Galois structure and de Rham invariants of elliptic curves

Let K be a number field with ring of integers OK. Suppose a finite group G acts numerically tamely on a regular scheme X over OK. One can then define a de Rham invariant class in the class group Cl(OK[G]), which is a refined Euler characteristic of the de Rham complex of X. Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois étale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive genus, we show that a given étale action of G on X extends to a numerically tame action on a regular model if and only if this is possible on the minimal model. Finally, we characterize the classes in Cl(OK[G]) which are realizable as the de Rham invariants for minimal models of elliptic curves when G has prime order.     

Homology of certain algebras defined by graphs

Let K(G) for a finite graph G with vertices v₁,...,v_n denote the K-algebra with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G. We describe centralizers of monomials, show that the centralizer of a monomial is again a graph algebra, prove a unique factorization theorem for factorizations of monomials into commuting factors, compute the homology of K(G), and show that K(G) is the homology ring of a certain loop space. We also construct a K(π, 1) explicitly where π is the group with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G.     

Computing invariants of algebraic groups in arbitrary characteristic

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring KG[X] in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes KG[X] in terms of a so-called colon-operation. From this, generators of KG[X] can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is KG[X]. Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal.     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K₀ is connected and the action of K₀ on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^∗(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.     

Strength of convergence in the orbit space of a transformation group

Let (G,X) be a second-countable transformation group with G acting freely on X. It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X/G provide equivalent ways of quantifying the extent of nonproperness of the action. These notions are linked via the representation theory of the transformation-group C^∗-algebra C₀(X)?G.     

Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group ${\bigcup_{n\in\mathbb {N}}G_n}We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group è_{n ? \mathbb N}G_n{\bigcup_{n\in\mathbb {N}}G_n} where the G _n are Banach Lie groups.     

On strong reality of the unipotent Lie-type subgroups over a field of characteristic 2

A group G is called strongly real if its every nonidentity element is strongly real, i.e. conjugate with its inverse by an involution of G. We address the classical Lie-type groups of rank l, with l ≤ 4 and l ≤ 13, over an arbitrary field, and the exceptional Lie-type groups over a field K with an element η such that the polynomial X ² + X + η is irreducible either in K[X] or K ₀[X] (in particular, if K is a finite field). The following question is answered for the groups under study: What unipotent subgroups of the Lie-type groups over a field of characteristic 2 are strongly real?     

Neighbourhoods of transitive graphs and GRR's

Let X be a vertex-transitive graph with complement $\text{X}$. We show that if both N, the neighbourhood of a vertex in X, and $\text{N}$, the neighbourhood of a vertex in $\text{X}$, are disconnected, then either X is isomorphic to K₃ × K₃ or both N and $\text{N}$ contain isolated vertices. We characterize the graphs which satisfy this last condition and show in consequence that they admit automorphisms of the form (12)(34). It follows that if X is a GRR for some graph G then at least one of N and $\text{N}$ is connected. (X is said to be a graphical regular representation, or GRR, for G if its automorphism group is isomorphic to G and acts regularly on its vertices.) Using this result we determine those groups generated by their involutions which do not have a GRR. The largest such group has order 18. As a corollary we conclude that all non-abelian simple groups have GRR's.     

Čech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory

Let X be a topological space equipped with the action of a finite group ∏. We may form the twisted group ring of ∏, coefficients being elements of the ring of continuous functions on X with values in the real numbers, complex numbers or quaternions. In this paper we show how the Witt groups of hermitian forms of various kinds over these twisted group rings can be described in terms of the real, complex or quaternionic equivariant K-theory of X.     

Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] ^G ]?=?[k[X]] ^G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ k[X] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?     

The alternating groups and K3 surfaces

In this note, we consider all possible extensions G of a non-trivial perfect group H acting faithfully on a K3 surface X. The pair (X,G) is proved to be uniquely determined by G if the transcendental value of G is maximum. In particular, we have , if H is the alternating group A₅ and normal in G.     

Topological K-Theory of Algebraic K-Theory Spectra

Let KX denote the algebraic K-theory spectrum of a regular Noetherian scheme X. Under mild additional hypotheses on X, we construct a spectral sequence converging to the topological K-theory of KX. The spectral sequence starts from the étale homology of X with coefficients in a certain copresheaf constructed from roots of unity. As examples we consider number rings, number fields, local fields, smooth curves over a finite field, and smooth varieties over the complex numbers.     

Extensions of zip rings

A ring R is called right zip provided that if the right annihilator r_R(X) of a subset X of R is zero, r_R(Y)=0 for a finite subset Y⊆X. Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; Characterize a ring R such that Mat_n(R) is right zip; When does R being a right zip ring imply R[G] being right zip when G is a finite group? In this note, we continue the study of the extensions of noncommutative zip rings based on Faith's questions.     

Symplectic <Emphasis Type="Italic">S</Emphasis><Subscript>5</Subscript> action on symplectic homotopy K3 surfaces

Let X be a symplectic homotopy K3 surface and G = S ₅ act on X symplectically. In this paper, we give a weak classification of the G action on X by discussing the fixed-point set structure. Besides, we analyse the exoticness of smooth structures of X under the action of G.     

Simplicity of Skew Group Rings of Abelian Groups

Necessary and sufficient conditions for simplicity of a general skew group ring A ?_σ G are not known. In this article, we show that a skew group ring A ?_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ?_σ G is simple.     

On the structure and representations of non-balanced quantum doubles

For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space (kX ^op )* ? kG (see Theorem 2.1), called a non-balanced quantum double and denoted by D _X (G). In this paper, some Hopf algebra properties of D _X (G) are given, the representation types of D _X (G) viewed as a k-algebra are discussed, the algebra structure and module category over D _X (G) are studied. Since the Hopf algebra structure of non-balanced quantum double D _X (G) generalizes the usual quantum double D(G) for a finite group G, all results about D _X (G) in this paper can also be used to describe D(G) as a special case and the universal R-matrix of D _X (G) provides more solutions of Yang-Baxter equation.     

Bounds on smooth matrix coefficients on L 2-spaces

Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and ${\mathfrak k}Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and \mathfrak k{\mathfrak k}-smooth matrix coefficients of the regular representation L ²(X) under an assumption about supp(L²(X)) ?[^(G)]_K{{\rm supp}(L^2(X)) \cap \hat G_K}. Furthermore, we show that this bound holds for unitary representations that are weakly contained in L ²(X). Our result generalizes a result of Cowling–Haagerup–Howe (J Reine Angew Math 387:97–110, 1988). As an example, we discuss the matrix coefficients of the O(p, q) representation L²(\mathbbR^p+q){L^2(\mathbb{R}^{p+q})}.     

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002