Stably Cayley Groups in Characteristic Zero

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

Degree 2 cohomological invariants of linear algebraic groups

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf.[4]).(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf.[4]).(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf.[6], [2], [4]).

This thesis contains two major parts. In the first part, I introduce a new theory of modules with iterative connection. This theory unifies the theory of modules with connection in characteristic zero as given by N. Katz and the theory of iterative differential modules in positive characteristic as given by B. H. Matzat und M. van der Put. The second part of this work is about the differential Abhyankar conjecture for iterative Picard-Vessiot extensions (IPV-extensions). This conjecture is concerned with the problem which linear algebraic groups occur as iterative differential Galois groups of IPV-extensions with restricted singular locus. In this thesis, I prove the differential Abhyankar conjecture for connected groups and give necessary and sufficient conditions for connected groups for being realisable with given singular points.

A D-derivation of a Lie algebra L is a linear map φ for which there exists a derivation D such that φ([x,y])=[φ(x),y]+[x,D(y)] for all x,y∈L. This paper presents explicit structural results concerning D-derivations in Lie algebras over arbitrary fields. It is established that the set of D-derivations forms a Lie algebra, which decomposes as the sum of derivations and centroids, intersecting precisely at the space of central derivations. For centerless Lie algebras, the inclusion chain for D-derivations within existing derivation classes is completed, resulting in a refined hierarchy. It is proven that for both perfect and centerless Lie algebras, D-derivations decompose as a direct sum of derivations and centroids. In particular, for semisimple Lie algebras, it is shown that DerD(L)=ad(L)⊕C(L), and for simple Lie algebras over an algebraically closed field of characteristic zero, DerD(L)=ad(L)⊕FidL. Furthermore, for any centerless Lie algebra, the Lie algebra of D-derivations is shown to be isomorphic to the semidirect product of the derivation and centroid algebras, with explicit descriptions provided for semisimple and solvable cases. Examples involving so(3), so(1,3), aff(1), and h3 confirm these decompositions and offer matrix realizations of their D-derivations, thereby supporting and illustrating the main theorems.

Linearity for actions on vector groups

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.

We solve the question about the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups over the algebraic closures of finite fields of positive characteristic.

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

Reductive subgroups of reductive groups in nonzero characteristic

SynopsisBy examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.

Let K be a differential field with algebraically closed field of constants CK . Let K P V ∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X ( K P V ∞ ) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X ( K P V ∞ ) = X ( K diff ) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K.

An associative unitary (respectively, Lie) algebra is said to be proper in case the intersection of all of its cofinite two-sided (respectively, Lie) ideals is zero.Using the Hopf algebra structure of UL, it is shown that over a field of characteristic zero a Lie algebra is proper if and only if its universal enveloping algebra is proper.(In the finite-dimensional case this provides a new proof of a result of Harish-Chandra.)The analogous result for Lie palgebras and their restricted universal enveloping algebras holds and is proved by the same technique.Notational conventions.Throughout this paper, p will denote a prime number.Barring mention to the contrary, we work over an arbitrary ground field K except that when Lie p-algebras (^restricted Lie algebras of characteristic p) are discussed, the ground field must have characteristic p (cf. [1, p. 78, Exercise 20] or [7, p. 187, Definition 4]).The unmodified words "algebra" and "coalgebra" shall, respectively, mean "associative unitary algebra" and "associative counitary coalgebra."If A is an algebra, (A) shall denote the Lie algebra whose underlying vector space coincides with that of A, and whose bracket is defined for all elements x and y of A by lx>y]c(A) = x y -y x where "" denotes the multiplication of A; while P(A) shall denote the Lie p-algebra whose underlying Lie algebra is (A) and whose p-mapping, x h-> z'pl, is the pth power mapping, x >-> xp, of the algebra A. For any algebra map f: A-* B, (f): (A) -> (B) (respectively, P(F): P(A) -P(B)) shall denote the map of Lie algebras (respectively, Lie p-algebras) which coincides with / as a vector space map.For any Lie algebra L, UL shall denote the universal enveloping algebra of L; while for any Lie p-algebra L, VL shall denote the restricted universal enveloping algebra of L; furthermore, i and j shall denote, respectively, the canonical injections i: L -> (UL) and j: L - P(VL).Occasionally, the vector space whose only element is zero shall be denoted by 0, and, occasionally, the end of proof shall be landmarked by the symbol D. DEFINITION 1.(a) If A is an algebra (respectively, a Lie algebra), denote by C.I.(A) the collection of all cofinite two-sided ideals (respectively, all cofinite Lie