
Author(s):  
Russell Fowler ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Rank ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Reductive Subgroup ◽  
Good For

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

scholarly journals On function field Mordell–Lang and Manin–Mumford

2016 ◽  
Vol 16 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Franck Benoist ◽  
Elisabeth Bouscaren ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Function Field ◽  
Positive Characteristic ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Morley Rank ◽  
Main Interest ◽  
Dichotomy Theorems ◽  
Finite Morley Rank ◽  
Crucial Ingredient

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.

scholarly journals Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

2018 ◽  
Vol 21 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Donna M. Testerman ◽  
Alexandre E. Zalesski
Keyword(s):  
Algebraic Groups ◽  
Jordan Block ◽  
Linear Algebraic Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Jordan Normal Form ◽  
Unipotent Element ◽  
Content Type ◽  
Simply Connected ◽  
Graphic Content

AbstractLetGbe a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed fieldFof characteristic{p\geq 0}, and let{u\in G}be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation ofG. Then the Jordan normal form of{\phi(u)}contains at most one non-trivial block if and only ifGis of type{G_{2}},uis a regular unipotent element and{\dim\phi\leq 7}. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

scholarly journals Harder–Narasimhan reduction of a principal bundle

2004 ◽  
Vol 174 ◽  
pp. 201-223 ◽  
Author(s):  
Indranil Biswas ◽  
Yogish I. Holla
Keyword(s):  
Algebraic Group ◽  
Principal Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Mild Assumption

AbstractLet E be a principal G–bundle over a smooth projective curve over an algebraically closed field k, where G is a reductive linear algebraic group over k. We construct a canonical reduction of E. The uniqueness of canonical reduction is proved under the assumption that the characteristic of k is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of k is positive.

Groups of finite Morley rank with transitive group automorphisms

1989 ◽  
Vol 54 (3) ◽  
pp. 1080-1082
Author(s):  
Ali Nesin
Keyword(s):  
Finite Order ◽  
Short Note ◽  
Algebraic Groups ◽  
Transitive Group ◽  
Simple Groups ◽  
Morley Rank ◽  
Nontrivial Center ◽  
Finite Morley Rank ◽  
Group Automorphisms

The aim of this short note is to prove the following result:Theorem. Let G be a group of finite Morley rank with Aut G acting transitively on G/{1}. Then G is either abelian or a bad group.Bad groups were first defined by Cherlin [Ch]: these are groups of finite Morley rank without solvable and nonnilpotent connected subgroups. They have been investigated by the author [Ne 1], Borovik [Bo], Corredor [Co], and Poizat and Borovik [Bo-Po]. They are not supposed to exist, but we are far from proving their nonexistence. This is one of the major obstacles to proving Cherlin's conjecture: infinite simple groups of finite Morley rank are algebraic groups.If the group G of the theorem is finite, then it is well known that G ≈ ⊕Zp for some prime p: clearly all elements of G have the same order, say p, a prime. Thus G is a finite p-group, so has a nontrivial center. But Aut G acts transitively; thus G is abelian. Since it has exponent p, G ≈ ⊕Zp.The same proof for infinite G does not work even if it has finite Morley rank, for the following reasons:1) G may not contain an element of finite order.2) Even if G does contain an element of finite order, i.e. if G has exponent p, we do not know if G must have a nontrivial center.

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