Isomorphisms between Semisimple Weighted Measure Algebras

It is shown that, if is an isomorphism between semisimple weightedmeasure algebras M(w₁) and M(w₂), then maps L¹(R⁺, w₁) ontoL¹(R⁺, w₂). This is used to describe all the automorphisms ofM(R⁺, w). A necessary and sufficient condition is given forM(w₁) and M(w₂) to be isomorphic.     

Lie Powers of Modules for Groups of Prime Order

Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Lⁿ(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Lⁿ(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J₁,...,J_p, where J_r has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J₁,...,J_p in the Lie powers Lⁿ(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J₁ or to an infinite sum of the form J_r J_{p-1} J_{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J₁,..., J_p in Lⁿ(J_p and Lⁿ(J_{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Lⁿ(J_r) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20.     

Some Remarks on p-Adic Analytic Groups

A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and p^e exactly divides |F|, then a Z_pF-latticeM is determined up to isomorphism by its finite quotient M/p^e+1M.If M is a free Z_p-module of rank d, this is equivalent to sayingthat representations of F in GL_d(Z_p) are determined up to equivalenceby their images modulo p^e+1. 1991 Mathematics Subject Classification20E18, 22E20.     

Algebraic Cycles and Even Unimodular Lattices

The number (up to isomorphism) of positive-definite, even, unimodularlattices of rank 8r grows rapidly with r. However, Bannai [1]has shown that, when counted according to weight, those withnon-trivial automorphisms make up a fraction of the whole, whichgoes rapidly to zero as r. Therefore it is of some interestto produce families of positive-definite, even, unimodular latticeswith large automorphism groups and unbounded ranks. Suppose that G is a finite group and V is an irreducible Q[G]-modulesuch that VR is still irreducible. Then, as observed by Gross[8], the space of G-invariant symmetric bilinear forms on Vis one-dimensional and is necessarily generated by a positive-definiteform, unique up to scaling by non-zero positive rationals. Thompson[23] showed that, if V is also irreducible modp for all primesp, then it contains an invariant lattice (unique up to scaling)which is even and unimodular with appropriate scaling of thequadratic form. Examples arising in this manner are the E₈-latticeof rank 8, the Leech lattice of rank 24 and the Thompson–Smithlattice of rank 248. Gow [6] has also constructed some examplesassociated with the basic spin representations of 2A_n and 2S_n.     

A Version of the Poincare-Birkhoff-Witt Theorem

In this note I shall prove that if L is a finite-dimensionalLie algebra over a field F of characteristic zero which is generatedas an algebra by a set of elements {e₁, e₂,...,e_k}, then theuniversal enveloping algebra U(L) of L is linearly generatedby monomials spanned by the elements {e_i} of an a priori boundedwidth. As an application, a criterion of Kostant for a leftideal of U(L) to be of finite codimension is proved by purelyalgebraic means.     

Fock Space and the Poisson Process

Using the Wiener–Poisson isomorphism, we show that if(F_t)_{0 t 1} is a real, bounded, predictable process adaptedto the filtration of a compensated Poisson process (X_t)_{0 t 1}, and if is the operator corresponding to multiplication by , then for any regular self-adjoint quantum semimartingale , the essentially self-adjoint quantumsemimartingale satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process toa measure space (M, M, µ) as an isometry I: L² (M, M,µ) L²(, F, P) and give a new construction of the generalisedWiener–Poisson isomorphism W_I: F₊ (L²(M)) L² (, F, P)using exponential vectors. Using C^*-algebra theory, given anymeasure space we construct a canonical generalised Poisson process.Unlike other constructions, we make no a priori use of Poissonmeasures. 2000 Mathematics Subject Classification 60G20, 60G35,46L53, 81S25.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

Oscillation theory and numerical solution of fourth-order Sturm--Liouville problems

A shooting method is developed to approximate the eigenvaluesand eigenfunc-tions of a fourth-order Sturm-Liouville problem.The main tool is a miss-distance function M(), which countsthe number of eigenvalues less than A. The method approximatesthe coefficients of the differential equation by piecewise-constantfunctions, which enables an exact solution to be found on eachmesh interval. In order to calculate N() for the approximateproblem, certain oscillation numbers N_L and N_R must be computed.These consist of sums of nullities (or rank deficiencies) of2 x 2 matrices obtained from the solutions of the approximatedifferential equation. Although these solutions can be foundexplicitly, the calculation of N_L and N_R is non-trivial, andis obtained by using certain properties of M().     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).     

Jordan triple derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Let A be a standard operator subalgebra (i.e., it contains all finite rank operators in AlgL) of AlgL and M■B(X) the Alg L-bimodule. It is shown that every linear Jordan triple derivation from A into M is a derivation, and that every generalized Jordan (triple) derivation from A into M is a generalized derivation.     

A Conformal Mapping Problem Arising in Elasticity, II

Suppose that B is an infinite right cylinder over a horizontalbase , which is a Jordan domain bounded by a smooth curve .We suppose that a normal pressure p₁(x, y) and a tangentialpressure p₂(x, y) are applied at every point of the boundaryof B, where p₁, p₂ lie in the plane of the cross-section andare independent of the vertical co-ordinate. It was shown by Cassisa (1982) that in this situation, givenan arbitrary normal pressure p₁, the tangential pressure p₂can be so chosen, that a purely plane stress tensor existswith the above hypotheses, provided that a certain 3 ? 3 matrixM, depending only on the curve , is non-singular. In a previous paper (Hayman, 1982) the author showed that thesingular case can occur and further that an arbitrary pointsymmetric curve can be approximated by singular point symmetriccurves. In the present paper it is shown that M can have rank1, 2 or 3. In the case of rank 2 p₁ must satisfy one additionalintegral condition and in the case of rank 1 two such conditions.If these conditions are satisfied p₂ can be chosen to dependon one or two arbitrary parameters respectively. Some geometrical conditions for the various cases are obtained.Thus for curves with line symmetry the rank can be two but notone, whereas curves with point symmetry can yield matrices ofrank 1, 2 or 3.     

Symmetric Polynomials on Rearrangement-Invariant Function Spaces

The exact representation of symmetric polynomials on Banachspaces with symmetric basis and also on separable rearrangement-invariantfunction spaces over [0, 1] and [0, ) is given. As a consequenceof this representation it is obtained that, among these spaces,l_2n, L_2n[0, 1], L_2n[0, ) and L_2n[0, )L_2m[0, ) where n, m areboth integers are the only spaces that admit separating polynomials.     

Some Approximations of the Characteristics of Second Order Hyperbolic Partial Differential Equations

The characteristic curves of the quasi-linear hyperbolic partialdifferential equation of the second order, having unequallyspaced, discreate initial conditions, are approximated by L₂parabolas, L₁ parabolas and cubic splines.     

Local Cohomology of BP*BP-Comodules

Given a spectrum X, we construct a spectral sequence of BP_*BP-comodulesthat converges to BP_*(L_nX), where L_nX is the Bousfield localizationof X with respect to the Johnson–Wilson theory E(n)_*.The E₂-term of this spectral sequence consists of the derivedfunctors of an algebraic version of L_n. We show how to calculatethese derived functors, which are closely related to local cohomologyof BP_*-modules with respect to the ideal I_n+1. 2000 MathematicsSubject Classification 55N22, 55P60, 16W30.     

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof an infinite set X, rank(T_X : U) is either 0, 1 or 2, or uncountable.In this paper, the relative ranks rank(T_X : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).     

The Marica-Schonheim Inequality in Lattices

The Marica-Schönheim Inequality says that if A is a finitefamily of sets, then |A–||A| where A–A=[A₁\A₂:A₁,A₂A]. For a finite lattice L and AL, we define a–b=(J_a\J_b)where J_a=[jL:ja and j is join-irreducible], and if AL then welet A–A=[a₁–a₂: a₁, a₂A]. Then the analogue of theMarica-Schöonheim Inequality is |A–A|A| for all AL.We prove that this is true if L is distributive or complementedand modular or L is a partition lattice.     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.     

Distribution of Brownian Local Time on Curves

For all smooth nondecreasing functions f on [0, 1], the localtime L_f spent by standard Brownian motion on the graph of fsatisfies P(L_f < x) P(2L₀ < x), for x > 0. The constant2 in this inequality may not be replaced with a smaller one.1991 Mathematics Subject Classification 60J55, 60J65.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.     

Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group

In [6] S. Shelah showed that in the endomorphism semi-groupof an infinitely generated algebra which is free in a varietyone can interpret some set theory. It follows from his resultsthat, for an algebra F which is free of infinite rank in avariety of algebras in a language L, if > |L|, then thefirst-order theory of the endomorphism semi-group of F, Th(End(F)),syntactically interprets Th(,L₂), the second-order theory ofthe cardinal . This means that for any second-order sentence of empty language there exists *, a first-order sentence ofsemi-group language, such that for any infinite cardinal >|L|, Th(,L₂)*Th(End(F)) In his paper Shelah notes that it is natural to study a similarproblem for automorphism groups instead of endomorphism semi-groups;a priori the expressive power of the first-order logic for automorphismgroups is less than the one for endomorphism semi-groups. Forinstance, according to Shelah's results on permutation groups[4, 5], one cannot interpret set theory by means of first-orderlogic in the permutation group of an infinite set, the automorphismgroup of an algebra in empty language. On the other hand, onecan do this in the endomorphism semi-group of such an algebra. In [7, 8] the author found a solution for the case of the varietyof vector spaces over a fixed field. If V is a vector spaceof an infinite dimension over a division ring D, then the theoryTh(, L₂) is interpretable in the first-order theory of GL(V),the automorphism group of V. When a field D is countable anddefinable up to isomorphism by a second-order sentence, thenthe theories Th(GL(V)) and Th(, L₂) are mutually syntacticallyinterpretable. In the general case, the formulation is a bitmore complicated. The main result of this paper states that a similar result holdsfor the variety of all groups.