Jones index theory by Hilbert C{${}^{*}$}-bimodules and K-theory
Tsuyoshi
Kajiwara;
Yasuo
Watatani
3429-3472
Abstract: In this paper we introduce the notion of Hilbert ${\mathrm{C}}^{*}$-bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert ${\mathrm{C}}^{*}$-bimodules and show that tensor products of minimal bimodules are also minimal. For an $A$-$A$ bimodule which is compatible with a trace on a unital ${\mathrm{C}}^{*}$-algebra $A$, its dimension (square root of Jones index) depends only on its $KK$-class. Finally, we show that the dimension map transforms the Kasparov products in $KK(A,A)$ to the product of positive real numbers, and determine the subring of $KK(A,A)$ generated by the Hilbert ${\mathrm{C}}^{*}$-bimodules for a ${\mathrm{C}}^{*}$-algebra generated by Jones projections.
Morse theory for the Yang-Mills functional via equivariant homotopy theory
Ursula
Gritsch
3473-3493
Abstract: In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds $\{ M_{2g} : g=0,1,2, \ldots \}$ with generic $SU(2)$-invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group $SU(2)$ with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker in A Morse theory for equivariant Yang-Mills, Duke Math. J. 66-2 (1992), 337-356 and Råde in Compactness theorems for invariant connections, submitted for publication.
A wall-crossing formula for the signature of symplectic quotients
David
S.
Metzler
3495-3521
Abstract: We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincaré polynomial and the Euler characteristic (equivalent to those of Kirwan can be expressed in the same recursive manner.
On cohomology algebras of complex subspace arrangements
Eva
Maria
Feichtner;
Günter
M.
Ziegler
3523-3555
Abstract: The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.
Algebraic gamma monomials and double coverings of cyclotomic fields
Pinaki
Das
3557-3594
Abstract: We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex ${\mathbb{SK} }$, to compute $H^*(\pm, {\mathbb{U} })$, where ${\mathbb{U} }$ is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension $K/F$, we define a double covering of $K/F$ to be an extension $\tilde{K}/K$ of degree $\leq 2$, such that ${\tilde{K}}/F$ is Galois. We demonstrate that each class ${\mathbf{a}}\in H^2(\pm, {\mathbb{U} })$ gives rise to a double covering of ${\mathbb{Q} }(\zeta_ \infty)/{\mathbb{Q} }$, by ${\mathbb{Q} }(\zeta_ \infty,\sqrt{\sin{\mathbf{a}}})/{\mathbb{Q} }(\zeta_ \infty)$. When ${\mathbf{a}}$ lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if ${\mathbf{a}}$ represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of ${\mathbb{Q} }(\zeta_ \infty,\sqrt{\sin{\mathbf{a}}})/{\mathbb{Q} }$ is abelian and hence $\sqrt{\sin{\mathbf{a}}} \in {\mathbb{Q} }(\zeta_ \infty)$. The $\sqrt{\sin{\mathbf{a}}}$ may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.
Counting solutions to trinomial Thue equations: a different approach
Emery
Thomas
3595-3622
Abstract: We consider the problem of counting solutions to a trinomial Thue equation -- that is, an equation \begin{equation*}\vert F(x,y)\vert = 1,\tag{$*$} \end{equation*} where $F$ is an irreducible form in $Z[x,y]$ with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the ``Thue-Siegel principle" and its relation to $(*)$. In this paper we give specific numerical bounds for the number of solutions to $(*)$ by a somewhat different approach, the difference lying in the initial step -- solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.
Geometric flow and rigidity on symmetric spaces of noncompact type
Inkang
Kim
3623-3638
Abstract: In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of $X$, where $X$ is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit.
The geography problem for irreducible spin four-manifolds
B. Doug
Park;
Zoltán
Szabó
3639-3650
Abstract: We study the geography problem for smooth irreducible simply-connected spin four-manifolds. For a large class of homotopy types, we exhibit both symplectic and non-symplectic representatives. We also compute the Seiberg-Witten invariants of all the four-manifolds we construct.
Multiscale decompositions on bounded domains
A.
Cohen;
W.
Dahmen;
R.
DeVore
3651-3685
Abstract: A construction of multiscale decompositions relative to domains $\Omega\subset \mathbb{R} ^d$ is given. Multiscale spaces are constructed on $\Omega$ which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.
Partial differential equations with matricial coefficients and generalized translation operators
N.
H.
Mahmoud
3687-3706
Abstract: Let $\Delta_{\alpha }$ be the Bessel operator with matricial coefficients defined on $(0,\infty )$ by \begin{equation*}\Delta_{\alpha }U(t)=U''(t)+\frac{2\alpha +I}{t}U'(t)\end{equation*} where $\alpha$ is a diagonal matrix and let $q$ be an $n\times n$ matrix-valued function. In this work, we prove that there exists an isomorphism $X$ on the space of even ${\mathcal C}^{\infty}$, $\mathbb{C} ^n$-valued functions which transmutes $\Delta_{\alpha}$and $(\Delta_{\alpha}+q)$. This allows us to define generalized translation operators and to develop harmonic analysis associated with $(\Delta_{\alpha}+q)$. By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.
Weak and Strong Density of Compositions
Luigi
De Pascale;
Eugene
Stepanov
3707-3721
Abstract: The convergence in various topologies of sequences of inner superposition (composition) operators acting between Lebesgue spaces and of their linear combinations is studied. In particular, the sequential density results for the linear span of such operators is proved for the weak, weak continuous and strong operator topologies.
Semiclassical analysis of general second order elliptic operators on bounded domains
E.
N.
Dancer;
J.
López-Gómez
3723-3742
Abstract: In this work we ascertain the semiclassical behavior of the fundamental energy and the ground state of an arbitrary second order elliptic operator, not necessarily selfadjoint, on a bounded domain. Our analysis provides us with substantial improvements of many previous results found in the context of quantum mechanics for $C^\infty$ perturbations of the Laplacian.
The problem of lacunas and analysis on root systems
Yuri
Berest
3743-3776
Abstract: A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.
Mean convergence of orthogonal Fourier series of modified functions
Martin
G.
Grigorian;
Kazaros
S.
Kazarian;
Fernando
Soria
3777-3798
Abstract: We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the $L^{p}$-metric for any $p > 2.$ We prove also that if $\Phi$ is a uniformly bounded ONS which is complete in all the spaces $L _ {[0,1]} ^{p} , 1 \leq p < \infty$, then there exists a rearrangement $\sigma$ of the natural numbers $\mathbf{N}$such that the system $\Phi _{\sigma }= \{ \phi _{\sigma (n)} \}_{n=1}^{\infty }$ has the strong $L^{p}$-property for all $p>2$; that is, for every $2 \leq p < \infty$ and for every $f \in L _ {[0,1]} ^{p}$ and $\epsilon > 0$there exists a function $f_ \epsilon \in L _ {[0,1]} ^{p}$ which coincides with $f$ except on a set of measure less than $\epsilon$ and whose Fourier series with respect to the system $\Phi _{\sigma }$ converges in $L _ {[0,1]} ^{p} .$
Scattering matrices for the quantum $N$ body problem
Andrew
Hassell
3799-3820
Abstract: Let $H$ be a generalized $N$ body Schrödinger operator with very short range potentials. Using Melrose's scattering calculus, it is shown that the free channel `geometric' scattering matrix, defined via asymptotic expansions of generalized eigenfunctions of $H$, coincides (up to normalization) with the free channel `analytic' scattering matrix defined via wave operators. Along the way, it is shown that the free channel generalized eigenfunctions of Herbst-Skibsted and Jensen-Kitada coincide with the plane waves constructed by Hassell and Vasy and if the potentials are very short range.
Hopf algebras of types $U_q(sl_n)'$ and $O_q(SL_n)'$ which give rise to certain invariants of knots, links and 3-manifolds
Shlomo
Gelaki;
Sara
Westreich
3821-3836
Abstract: In this paper we determine when Lusztig's
Extensions of Hopf Algebras and Lie Bialgebras
Akira
Masuoka
3837-3879
Abstract: Let $\mathfrak{f}$, $\mathfrak{g}$ be finite-dimensional Lie algebras over a field of characteristic zero. Regard $\mathfrak{f}$ and $\mathfrak{g} ^*$, the dual Lie coalgebra of $\mathfrak{g}$, as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair $(\mathfrak{f} , \mathfrak{g} ^*)$of Lie bialgebras is given, which has structure maps $\rightharpoonup , \rho$. Then it induces a matched pair $(U\mathfrak{f}, U\mathfrak{g}^{\circ},\rightharpoonup ', \rho ')$ of Hopf algebras, where $U\mathfrak{f}$ is the universal envelope of $\mathfrak{f}$ and $U\mathfrak{g}^{\circ}$ is the Hopf dual of $U\mathfrak{g}$. We show that the group $\mathrm{Opext} (U\mathfrak{f},U\mathfrak{g}^{\circ })$of cleft Hopf algebra extensions associated with $(U\mathfrak{f}, U\mathfrak{g} ^{\circ}, \rightharpoonup ', \rho ' )$ is naturally isomorphic to the group $\operatorname{Opext}(\mathfrak{f},\mathfrak{g} ^*)$of Lie bialgebra extensions associated with $(\mathfrak{f}, \mathfrak{g}^*, \rightharpoonup , \rho )$. An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If $\mathfrak{g} =[\mathfrak{g} , \mathfrak{g}]$, there follows a bijection between the set $\mathrm{Ext}(U\mathfrak{f} , U\mathfrak{g}^{\circ })$of all cleft Hopf algebra extensions of $U\mathfrak{f}$ by $U\mathfrak{g}^{\circ }$ and the set $\mathrm{Ext}(\mathfrak{f}, \mathfrak{g}^*)$ of all Lie bialgebra extensions of $\mathfrak{f}$ by $\mathfrak{g} ^*$.
Galois embeddings for linear groups
Shreeram
S.
Abhyankar
3881-3912
Abstract: A criterion is given for the solvability of a central Galois embedding problem to go from a projective linear group covering to a vectorial linear group covering.