The Classification of the Simple Modular Lie Algebras: VI. Solving the Final Case

We investigate the structure of simple Lie algebras over an algebraically closed field of characteristic . Let denote a torus in the -envelope of in of maximal dimension. We classify all for which every 1-section with respect to every such torus is solvable. This settles the remaining case of the classification of these algebras.

     

Picard groups and infinite matrix rings

We describe a connection between the Picard group of a ring with local units and the Picard group of the unital overring . Using this connection, we show that the three groups , , and are isomorphic for any unital ring . Furthermore, each element of arises from an automorphism of , which yields an isomorphsm between and . As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group is abelian for any commutative unital ring .

     

Boundary slopes of punctured tori in 3-manifolds

Let be an irreducible 3-manifold with a torus boundary component , and suppose that are the boundary slopes on of essential punctured tori in , with their boundaries on . We show that the intersection number of and is at most . Moreover, apart from exactly four explicit manifolds , which contain pairs of essential punctured tori realizing and 6 respectively, we have . It follows immediately that if is atoroidal, while the manifolds obtained by - and -Dehn filling on are toroidal, then , and unless is one of the four examples mentioned above.

Let be the class of 3-manifolds such that is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that , where has a torus component , and . Let be slopes on such that . Then . The exterior of the Whitehead sister link shows that this bound is best possible.

     

Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map

Let be a locally compact group and the Fourier-Stieltjes algebra of . We study the problem of how weak*-closedness of some translation invariant subspaces of is related to the structure of . Moreover, we prove that for a closed subgroup of , the restriction map from to is weak*-continuous only when is open in .

     

A weak-type inequality for differentially subordinate harmonic functions

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder for two harmonic functions and . That is, we prove the sharp weak-type inequality under the assumptions that , and the extra assumption that . Here is the harmonic measure with respect to and the constant is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

Comultiplications on free groups and wedges of circles

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group . We consider individual comultiplications on and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of . For a comultiplication of we define a subset of quasi-diagonal elements which is basic to our investigation of associativity. The subset can be determined algorithmically and contains the set of diagonal elements . We show that is a basis for the largest subgroup of on which is associative and that is a free factor of . We also give necessary and sufficient conditions for a comultiplication on to be a coloop in terms of the Fox derivatives of with respect to a basis of . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group on the set of comultiplications of . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.

     

Generators and relations of direct products of semigroups

The purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup of order 11 such that is finitely generated but not finitely presented for every finitely generated infinite semigroup . By way of contrast we show that, if and belong to a wide class of semigroups, then is finitely presented if and only if both and are finitely presented, exactly as in the case of groups and monoids.

     

On the averages of Darboux functions

Let be the family of functions which can be written as the average of two comparable Darboux functions. In 1974 A. M. Bruckner, J. G. Ceder, and T. L. Pearson characterized the family and showed that if , then is the family of the averages of comparable Darboux functions in Baire class . They also asked whether the latter result holds true also for . The main goal of this paper is to answer this question in the negative and to characterize the family of the averages of comparable Darboux Baire one functions.

     

Factorisation in nest algebras. II

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator for the existence of an operator in the nest algebra of a nest satisfying (resp. . In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator has the property that there exists for every nest an operator in satisfying (resp. ) if and only if is a Fredholm operator. In Section 4 we show that for a given operator in there exists an operator in satisfying if and only if the range of is equal to the range of some operator in . We also determine the algebraic structure of the set of ranges of operators in . Let be the set of positive operators for which there exists an operator in satisfying . In Section 5 we obtain information about this set. In particular we discuss the following question: Assume and are positive operators such that and belongs to . Which further conditions permit us to conclude that belongs to ?

     

A new degree bound for vector invariants of symmetric groups

Let be a commutative ring, a finitely generated free -module and a finite group acting naturally on the graded symmetric algebra . Let denote the minimal number , such that the ring of invariants can be generated by finitely many elements of degree at most .

For and , the -fold direct sum of the natural permutation module, one knows that , provided that is invertible in . This was used by E. Noether to prove if .

In this paper we prove for arbitrary commutative rings and show equality for a prime power and or any ring with . Our results imply

for any ring with .

     

Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Generalized Hestenes' Lemma and extension of functions

Suppose we have an -jet field on which is a Whitney field on the nonsingular part of . We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on , if the field is flat enough at the singular part , then it is a Whitney field on (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when is subanalytic. In Section II, we show that a function on can be extended to one on if the differential goes to faster than the order of divergence of the principal curvatures of and if the first covariant derivative of is sufficiently flat. For the general case of functions with , we give a similar result for in Section III.

     

Widths of Subgroups

We say that the width of an infinite subgroup in is if there exists a collection of essentially distinct conjugates of such that the intersection of any two elements of the collection is infinite and is maximal possible. We define the width of a finite subgroup to be . We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic -manifolds satisfy the -plane property for some .

     

Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Let denote the classical equilibrium distribution (of total charge ) on a convex or -smooth conductor in with nonempty interior. Also, let be any th order ``Fekete equilibrium distribution' on , defined by point charges at th order ``Fekete points'. (By definition such a distribution minimizes the energy for -tuples of point charges on .) We measure the approximation to by for by estimating the differences in potentials and fields,

both inside and outside the conductor . For dimension we obtain uniform estimates at distance from the outer boundary of . Observe that throughout the interior of (Faraday cage phenomenon of electrostatics), hence on the compact subsets of . For the exterior of the precise results are obtained by comparison of potentials and energies. Admissible sets have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions of equal point charges on . In that context there is an important open problem for the sphere which is discussed at the end of the paper.

     

An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.

     

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .

     

Poisson transforms on vector bundles

Let be a connected real semisimple Lie group with finite center, and a maximal compact subgroup of . Let be an irreducible unitary representation of , and the associated vector bundle. In the algebra of invariant differential operators on the center of the universal enveloping algebra of induces a certain commutative subalgebra . We are able to determine the characters of . Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to the trivial representation.