Krull-Schmidt theorems in dimension 1
Lawrence
S.
Levy;
Charles
J.
Odenthal
3391-3455
Abstract: Let $\Lambda$ be a semiprime, module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated $\Lambda$-modules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree $\Lambda$-modules (called ``$\Lambda$-lattices'' in integral representation theory, and ``maximal Cohen-Macaulay modules'' in the dimension-one situation in commutative algebra).
Package deal theorems and splitting orders in dimension 1
Lawrence
S.
Levy;
Charles
J.
Odenthal
3457-3503
Abstract: Let $\Lambda$ be a module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations $\Lambda _{\mathbf {m}}$, at maximal ideals of $R$, is the family of all localizations $M_{\mathbf {m}}$ of a finitely generated $\Lambda$-module $M$. When $R$ is semilocal we also determine which finitely generated modules over the $J(R)$-adic completion of $\Lambda$ are completions of finitely generated $\Lambda$-modules. If $\Lambda$ is an $R$-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order'' of $\Lambda$ that can replace maximal orders in many situations in which maximal orders do not exist.
Drinfel´d algebra deformations, homotopy comodules and the associahedra
Martin
Markl;
Steve
Shnider
3505-3547
Abstract: The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra $A$ and thus finish the program which began in [13], [14]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel'd algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12].
Homology and some homotopy decompositions for the James filtration on spheres
Paul
Selick
3549-3572
Abstract: The filtrations on the James construction on spheres, $J_{k}\left (S^{2n}\right )$, have played a major role in the study of the double suspension $S^{2n-1}\to \Omega ^2 S^{2n+1}$ and have been used to get information about the homotopy groups of spheres and Moore spaces and to construct product decompositions of related spaces. In this paper we calculate $H_*\left ( \Omega J_{k}\left (S^{2n}\right ); {\mathbb {Z}}/p{\mathbb {Z}}\right )$ for odd primes $p$. When $k$ has the form $p^t-1$, the result is well known, but these are exceptional cases in which the homology has polynomial growth. We find that in general the homology has exponential growth and in some cases also has higher $p$-torsion. The calculations are applied to construct a $p$-local product decomposition of $\Omega J_{k}\left (S^{2n}\right )$ for $k<p^2-p$ which demonstrates a mod $p$ homotopy exponent in these cases.
Constructing product fibrations by means of a generalization of a theorem of Ganea
Paul
Selick
3573-3589
Abstract: A theorem of Ganea shows that for the principal homotopy fibration $\Omega B\to F\to E$ induced from a fibration $F\to E\to B$, there is a product decomposition $\Omega (E/F)\approx \Omega B\times \Omega (F*\Omega B)$. We will determine the conditions for a fibration $X\to Y\to Z$ to yield a product decomposition $\Omega (Z/Y)\approx X\times \Omega (X*Y)$ and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational methods. The results are then applied to produce, after localization at an odd prime $p$, homotopy decompositions for $\Omega {J_{k}\left (S^{2n}\right )}$ for some $k$ which include the cases $k=p^{t}$. The factors of $\Omega {J_{p^{t}}\left (S^{2n}\right )}$ consist of the homotopy fibre of the attaching map $S^{2np^{t}-1}\to {J_{p^{t}-1}\left (S^{2n}\right )}$ for ${J_{p^{t}}\left (S^{2n}\right )}$ and combinations of spaces occurring in the Snaith stable decomposition of $\Omega ^{2} S^{2n+1}$.
Combinatorial $B_n$-analogues of Schubert polynomials
Sergey
Fomin;
Anatol
N.
Kirillov
3591-3620
Abstract: Combinatorial $B_{n}$-analogues of Schubert polynomials and corresponding symmetric functions are constructed and studied. The development is based on an exponential solution of the type $B$ Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group.
Continuous-trace groupoid $C^*$-algebras. III
Paul
S.
Muhly;
Jean
N.
Renault;
Dana
P.
Williams
3621-3641
Abstract: Suppose that ${\mathcal {G}}$ is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid $C^{\displaystyle *}$-algebra $C^{\displaystyle *} ({\mathcal {G}},\lambda )$ has continuous trace if and only if there is a Haar system for the isotropy groupoid ${\mathcal {A}}$ and the action of the quotient groupoid ${\mathcal {G}}/{\mathcal {A}}$ is proper on the unit space of ${\mathcal {G}}$.
Exact controllability and stabilizability of the Korteweg-de Vries equation
David
L.
Russell;
Bing-Yu
Zhang
3643-3672
Abstract: In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation \begin{equation*}% \partial _t % u + % u % \partial _x u + % \partial _x^3 u = f \tag {i}\label {star} \end{equation*} on the interval $0\leq x\leq 2\pi , \, t\geq 0$, with periodic boundary conditions \begin{equation*}\partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii}\label {2star} \end{equation*} where the distributed control $f\equiv f(x,t)$ is restricted so that the ``volume'' $\int ^{2\pi }_0 u(x,t) dx$ of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control $f$ is allowed to act on the whole spatial domain $(0,2\pi )$, it is shown that the system is globally exactly controllable, i.e., for given $T> 0$ and functions $\phi (x)$, $\psi (x)$ with the same ``volume'', one can alway find a control $f$ so that the system (i)--(ii) has a solution $u(x,t)$ satisfying \begin{displaymath}u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) .\end{displaymath} If the control $f$ is allowed to act on only a small subset of the domain $(0,2\pi )$, then the same result still holds if the initial and terminal states, $\psi$ and $\phi$, have small ``amplitude'' in a certain sense. In the case of closed loop control, the distributed control $f$ is assumed to be generated by a linear feedback law conserving the ``volume'' while monotonically reducing $\int ^{2\pi }_0 u(x,t)^2 dx$. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.
Optimal natural dualities. II: General theory
B.
A.
Davey;
H.
A.
Priestley
3673-3711
Abstract: A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set $R$ of finitary algebraic relations yields a duality on a class of algebras $\mathcal {A} = \operatorname {\mathbb {I}\mathbb {S}\mathbb {P}}( \underline {M})$, those subsets $R'$ of $R$ which yield optimal dualities are characterised. Further, the manner in which the relations in $R$ are constructed from those in $R'$ is revealed in the important special case that $\underline {M}$ generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on $\underline {M}$. Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties $\mathbf {B}_{n}$ of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.
Minimal isometric immersions of inhomogeneous spherical space forms into spheres--- a necessary condition for existence
Christine
M.
Escher
3713-3732
Abstract: Although much is known about minimal isometric immersions into spheres of homogeneous spherical space forms, there are no results in the literature about such immersions in the dominant case of inhomogeneous space forms. For a large class of these, we give a necessary condition for the existence of such an immersion of a given degree. This condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function. Evaluating this function shows that neither $L(5,2)$ nor $L(8,3)$ admit a minimal isometric immersion into any sphere if the degree of the immersion is less than $28$, or less than $20$, respectively.
On representations of affine Kac-Moody groups and related loop groups
Yu
Chen
3733-3743
Abstract: We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type $A_{n}^{(1)}$ for some $n\ge 1$. Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.
Maximal subgroups in finite and profinite groups
Alexandre
V.
Borovik;
Laszlo
Pyber;
Aner
Shalev
3745-3761
Abstract: We prove that if a finitely generated profinite group $G$ is not generated with positive probability by finitely many random elements, then every finite group $F$ is obtained as a quotient of an open subgroup of $G$. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203--220, we thenprove that a finite group $G$ has at most $|G|^c$ maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.
Seifert manifolds with fiber spherical space forms
Jong Bum
Lee;
Kyung
Bai
Lee;
Frank
Raymond
3763-3798
Abstract: We study the Seifert fiber spaces modeled on the product space $S^3 \times \mathbb {R}^2$. Such spaces are ``fiber bundles'' with singularities. The regular fibers are spherical space-forms of $S^3$, while singular fibers are finite quotients of regular fibers. For each of possible space-form groups $\Delta$ of $S^3$, we obtain a criterion for a group extension $\varPi$ of $\Delta$ to act on $S^3 \times \mathbb {R}^2$ as weakly $S^3$-equivariant maps, which gives rise to a Seifert fiber space modeled on $S^3 \times \mathbb {R}^2$ with weakly $S^3$-equivariant maps $\mathrm {TOP}_{S^3}(S^3 \times \mathbb {R}^2)$ as the universal group. In the course of proving our main results, we also obtain an explicit formula for $H^2(Q; \mathbb {Z})$ for a cocompact crystallographic or Fuchsian group $Q$. Most of our methods for $S^3$ apply to compact Lie groups with discrete center, and we state some of our results in this general context.
Rook theory, compositions, and zeta functions
James
Haglund
3799-3825
Abstract: A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in $\mathrm {Re}(s)>1/2$. Some identities in the ring of formal power series involving rook theory and continued fractions are developed.
On the ordering of $n$-modal cycles
Chris
Bernhardt
3827-3834
Abstract: The forcing relation on $n$-modal cycles is studied. If $\alpha$ is an $n$-modal cycle then the $n$-modal cycles with block structure that force $\alpha$ form a $2^n$-horseshoe above $\alpha$. If $n$-modal $\beta$ forces $\alpha$, and $\beta$ does not have a block structure over $\alpha$, then $\beta$ forces a $2$-horseshoe of simple extensions of $\alpha$.