Showing papers in "American Journal of Mathematics in 1990"

Forms in odd degree extensions and self-dual normal bases

Abstract: Introduction. Let K be a field. Springer has proved that an anisotropic quadratic form over K is also anisotropic over any odd degree extension of K (see [31], [14]). If the characteristic of K is not 2, this implies that two nonsingular quadratic forms that become isomorphic over an extension of odd degree of K are already isomorphic over K (see [31]). In [27], Serre reformulated the latter Statement äs follows: if O is an orthogonal group over K, then the canonical map of Galois cohomology sets

Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures

Abstract: Introduction. Let L be a finite dimensional Lie algebra over the real numbers, R, and let L* be its dual vector space. It is well-known [24] that the Lie algebra structure on L defines a natural Poisson structure on L*-in fact this was already known to Lie [24]-and these Poisson structures are exactly what are now called the linear Poisson structures. Given a manifold M equipped with a Poisson structure, { , }, one can seek deformation quantizations "in the direction of { , }", as first studied in [3]. These are, loosely speaking, one-parameter families, {*h}lER, of deformations of the pointwise multiplication on C<(M) (or an appropriate subalgebra), such that

Courbes tracees sur un germe d'hypersurface

Abstract: Soit k un corps, A une k-algebre locale, noetherienne, complete pour la topologie definie par son ideal maximal M et (V, v) = Spec A le germe de k-variete algebroide associee. Dans tout ce qui suit, nous supposons v rationnel sur k. Peu apres la publication de [H1], J. Nash entreprit, dans un preprint non publie, l'etude de l'espace N des arcs traces sur (V, v). Rappelons qu'il designe par "arc", une courbe parametree tracee sur (V, v), c'est-'a-dire, k[[t]] etant l'anneau des series formelles a une indeterminee sur k, un k-morphisme local continu 0V,v -> k[[t]]. Plus precisement, il commencait l'investigation des liens entre la structure de WC et celle de l'image reciproque de v dans une resolution des singularites quelconque de (V, v). WC ne peut etre muni d'une structure algebrique mais porte naturellement une structure pro-algebrique intrinseque. Pour tout entier i : 1, designons par "V-arc" (ou "V-courbe") a l'ordre i un k-morphisme local Cv,v -> k[[t]]/(t)" 1 et par Xi l'ensemble des "V-arcs" 'a l'ordre i. Xi a une structure naturelle d'ensemble algebrique affine (voir Section 0 pour une definition detaillee) et si 1 S j S i, la projection canonique:

Degree of the bicanonical map of a surface of general type

Abstract: Let 5 be a smooth complex projective surface of general type, rt: 5 ?> FPn~l be the map defined by nKs. <&? is called pluricanonical map for n ^ 2. Due to the works by Bombieri and others in the 1970s, the nature of $n is determined except when n = 2 and for some low values of invariants of 5 (see [1] for an up-to-date treatment and references). These maps play an important role in the classification of surfaces of general type. Recently, the remaining case of the bicanonical map $2 for low values of invariants has aroused new research interests, due to the work of Francia [8], and especially the remarkable method of Reider ([15], see also [6], [14], [4]). They have proved the following result (among other things).

Kazhdan groups, cocycles and trees

Abstract: We study cocycles of Kazhdan group actions with values in groups acting on trees. In particular we show that all cocycles of finite measure preserving actions of Kazhdan groups taking values in a free group are cohomologically trivial. Analogous techniques show that a Kazhdan equivalence relation is not treeable. 1. Introduction. Let G be a locally compact group. Recall (Zi) that a unitary representation p of G on a Hilbert space We almost has invariant vectors if for all E > 0 and compact subsets K of G there is a unit vector v E We such that 11p(g)v - vllx < E for all g E K. We say that G has Kazhdan's property T or that G is a Kazhdan group if all unitary representations of G that almost have invariant vectors actually

Deformations of Lie Algebra Representations

Abstract: Let G be a reductive algebraic group defined over an algebraically closed field k of positive characteristic p. In this paper, we continue our previous investigation [7] of the representation theory of the Lie algebra f of G. A key ingredient in our earlier work involved the consideration of a family {A,}X,,(s) of finite dimensional algebras parameterized by the set 0(QS) = 1HomkA1g(Y('b), k) of characters on a certain central subalgebra 0('6) of the universal enveloping algebra U('6) of G. Each irreducible %-module is naturally an irreducible module for exactly one such Ax, with the special case X = 0 corresponding to restricted%-modules. We introduce a so-called "deformation algebra" s4 intermediate between the universal enveloping algebra U(%) and certain of the algebras Ax. Here, sl is a quotient of U(Q) such that each Ax containing the restricted enveloping algebra V(a) of a fixed Borel subalgebra 2A is in turn a quotient of s4. Given a module MX for such an A, we introduce the important notion of a deformation family of MX in Definition 1.3. The hypothesis that the behavior of Mx is related to that of the corresponding modules for "near-by" algebras At motivates the present study. A basic result [7; Theorem 3.2] reduces the representation theory of AX for X =$ 0 to that of an associated A; for a subalgebra of % to which the restriction t of X is "nilpotent" (in the sense of [121). Thus, a study of the representation theory of the algebras AX for X a nilpotent character subsumes the classical problem of determining the irreducible representations of the Lie algebra %. Each G-conjugacy class in the "nullcone" X of nilpotent elements of 0(Y(f) defines a family of isomorphic algebras Ax. Presumably, the restricted enveloping algebra V(%) = Ao corresponding to the conjugacy class {O} is the most complicated algebra of this kind. Its representation theory has been extensively investigated in recent years because of its intimate connections

The Group of Automorphisms of a Semisimple Hopf Algebra Over a Field of Characteristic 0 is Finite

Abstract: Introduction. Suppose that A is a semisimple Hopf algebra over a field of characteristic 0, or that A is a semisimple cosemisimple involutory Hopf algebra over a field k of characteristic p > dim A. In this paper we prove that the group AutHopf(A) of Hopf algebra automorphisms of A is finite. We show that the semigroup EndHOpf(A) of Hopf algebra endomorphisms of A is also finite as a corollary. Generally the group of Hopf algebra automorphisms of a finite-dimensional Hopf algebra need not be finite. For any positive integer n and any field k, we construct a family of finite-dimensional Hopf algebras over k with automorphism group GLn(k). The fact that AutHopf(A) is finite is a consequence of a theorem of

ON THE POLES OF p-ADIC COMPLEX POWERS AND THE b- FUNCTIONS OF PREHOMOGENEOUS VECTOR SPACES

Abstract: Introduction. In the real case, it is well known that there exists an intimate relation between poles of complex powers of polynomial functions and the roots of the b-functions (cf. [B]). Many examples suggest that a similar relation between them would exist also in the p-adic case. Recently the case of polynomials over a p-adic field in two variables has been closely examined (cf. F. Loeser [L], W. Veys [V]). In this paper we shall deal with the case of relative invariants of prehomogeneous vector spaces over a p-adic field. Let us explain our problem more precisely. Let K be a p-adic field, i.e., a finite algebraic extension of Qp. We denote by ?)K the maximal compact subring of K, by rrT)K the ideal of nonunits of )K, and put q = #(DKI/'XC)K). We denote by | IKthe absolute value on K normalized as ITWK = q-. Let fQ(KX) be the set of quasi-characters of KX. Then the identity component fl(KX)O of fl(KX) consists of w,(t) = Itls for all s in C and fl(Kx)o CX under ws q-S. Let (G, p, V) be a prehomogeneous vector space (abbrev. P.V.) defined over K, where we assume that

Nonlinear Elliptic Boundary Value Problems at Strong Resonance

Abstract: Let A be a linear elliptic partial differential operator on a domain Ω⊂R n such that the boundary value problem (1.1) Au=λu in Ω, Bu=0 on ∂Ω is selfadjoint and has a discrete set of isolated eigenvalues bounded from below and no other spectrum

One-Parameter Automorphism Groups of the Injective II$_1$ Factor Arising from the Irrational Rotation C^*-Algebra

Abstract: We show a certain one-parameter automorphism group of the injective II1 factor R arising from the irrational rotation C∗-algebra Aθ is cocycle conjugate to an infinite tensor product type action, hence, unique up to cocycle conjugacy. An SL(2,Z)-action on Aθ, a Rieffel projection in Aθ, and central sequence technique in R are used. §0 Introduction In the irrational rotation C∗-algebra Aθ with uv = evu, consider the following one-parameter automorphism group αt: αt(u) = eu, αt(v) = ev. Here λ and μ are non-zero real numbers with λ/μ / ∈ Q. We extend this one-parameter automorphism group to the weak closure R of Aθ with respect to the trace τ , which is the AFD (approximately finite dimensional) II1 factor. We will show this one-parameter automorphism group is cocycle conjugate to an infinite tensor product type one-parameter automorphism group with full Connes spectrum R if and only if λ/μ is not in the GL(2,Q) orbit of θ. Then such a one-parameter automorphism group is unique up to cocycle conjugacy by our previous result [12]. *Alfred P. Sloan Doctoral Dissertation Fellow

Toposes are cohomologically equivalent to spaces

Abstract: Please be advised that this information was generated on 2017-02-10 and may be subject to change.

On the Local Solvability for Top-Degree Forms in Hypo-Analytic Structures

Abstract: (L is the differential operator in the differential complex defined by the hypo-analytic structure) for "most" I' differential forms of top degree m + n (the solution u should be an equivalence class of currents of bidegree (m, n 1)). In the same paper it is also shown that the nonexistence of hypoanalytic functions h whose real parts have local extrema ensures the local solvability of (1), at least in the cases where either the tangent or the cotangent structure bundles have rank exactly equal to one and are real-analytic vector bundles. In the present paper we show that this remains true in the general '( case. When the rank of the cotangent structure bundle is equal to 1 this is established by modifying the method in [C-H]. In this case the existence of hypo-analytic functions whose real parts have local extrema is seen to be equivalent to the nonvanishing of the (n 1)-dimensional homology of the fibres of the structure (see end of Section 2 and cf. [T2]). When it is the rank of the tangent structure bundle which is equal to 1 the result is derived (in Section 3) by reinterpreting Moyer's proof of the necessity of Condition (NV). I wish to thank Paulo Cordaro and Jorge Hounie for looking at the manuscript and pointing out errors and misprints. Needless to say, if there are any left it is entirely my fault.

On the closure of triangular algebras

Abstract: algebras of operators on Hilbert space introduced by Kadison and Singer in [8]. If sA is a maximal abelian self-adjoint algebra (m.a.s.a.) in B(NJ) then an algebra of operators, 9T, is said to be triangular with diagonal, S, if S-T n o* = s4. This definition transfers naturally to a subalgebra, T, of a von Neumann algebra, M, for which T n T* is a m.a.s.a. in M. Recently, Muhly, Saito and Solel [9] have made an extensive study of a-weakly closed triangular subalgebras of von Neumann algebras when the diagonal m.a.s.a. is a Cartan subalgebra of M. In particular, they have obtained a structure theorem for cr-weakly closed maximal triangular algebras which generalizes the most regular behaviour of triangular algebras in B(NW). However, the most general results on triangular algebras (by which we mean results for which as little as possible is assumed about the algebra beyond its triangularity) have proved elusive. One natural question of this type, which was raised by Erdos in [5], is; Question 1. Is the norm closure of a triangular algebra necessarily triangular? It follows from a straightforward Zorn's Lemma argument that every triangular algebra in B(C) (resp. a von Neumann algebra, M) is contained in a maximal triangular algebra. Thus, Question 1 is equivalent to,