Cohomological construction of quantized universal enveloping algebras

Given an associative algebra and the category of its finite dimensional modules, additional structures on the algebra induce corresponding ones on the category . Thus, the structure of a rigid quasi-tensor (braided monoidal) category on is induced by an algebra homomorphism (comultiplication), coassociative up to conjugation by (associativity constraint) and cocommutative up to conjugation by (commutativity constraint), together with an antiautomorphism (antipode) of satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element with suitable induced actions on , and . Drinfeld defined such a structure on for any semisimple Lie algebra with the usual comultiplication and antipode but nontrivial and , and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), .

In the paper we give a direct cohomological construction of the which reduces to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of , in particular, gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements , and can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to with depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Boundary limits and non-integrability of )

In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball in that are subharmonic with respect to the Laplace-Beltrami operator on . Since the operator is invariant under the group of holomorphic automorphisms of , functions that are subharmonic with respect to are usually referred to as -subharmonic functions. Our main result is as follows: Let be a non-negative -subharmonic function on satisfying

for some and some , where is the -invariant measure on . Suppose . Then for a.e. ,

uniformly as in each , where for ( when )

We also prove that for the only non-negative -subharmonic function satisfying the above integrability criteria is the zero function.

     

Randomness and semigenericity

Let contain only the equality symbol and let be an arbitrary finite symmetric relational language containing . Suppose probabilities are defined on finite structures with `edge probability' . By , the almost sure theory of random -structures we mean the collection of -sentences which have limit probability 1. denotes the theory of the generic structures for (the collection of finite graphs with hereditarily nonnegative). . , the almost sure theory of random -structures, is the same as the theory of the -generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

     

The quantum analog of a symmetric pair: a construction in type

Let be the ideal in the enveloping algebra of generated by the maximal compact subalgebra of . In this paper we construct an analog of in the quantized enveloping algebra corresponding to a type diagram at generic . We find generators for and explicit bases for .

     

Symmetric powers of complete modules over a two-dimensional regular local ring

Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

     

A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

Liouville type theorems for fourth order elliptic equations in a half plane

Consider an elliptic equation in the half plane with boundary conditions if and if where are second and third order differential operators. It is proved that if and, for some , if if where for some nonnegative integer , then . Results of this type are also established in case under different conditions on and ; furthermore, in one case has a lower order term which depends nonlocally on . Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) and (ii) ; both methods can be extended to other linear elliptic boundary value problems in a half plane.

     

Signed Quasi-Measures

Let be a compact Hausdorff space and let denote the subsets of which are either open or closed. A quasi-linear functional is a map which is linear on singly generated subalgebras and such that for some . There is a one-to-one correspondence between the quasi-linear functional on and the set functions such that i) , ii) If with and disjoint, then , iii) There is an such that whenever are disjoint open sets, , and iv) if is open and , there is a compact such that whenever is open, then . The space of quasi-linear functionals is investigated and quasi-linear maps between two spaces are studied.

     

One and two dimensional Cantor-Lebesgue type theorems

Let be any function which grows more slowly than exponentially in i.e., There is a double trigonometric series whose coefficients grow like and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like and which has the everywhere convergent partial sum subsequence For any there is a one dimensional trigonometric series whose coefficients grow like and which has the everywhere convergent partial sum subsequence All these examples exhibit, in a sense, the worst possible behavior. If is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence

     

Absolute Borel sets and function spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of is determined by the uniform structure of the space of continuous functions ; however the case of absolute metric spaces is still open. More precisely, we prove that, for metrizable spaces and , if is a uniformly continuous surjection and is an absolute Borel set of multiplicative (resp., additive) class , , then is also an absolute Borel set of the same class. This result is new even if is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space is determined by the linear structure of .

     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Unramified cohomology and Witt groups of anisotropic Pfister quadrics

The unramified Witt group of an anisotropic conic over a field , with , defined by the form is known to be a quotient of the Witt group of and isomorphic to . We compute the unramified cohomology group , where is the three dimensional anisotropic quadric defined by the quadratic form over . We use these computations to study the unramified Witt group of .

     

The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua

A homeomorphism of a compactum with metric is expansive if there is such that if and , then there is an integer such that . It is well-known that -adic solenoids () admit expansive homeomorphisms, each is an indecomposable continuum, and cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each , does there exist a plane continuum so that admits an expansive homeomorphism and separates the plane into components? For the case , the typical plane continua are circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if is an expansive homeomorphism of a circle-like continuum , then is itself weakly chaotic in the sense of Devaney.

     

Automorphism Groups and Invariant Subspace Lattices

Let be a - dynamical system and let be the analytic subalgebra of . We extend the work of Loebl and the first author that relates the invariant subspace structure of for a -representation on a Hilbert space , to the possibility of implementing on We show that if is irreducible and if lat is trivial, then is ultraweakly dense in We show, too, that if satisfies what we call the strong Dirichlet condition, then the ultraweak closure of is a nest algebra for each irreducible representation Our methods give a new proof of a ``density' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid -algebras.

     

Group actions on arrangements of linear subspaces and applications to configuration spaces

For an arrangement of linear subspaces in that is invariant under a finite subgroup of the general linear group we develop a formula for the -module structure of the cohomology of the complement . Our formula specializes to the well known Goresky-MacPherson theorem in case , but for the formula shows that the -module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space . Our motivating examples are arrangements in that are invariant under the action of by permuting coordinates. A particular case is the ``-equal' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases and are spaces of ordered and unordered point configurations in many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in and provide explicit results for the ``-equal' and the ``-divisible' cases.

     

Boundary value maps, Szegö maps and intertwining operators

We consider one series of unitarizable representations, the cohomological induced modules with dominant regular infinitesimal character. The minimal -type of determines a homogeneous vector bundle . The derived functor modules can be realized on the solution space of a first order differential operator on . Barchini, Knapp and Zierau gave an explicit integral map from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle . In this paper we study the asymptotic behavior of elements in the image of . We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map and a passage to boundary values.

     

Coherent functors, with application to torsion in the Picard group

Let be a commutative noetherian ring. We investigate a class of functors from commutative -algebras to sets, which we call coherent. When such a functor in fact takes its values in abelian groups, we show that there are only finitely many prime numbers such that is infinite, and that none of these primes are invertible in . This (and related statements) yield information about torsion in . For example, if is of finite type over , we prove that the torsion in is supported at a finite set of primes, and if is infinite, then the prime is not invertible in . These results use the (already known) fact that if such an is normal, then is finitely generated. We obtain a parallel result for a reduced scheme of finite type over . We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.

     

The second variation of nonorientable minimal submanifolds

Suppose is a complete nonorientable minimal submanifold of a Riemannian manifold . We derive a second variation formula for the area of with respect to certain perturbations, giving a sufficient condition for the instability of . Some simple applications are given: we show that the totally geodesic is the only stable surface in , and we show the non-existence of stable nonorientable cones in . We reproduce and marginally extend some known results in the truly non-compact setting.