The Jacobson radical of group rings of locally finite groups
D.
S.
Passman
4693-4751
Abstract: This paper is the final installment in a series of articles, started in 1974, which study the semiprimitivity problem for group algebras $K[G]$ of locally finite groups. Here we achieve our goal of describing the Jacobson radical ${\mathcal{J}}K[{G}]$ in terms of the radicals ${\mathcal{J}}K[{A}]$ of the group algebras of the locally subnormal subgroups $A$ of $G$. More precisely, we show that if $\operatorname{char} K=p>0$ and if $\mathbb{O}_{p}(G)=1$, then the controller of ${\mathcal{J}}K[{G}]$ is the characteristic subgroup $\mathbb{S}^{p}(G)$ generated by the locally subnormal subgroups $A$ of $G$ with $A=\mathbb{O}^{p'}(A)$. In particular, we verify a conjecture proposed some twenty years ago and, in so doing, we essentially solve one half of the group ring semiprimitivity problem for arbitrary groups. The remaining half is the more difficult case of finitely generated groups. This article is effectively divided into two parts. The first part, namely the material in Sections 2-6, covers the group theoretic aspects of the proof and may be of independent interest. The second part, namely the work in Sections 7-12, contains the group ring and ring theoretic arguments and proves the main result. As usual, it is necessary for us to work in the more general context of twisted group algebras and crossed products. Furthermore, the proof ultimately depends upon results which use the Classification of the Finite Simple Groups.
Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds
Bruno
Bianchini;
Marco
Rigoli
4753-4774
Abstract: We prove nonexistence and uniqueness of positive $C^{2}$-solutions of the elliptic equation $\Delta u +a(x)u - K(x)u^{\sigma }=0$, $ \sigma >1$, on a nonpositively curved, complete manifold $(M,g)$ .
Restriction of stable bundles in characteristic $p$
Tohru
Nakashima
4775-4786
Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a nonsingular projective variety defined over $k$ and $H$ an ample line bundle on $X$. We shall prove that there exists an explicit number $m_{0}$ such that if $E$ is a $\mu$-stable vector bundle of rank at most three, then the restriction $E_{\vert D}$ is $\mu$-stable for all $m\geq m_{0}$ and all smooth irreducible divisors $D\in \vert mH\vert$. This result has implications to the geometry of the moduli space of $\mu$-stable bundles on a surface or a projective space.
A theorem of the Dore-Venni type for noncommuting operators
Sylvie
Monniaux;
Jan
Prüss
4787-4814
Abstract: A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations $\dot{u}(t)+L(t)u(t)=f(t)$ and evolutionary integral equations $u(t)+\int _0^ta(t-s)L(s)u(s)ds = g(t)$ which are nonautonomous. The domains of the involved operators $L(t)$ may depend on $t$, but $L(t)^{-1}$ is required to satisfy a certain smoothness property. The results are then applied to parabolic partial differential and integro-differential equations.
On measures ergodic with respect to an analytic equivalence relation
Alain
Louveau;
Gabriel
Mokobodzki
4815-4823
Abstract: In this paper, we prove that the set of probability measures which are ergodic with respect to an analytic equivalence relation is an analytic set. This is obtained by approximating analytic equivalence relations by measures, and is used to give an elementary proof of an ergodic decomposition theorem of Kechris.
Lexicographic TAF Algebras
Justin
R.
Peters;
Yiu
Tung
Poon
4825-4855
Abstract: Lexicographic TAF algebras constitute a class of triangular AFalgebras which are determined by a countable ordered set $\Omega$, a dimension function, and a third parameter. While some of the important examples of TAF algebras belong to the class, most algebras in this class have not been studied. The semigroupoid of the algebra, the lattice of invariant projections, the Jacobson radical, and for some cases the automorphism group are computed. Necessary and sufficient conditions for analyticity are given. The results often involve the order properties of the set $\Omega$.
Frobenius extensions of subalgebras of Hopf algebras
D.
Fischman;
S.
Montgomery;
H.-J.
Schneider
4857-4895
Abstract: We consider when extensions $S\subset R$ of subalgebras of a Hopf algebra are $\beta$-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra $H$, we show that when $S\subset R$ are Hopf algebras in the Yetter-Drinfeld category for $H$, the extension is $\beta$-Frobenius provided $R$ is finite over $S$ and the extension of biproducts $S\star H\subset R\star H$ is cleft. More generally we give conditions for an extension to be $\beta$-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants. We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.
Dehn surgery on knots in solid tori creating essential annuli
Chuichiro
Hayashi;
Kimihiko
Motegi
4897-4930
Abstract: Let $M$ be a $3$-manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when $M$ contains a separating essential annulus. It is shown that $M$ does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the $3$-sphere which are not periodic knots of period $2$ satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the $2$-sphere with exactly three exceptional fibres.
On the rational homotopy type of function spaces
Edgar
H.
Brown Jr.;
Robert
H.
Szczarba Jr.
4931-4951
Abstract: The main result of this paper is the construction of a minimal model for the function space $\mathcal {F}(X,Y)$ of continuous functions from a finite type, finite dimensional space $X$ to a finite type, nilpotent space $Y$ in terms of minimal models for $X$ and $Y$. For the component containing the constant map, $\pi _{*}(\mathcal {F}(X,Y))\otimes Q =\pi _{*}(Y)\otimes H^{-*}(X;Q)$ in positive dimensions. When $X$ is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for $Y$ and the coproduct of $H_{*}(X;Q)$. We also give a version of the main result for the space of cross sections of a fibration.
Lower bounds for derivatives of polynomials and Remez type inequalities
Tamás
Erdélyi;
Paul
Nevai
4953-4972
Abstract: P. Turán [Über die Ableitung von Polynomen, Comositio Math. 7 (1939), 89-95] proved that if all the zeros of a polynomial $p$ lie in the unit interval $I% \overset {\text {def}}{=} [-1,1]$, then $\|p'\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\;$. Our goal is to study the feasibility of $\lim _{{n\to \infty }% }{\|p_{n}'\|_{X}} / {\|p_{n}\|_{Y}} =\infty$ for sequences of polynomials $\{p_{n}\}_{n\in \mathbb N% }$ whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.
Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers
Steven
Kleiman;
Bernd
Ulrich
4973-5000
Abstract: Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade $2$ as those with a Hilbert-Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade $1$ can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade $1$ that are birational onto their image, on the one hand, and self-linked perfect ideals of grade $2$ that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.
Partition identities involving gaps and weights
Krishnaswami
Alladi
5001-5019
Abstract: We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers-Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.
On the number of geodesic segments connecting two points on manifolds of non-positive curvature
Paul
Horja
5021-5030
Abstract: We prove that on a complete Riemannian manifold $M$ of dimension $n$ with sectional curvature $K_M < 0$, two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least $2n+1$ geodesic segments. A simpler version of the argument shows that if one of the points is fixed and $K_M \leq 0$ then the two points are connected by at least $n+1$ geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.
Covering Sato-Levine invariants
Gui-Song
Li
5031-5042
Abstract: Two covering versions of the Sato-Levine invariant are constructed which provide obstructions to certain two-component oriented links in the 3-sphere being link concordant to boundary links. These covering invariants are rational functions one of which detects both nonamphicheirality and noninvertibility of oriented links.
Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem
William
Cherry;
Zhuan
Ye
5043-5071
Abstract: Cartan's method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects.
Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras
Yiftach
Barnea;
Aner
Shalev
5073-5091
Abstract: Every finitely generated profinite group can be given the structure of a metric space, and as such it has a well defined Hausdorff dimension function. In this paper we study Hausdorff dimension of closed subgroups of finitely generated pro-$p$ groups $G$. We prove that if $G$ is $p$-adic analytic and $H \le _c G$ is a closed subgroup, then the Hausdorff dimension of $H$ is $\dim H/\dim G$ (where the dimensions are of $H$ and $G$ as Lie groups). Letting the spectrum $% \operatorname {Spec}(G)$ of $G$ denote the set of Hausdorff dimensions of closed subgroups of $G$, it follows that the spectrum of $p$-adic analytic groups is finite, and consists of rational numbers. We then consider some non-$p$-adic analytic groups $G$, and study their spectrum. In particular we investigate the maximal Hausdorff dimension of non-open subgroups of $G$, and show that it is equal to $1 - {1 \over {d+1}}$ in the case of $G = SL_d(F_p[[t]])$ (where $p > 2$), and to $1/2$ if $G$ is the so called Nottingham group (where $p >5$). We also determine the spectrum of $SL_2(F_p[[t]])$ ($p>2$) completely, showing that it is equal to $[0,2/3] \cup \{ 1 \}$. Some of the proofs rely on the description of maximal graded subalgebras of Kac-Moody algebras, recently obtained by the authors in joint work with E. I. Zelmanov.
On a General Form of the Second Main Theorem
Min
Ru
5093-5105
Abstract: We give a proof of a general form of the Second Main Theorem for holomorphic curves with a good error term. Two applications of this general form are also provided.