Superstability and central extensions of algebraic groups

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

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.

Galois Theory of Differential Equations, Algebraic Groups and Lie Algebras

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

Let G be a simple group of Lie type over an infinite locally finite field F. For any field K, we prove that the group algebra K [ G ] K[G] is semiprimitive. The argument here is a mixture of combinatorial and topological methods. Combined with earlier results, it now follows that any group algebra of an infinite locally finite simple group is semiprimitive. Furthermore, if the group is countably infinite, then the group algebra is primitive. In particular, if G is a simple group of Lie type over the field F, then K [ G ] K[G] is a primitive ring.

We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group $G$ is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.

A signalizer functor theorem for groups of finite Morley rank

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

We establish regularity conditions for ‐boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander–Mikhlin (HM) criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard, and de la Salle for . Our approach is partly based on a sharp local HM theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular non‐Toeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle‐based approach to Fourier multipliers in group algebras by Junge, Mei, and Parcet. A few related open problems are also discussed.

David Marker, Introduction to the model theory of fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 1–37. - David Marker. Model theory of differential fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 38–113. - Anand Pillay. Differential algebraic groups and the number of countable differentially closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 114–134. - Margit Messmer. Some model theory of separably closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 135–152. - Volume 63 Issue 2

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from mathbf {PSL}_n(q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is mathbf {PSp}_{2n}(q), mathbf {P}{varvec{Omega }}^+_{4n}(q), mathbf {P}{varvec{Omega }}^-_{4n}(q), ^3D_4(q), E_7(q), E_8(q), F_4(q), or G_2(q) with q even is the group algebra.

Semiprimitivity of group algebras of locally finite groups. II

We classify elementary abelian 2-subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian \(p\)-subgroups of compact simple Lie groups (equivalently, complex linear algebraic simple groups) of adjoint type started in Griess (Geom Dedicata 39(3):253–305, 1991).

Let K[?] denote the group algebra of an infinite locally finite group?. In recent years, the lattice of ideals of K[?] has been extensively studied under the assumption that? is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let B be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If B acts nontrivially on V by way of the homomorphism Φ: G → GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal ωK[V] is the unique proper G-stable ideal of K[V] when char K ¬= p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups G of D*, and the action of G on the group algebra K[A], where A is the additive group D + . In particular, properties of the quasi-simple group G come into play only in the final section of this paper.